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                                 HYPERGEO
                                Version 1.1





                A program for the investigation of the geometrical
                properties of four-dimensional hyperspace.









                   USER'S GUIDE AND PROGRAM REFERENCE MANUAL













                               Stuart Esten

                              33 Banks Avenue
                            Lexington, MA  02173
                                   USA


        HYPERGEO    Version 1.1                             Page ii










    COPYRIGHT NOTICE


            HYPERGEO is Copyrighted (c) 1992 by Stuart Esten.

            This document is Copyrighted (c) 1992 by Stuart Esten.

            ALL RIGHTS RESERVED.









    DISCLAIMER OF WARRANTIES AND LIMITATION OF LIABILITIES


        Due to the nature of computer programming it is impossible to
        guarantee the performance of this product.  The author makes no
        warranty, expressed or implied, in regard to this program and its
        accompanying data files and documentation.  In no event shall the
        author of this program be liable for incidental or consequential
        damages relating to or arising out of its use.


        HYPERGEO    Version 1.1                             Page iii








    LICENSE


        HYPERGEO is a copyrighted computer program.  It is not public domain
        or free software.

        Copies of the program are made available on a trial basis to
        prospective users according to the distribution system known as
        "shareware".  Persons acquiring a shareware copy of HYPERGEO are
        granted a limited license to use the program for an evaluation
        period not to exceed 30 days.  At the end of that time they must
        either become registered users of HYPERGEO by paying the requisite
        registration fee or cease using the program.  Recipients of the
        shareware copy of HYPERGEO may make duplicate copies of the program
        material for the express purpose of distribution to other
        prospective users, provided that:

            *   Each copy contains all HYPERGEO program files, data files,
                and documentation files complete and unmodified;

            *   No payment is charged for the shareware copies beyond normal
                charges for the cost of the diskette and its shipment and
                handling.


        Registered users of HYPERGEO are granted license to use the software
        without time limit on a single computer.  A registered copy of the
        HYPERGEO program material may not be duplicated other than for
        back-up purposes.  A registered copy of HYPERGEO may not be
        distributed to others.


        HYPERGEO    Version 1.1                             Page iv






                          _______
                     ____|__     |               (R)
                  --|       |    |-------------------
                    |   ____|__  |  Association of
                    |  |       |_|  Shareware
                    |__|   o   |    Professionals
                  -----|   |   |---------------------
                       |___|___|    MEMBER







    ASSOCIATION OF SHAREWARE PROFESSIONALS OMBUDSMAN POLICY


        This program is produced by a member of the Association of Shareware
        Professionals (ASP).  ASP wants to make sure that the shareware
        principle works for you.  If you are unable to resolve a
        shareware-related problem with an ASP member by contacting the
        member directly, ASP may be able to help.  The ASP Ombudsman can
        help you resolve a dispute or problem with an ASP member, but does
        not provide technical support for members' products.  Please write
        to the ASP Ombudsman at 545 Grover Road, Muskegon, MI 49442-9427 or
        send a CompuServe message via CompuServe Mail to ASP Ombudsman
        70007,3536.


        HYPERGEO    Version 1.1                             Page v




    Table of Contents


    Section 1.  Introduction.......................................  1-1

    Section 2.  From Hyperspace to the Video Screen................  2-1
        2.1         From Four Dimensions to Three..................  2-1
        2.1.1           Projection.................................  2-1
        2.1.2           Intersection...............................  2-2
        2.2         From Three Dimensions to Two...................  2-2

    Section 3.  The HYPERGEO Program...............................  3-1
        3.1         General Structure..............................  3-1
        3.2         The HYPERGEO Display Screen....................  3-3
        3.3         Geometry Manipulations.........................  3-6
        3.3.1           Rotations..................................  3-7
        3.3.2           Translation of the Intersecting Hyperplane.  3-10
        3.3.3           View Control...............................  3-11
        3.4         Special 3-D Geometry Definition................  3-12

    Section 4.  Inner Workings.....................................  4-1
        4.1         Units..........................................  4-1
        4.2         Verification of the Geometry Definition........  4-1
        4.3         Calculating 4-D to 3-D Projections.............  4-2

    Section 5.  Restrictions and Limitations.......................  5-1
        5.1         Geometry Definition File.......................  5-1
        5.2         Mathematical Precision.........................  5-2

    Section 6.  Supplied Geometry Definition Files.................  6-1

    Appendix A. Geometry Definition File...........................  A-1

    Appendix B. Configuration File.................................  B-1

    Appendix C. Command Line Arguments.............................  C-1

    Appendix D. Interactive Commands...............................  D-1
        D.1         Keyboard Commands and the Modifier Keys........  D-1
        D.2         Mouse Selected Interactive Commands............  D-2
        D.3         Interactive Parameter Specification............  D-3
        D.4         HYPERGEO Interactive Keyboard Commands.........  D-4

    Appendix E. Running HYPERGEO under Microsoft Windows...........  E-1

    Appendix F. Registration Form..................................  F-1


        HYPERGEO    Version 1.1                             Page 1-1




    Section 1.    Introduction


        "I tried turning the hypercube around, moving it away, bringing
        it up close, turning it around another way.  Suddenly I could
        feel it!  The hypercube had leaped into palpable reality, as I
        learned how to manipulate it, feeling in my fingertips the power
        to change what I saw and change it back again.  The active control
        at the computer console created a union of kinesthetics and visual
        thinking which brought the hypercube up to the level of intuitive
        understanding."

                                        "The Mathematical Experience"
                                        Philip J. Davis & Reuben Hersh



    For reasons known only to the Great Cosmic Programmer, the universe in
    which we dwell possesses three spatial dimensions.  Our senses of
    physical perception operate intuitively and naturally in these three
    dimensions.  Indeed, it requires an act of intellectual imagination
    to consider the nature of alternative dimensional realities.  With some
    contrivance we can picture the peculiar qualities of a world aware of
    only two dimensions.  This is the substance of Edwin Abott Abott's
    famous fantasy "Flatland" (written from the viewpoint of Abott's two-
    dimensional narrator A Square).  We can make the mental adjustment
    required to conceive of a two-dimensional universe by imagining one of
    our three dimensions to be somehow squeezed down to zero extent and thus
    out of existence.  It is a simple matter to draw pictures of
    two-dimensional objects; in fact, the nature of writing surfaces and
    writing implements is such that all our drawings are already limited to
    two dimensions (it is only the insistently three-dimensional nature of
    our sense of sight that causes us to "see" depth in our flat artwork).
    Perceptions in two dimensions are accessible because they already exist
    within our three-dimensional space and because we have at hand ready
    physical analogs in the form of flat surfaces of solid objects.

    In "Flatland", Abott makes his narrator aware of the existence of
    three-dimensional "Spaceland" although A Square is unable to fully
    conceive of its nature.  He further contemplates the analogous
    relationship between Spaceland and a still higher universe possessing
    four dimensions.  In this scenario we, the inhabitants of our familiar
    three-dimensional reality, become the Flatlanders, constitutionally
    unable to perceive the totality of the encompassing higher realm.

    When we try to imagine the shape and configuration of "hyperobjects" -
    that is, objects constructed in four dimensions - we are stopped short
    by the inability of our conceptual powers to find a place for the fourth
    dimension.  For whatever reason, there simply is no way to twist or


        HYPERGEO    Version 1.1                             Page 1-2


    rearrange four separate orthogonal dimensions so they make spatial sense
    to us.

    Our best hope for grasping the essence of four-dimensional reality lies
    in somehow reducing the full complexity of hyperobjects to a simplified
    image that can be represented in three dimensions.  Although the full
    four-dimensional "depth" of the hyperobject's geometry cannot be fully
    conveyed this way in a single image, it is possible by generating
    sequences of such images - each corresponding to a different orientation
    of the hyperobject in four-dimensional space - to begin to comprehend
    the shape of the fourth dimension.  Naturally, computers and the
    capabilities of computer graphics are ideally suited for use as tools in
    generating such images, and it is well within the powers of the typical
    personal computer equipped with a standard display device to serve as a
    research platform for the investigation of hyperspace.


        HYPERGEO    Version 1.1                             Page 2-1




    Section 2.    From Hyperspace to the Video Screen


    When a computer is used to generate images derived from a
    four-dimensional hyperobject, two separate stages of geometric
    transformation must be performed.  First, the four-dimensional object
    must be reduced to a particular three-dimensional image based on the
    object's orientation and position in hyperspace.  Second, the resulting
    three-dimensional image must be represented on the flat, two-dimensional
    surface of the computer's display device.  Because each of these
    transformations entails the loss of dimensional information, it is
    desirable to provide the computer user with as much control as possible
    over the manner in which the images are constructed and displayed.


    2.1  From Four Dimensions to Three

    There are two basic approaches for generating a three-dimensional image
    from a four-dimensional object: projection and intersection.  Each is
    highly analogous to the same technique applied to the reduction of a
    three-dimensional object to a flat, two-dimensional representation.


    2.1.1  Projection

    Projection is the mathematically simpler of the two.  It entails the
    construction of an imaginary "viewscreen" in hyperspace and the
    projection of each point in the hyperobject onto that viewscreen.  The
    calculations are performed by considering the straight lines that run
    between each point in the hyperobject and a fixed four-dimensional
    viewpoint; the desired projection points are the intersections of these
    lines with the viewscreen.  But because this is four-dimensional
    hyperspace, the viewscreen is itself a three-dimensional "hyperplane"
    (not just a simple two-dimensional plane), and the projection points all
    possess three dimensions.  The positions of the viewpoint and viewscreen
    in hyperspace relative to the hyperobject determine the degree of
    "perspective" in the resulting three-dimensional image.  (This process
    is entirely analogous to the projection of a 3-D object onto a flat 2-D
    viewscreen; in fact, just such a 3-D to 2-D projection is used
    subsequently to generate the final graphical output.)

    In a 4-D to 3-D projection, every point in the original hyperobject is
    always visible as a corresponding projected point in the 3-D image
    (although, of course, a point may from time to time happen to be hidden
    behind other points or lines).  The overall 3-D image that results is a
    network of connected lines corresponding to the vertexes and edges of
    the 4-D hyperobject; it is not itself a simple 3-D polyhedron.  The
    complexity of this network relates directly to the four-dimensional
    structure of the hyperobject.


        HYPERGEO    Version 1.1                             Page 2-2


    2.1.2  Intersection

    The second approach for reducing a four-dimensional hyperobject to a
    three-dimensional image involves calculating the intersection of the
    hyperobject with a three-dimensional hyperplane.  This can be thought of
    as cutting a "slice" through the hyperobject and viewing the resulting
    cross-section.  But again, because the cutting instrument is a
    hyperplane in the case of the 4-D to 3-D intersection, the object that
    is determined by the cut is three-dimensional and forms a simple 3-D
    polyhedron - assuming that the original 4-D hyperobject is itself simple
    and is externally convex.  (The 3-D to 2-D analogy involves making a
    planar cut through the 3-D geometry and viewing the flat polygonal
    outline of the cross-section.)

    In a 4-D to 3-D intersection, the only portion of the hyperobject that
    contributes to the 3-D image is that lying exactly on the intersecting
    hyperplane; all the rest is lost and invisible.  For this reason, an
    intersection can result in very minimal 3-D polyhedra such as pyramids
    and box shapes, even from a complex hyperobject.  In fact, if the
    intersecting hyperplane is positioned (in hyperspace) entirely outside
    the extents of the hyperobject, the resulting 3-D image will be empty.
    It is generally most revealing to examine the sequence of intersections
    that result when the cutting hyperplane is gradually moved through the
    hyperobject, starting at one face or edge and working across to the
    opposite.  By taking a number of such sequences for various orientations
    of the hyperobject in four-space, the user can achieve a sense of its
    "extradimensional" structure.


    2.2  From Three Dimensions to Two

    All the options available in the HYPERGEO program for converting a
    three-dimensional image into a viewable two-dimensional picture on the
    computer's display are variations of the projection technique referred
    to above.  Each 3-D point is projected onto a 2-D viewscreen by
    calculating where the line that runs between the 3-D point and a fixed
    3-D viewpoint intersects the viewscreen.  However, unlike the case for
    4-D to 3-D projections, for 3-D to 2-D projections the viewscreen and
    viewpoint don't have to be invented; they correspond to the surface of
    the real display device and the eye of the (presumably real) viewer.
    Given this basic technique for projecting a 3-D image onto a flat
    screen, a great deal can be done to enhance the meaningfulness of the
    resulting 2-D picture.  The following alternatives are available:


        *  Orthographic Projection

        This is the simplest and least revealing form of 2-D image.  It is
        generated by assuming the viewpoint to be at infinity and projecting
        all 3-D points orthogonally onto the screen.  The resulting picture
        is a "wire-frame" representation of the 3-D object's edges.


        HYPERGEO    Version 1.1                             Page 2-3



        *  Perspective Projection

        This is a straightforward linear projection similar to orthographic
        except that it uses a finite value for the distance between the
        viewpoint and the screen.  It also generates a wireframe
        representation but provides a sense of perspective which gives some
        sense of 3-D depth.  It is enhanced by allowing the user to modify
        the apparent eye-to-screen distance, thus adjusting the amount of
        perspective in the picture.


        *  Stereoscopic Projection (Anaglyph)

        This uses the realistic eye-to-screen distance of the simple
        perspective approach, but two separate images are generated which
        correspond to the projections as perceived by the right and left eye
        (the calculation of the projections takes into account the
        distance between the eyes).  When the two images are displayed in
        contrasting colors (typically red and blue or red and green), they
        can be viewed through filtering "3-D" glasses which cause the brain
        to perceive the image with full three-dimensional depth.  Although
        wearing the colored glasses is somewhat cumbersome, this method
        produces far and away the most vivid and revealing impressions of
        the structure of the 3-D object.

        (The general technique of generating dual projections that
        correspond to each eye's separate view is known as "stereoscopic"
        projection.  The particular instance of stereoscopic viewing that
        uses different colors for the two projections with matching colored
        filters for the eyes is called an "anaglyph".)


        *  Hidden-line Projection

        This approach begins with the wire-frame representation generated by
        a normal perspective projection.  Additional calculations are
        performed that prevent the display of any edge lying behind a face
        of the 3-D object.  This causes the object to appear opaque.  The
        calculations required to determine which edges (or portions of an
        edge) to hide are extensive and slow the program down considerably.


        *  Solid Projection

        A solid projection performs all the processing required by a
        hidden-line projection plus some additional calculations to
        determine the orientation of each visible face of the 3-D object.
        Rather than drawing the edge outlines of the object, the faces are
        filled in using either a density of fill pattern or a particular


        HYPERGEO    Version 1.1                             Page 2-4


        shade of color that corresponds to the obliqueness of the face as
        seen from a theoretical light source; faces that are most directly
        illuminated are filled the brightest.  The result is a fairly
        realistic rendering of the 3-D object as solid.  The calculations
        required make solid projections even slower than hidden-line
        projections.


        (Note: since hidden-line and solid projections require the 3-D
        object to be a simple polyhedron, they can only be used with 3-D
        images created by a 4-D intersection, not those created by 4-D
        projections.)


        HYPERGEO    Version 1.1                             Page 3-1



    Section 3.    The HYPERGEO Program


    HYPERGEO is a program written for the IBM-PC and compatible computers
    running the PC-DOS or MS-DOS operating system.  It requires about 180K
    bytes of memory plus whatever is needed for the data requirements of the
    particular geometry currently active in the program.  For most ordinary
    objects (including the basic hypercube) memory requirements are quite
    modest, but they can increase tremendously for very large and complex
    four-dimensional geometries.  The program does not make use of expanded
    or extended memory, so the 640K DOS barrier is the absolute limiting
    factor on the size of geometries that can be handled by HYPERGEO.

    The program will generate graphical output for EGA, VGA, or Hercules
    display devices.  It does not support CGA graphics.  For Hercules (and
    optionally for the others) the graphics are monochrome, and the
    stereoscopic anaglyph display mode is unavailable.  The performance of
    HYPERGEO is identical on EGA and VGA systems with the exception of two
    areas in which the VGA's higher performance is utilized: first, all
    graphics are drawn with the higher VGA display resolution (640x480
    pixels); and second, the enhanced VGA color capabilities are used to
    provide superior solid surface rendering.  No non-standard, so-called
    "super-VGA" display modes are supported in this version.

    HYPERGEO performs extensive floating-point calculations, and the speed
    of the program will be enhanced considerably by the presence of a math
    coprocessor chip such as an 80287 or 80387.  If the system has a math
    coprocessor, HYPERGEO will detect its presence automatically.  All
    features are functional without a coprocessor, but performance will be
    as much as several times slower.

    If a mouse pointing device is on the system, it can be used to invoke
    the interactive rotation commands (see below).  However, a mouse is not
    required since all commands can be executed through keyboard entry.
    Only the left and right mouse buttons are used, so a two-button mouse is
    as good as a three-button mouse.


    3.1  General Structure

    The HYPERGEO program accepts a description of the topology of a
    four-dimensional hyperobject.  This description is contained in a
    "geometry definition file" which is a plain text file that can be edited
    by the user.  It contains coordinate information for each vertex point
    of the hyperobject plus connectivity information that indicates which
    vertexes are joined by edges.  See Appendix A for a complete discussion
    of the format of this file.  The program analyzes the topology and
    determines which edges form external faces and, in turn, which faces
    form external hyperfaces.


        HYPERGEO    Version 1.1                             Page 3-2


    The program also accepts various parameters that control the manner in
    which the hyperobject is displayed.  There are two ways in which these
    parameters are given to the program: through a special configuration
    file or as command-line arguments.  (Note: There is not a complete
    correspondence between all possible configuration file parameters and
    all possible command-line arguments.  Some parameters can be specified
    only in the configuration file and some only as command-line arguments.)
    See Appendix B for a full description of the format of the configuration
    file and Appendix C for a description of the available command-line
    arguments.

    A section of configuration parameters can also be appended to the end of
    a geometry definition file following all topology information; the
    format is the same as in the primary configuration file.  In effect,
    this acts as a secondary configuration file which will apply only to
    that particular geometry file.

    The order of precedence for conflicting parameters is as follows.
    Command-line arguments take the highest precedence and will always
    override a corresponding parameter in the configuration file.  Also,
    configuration parameters contained in a geometry definition file take
    precedence over corresponding parameters in the primary configuration
    file.  Finally, every parameter has a pre-defined default value which
    will be used in the absence of any specification via the configuration
    file or the command line.  The default values for all parameters are
    listed in the appendixes.  To summarize the order of precedence for the
    various avenues of parameter specification, they are, from highest to
    lowest:

        *   Command line arguments

        *   Configuration parameters in current geometry definition file

        *   Configuration parameters in primary configuration file

        *   Program's internal default values

    It should be noted that no configuration file or command-line arguments
    are absolutely required.  The program will operate without them using
    reasonable default values.

    Once all geometry and configuration information has been read in and
    processed, the HYPERGEO program creates a display of the image of the
    hyperobject.  At this point, the user has access to a set of commands
    which facilitate the interactive manipulation of the hyperobject's
    display.  These interactive commands are described throughout subsequent
    sections of this document and are summarized in Appendix D.


        HYPERGEO    Version 1.1                             Page 3-3



    3.2  The HYPERGEO Display Screen

    The screen displayed by the HYPERGEO program consists of two areas: the
    primary graphics area where the image of the geometrical object being
    examined is displayed, and an optional menu and mouse control pad area.
    The user may select whether or not to display the menu area.  If it is
    not displayed, the primary graphics area is the entire screen.   When
    the menu area is displayed, it occupies the right edge of the screen,
    and the primary graphics area fills the entire remaining screen.

    The menu area lists a number of display parameters that may be modified
    interactively by the user.  It also contains a panel of command
    selection boxes that may be activated by the mouse to perform various
    rotations of the object being displayed.  A mouse is not required for
    the HYPERGEO program; all mouse functions may also be invoked through
    keyboard commands.  However, the mouse does provide a more naturally
    intuitive feeling of control over the manipulation of the geometry being
    studied.  The mouse functions are only available when the menu area is
    displayed.  (All keyboard commands are always available regardless of
    whether the menu area is displayed or not.)

    The HYPERGEO program automatically scales and centers the geometry
    display within the primary graphics area.  The user may subsequently
    modify the scale and position of the display as desired.

    Some of the interactive commands will prompt for additional keyboard
    input (such as a numeric value, a color name, or a file name).  This is
    done using a special prompting window which pops up in the middle of the
    primary graphics area.  The user should type in the appropriate entry
    which will echo in the prompting window, and should terminate the entry
    by pressing <Enter>; this indicates that the entry is complete and
    correct and initiates the particular command.  Pressing <Esc> at any
    time while a prompting window is displayed will dismiss the window and
    cancel the command.

    A similar message window will appear whenever an error condition (such
    as invalid command input) must be reported.  No user input is asked for;
    the window serves only to convey a message.  Once it has been read, the
    user may dismiss the window by pressing any key.

    A special message window is the HYPERGEO Information Window.  This
    window contains some basic data describing the geometry and topology of
    the currently active object, and it also contains the HYPERGEO copyright
    statement.  By default, the Information Window is displayed
    automatically upon the initial start up of the program and every time a
    new geometry definition is read in.  However, the program may be
    configured to bypass this.  An interactive command exists to display the
    Information Window at any time during normal operation of the HYPERGEO
    program.  The Information Window is a normal message window and may be
    dismissed by pressing any key.


        HYPERGEO    Version 1.1                             Page 3-4


    On-line help is available at all times within the HYPERGEO program.  An
    interactive command brings up the display of a help window that
    temporarily overwrites the entire screen.  The help window contains a
    brief description of each of the interactive commands that the user may
    invoke to manipulate the active geometry.  Pressing any key dismisses
    the help window and returns the HYPERGEO display to its previous state.

    Note: The HYPERGEO program does NOT use the (more or less) standard F1
    key to bring up its on-line help window.  Instead it uses the '?' key.
    This frees the function keys (including F1) to accommodate other
    interactive commands.

    For systems with color EGA or VGA displays, the user has control over
    the colors used to draw the various components which comprise the
    overall graphical output.  The colors are specified using statements in
    the configuration file and may be set to any one of the sixteen standard
    EGA colors (listed in Appendix B).  In addition, the color used to draw
    the actual edge lines of the geometrical object itself can be modified
    interactively from within the HYPERGEO program.  In the absence of any
    user specification, HYPERGEO uses a set of default colors which are
    believed to provide an attractive and comfortably viewable appearance
    (although, of course, such matters are subject to the contentiousness of
    individual tastes).

    There are eight separate components that together make up the total
    HYPERGEO display, and each can be assigned a different color:

        *   Primary background

            This is the background color which covers the entire screen
            (including the optional menu area) before any graphics are
            drawn.  It may be any color, but a dark shade such as black or
            the default dark gray is recommended for visibility.  This is
            especially so when using the stereoscopic (anaglyph) display
            mode, which requires a dark background to produce a good 3-D
            effect.  (Black is probably the most vivid in this respect, but
            dark gray is a bit easier on the eyes for extended viewing.)

            Default: dark gray


        *   Primary foreground

            This is the color used to draw the actual edge lines of the
            geometrical object as it is projected onto the computer's
            display screen.  This color is also used for the transient data
            values displayed in the menu area.  Naturally, the primary
            foreground color should give a good contrast with the primary
            background color.

            Default: white


        HYPERGEO    Version 1.1                             Page 3-5



        *   Menu

            This color is used for drawing the outlines and fixed labels of
            the data items and the mouse control pad section in the menu area
            of the HYPERGEO program.  It is also used for the display of the
            name of the geometry definition file which the user may select
            to have appear in the upper, left corner of the primary graphics
            area.

            Default: yellow


        *   Geometry axis vectors

            As the user rotates the geometrical object in hyperspace, the
            HYPERGEO program keeps track of the orientation of each of the
            axes of the object's original coordinate system.  A projection
            onto 3-space of the unit vector drawn from the origin along each
            axis may optionally be displayed as part of the primary geometry
            graphics.  This assists in visualizing the displacement of the
            original orientation of the object in all the dimensions of
            hyperspace.

            Default: cyan


        *   Message window background

            This is the background color for all prompting and message
            windows.

            Default: brown


        *   Message window text

            This is the text color for all prompting and message windows.

            Default: white


        *   Help window background

            This is the background color used in the on-line help window.

            Default: light gray


        HYPERGEO    Version 1.1                             Page 3-6



        *   Help window text

            This is the color used for the text in the on-line help window.

            Default: black


    Two other colors are used in the HYPERGEO primary graphics, the red and
    blue used by the stereoscopic anaglyph display.  However, these colors
    are fixed by the requirements of the "3-D" viewing glasses and are not
    modifiable by the user.

    Color is also used in the surface rendering of the solid display mode,
    which may be used with images produced by 4-D intersections, but its use
    and configurability are complicated by the differing capabilities of EGA
    and VGA display devices.  For EGA systems, the colors used for solid
    surface rendering are fixed by the program and are not modifiable by the
    user; these colors give an acceptable gradation of shades from light to
    dark and are not colors available in the sixteen standard EGA colors.

    For VGA, the program takes full advantage of the enhanced color
    capabilities of the display device, in particular its ability to
    generate over 256,000 discrete shades.  HYPERGEO uses eight variations -
    ranging from very light to very dark - of a single VGA color to produce
    the consistent shadings necessary for realistic solid surface rendering.
    The user can specify the base color (in terms of a VGA red/green/blue
    color definition) from which HYPERGEO will generate the eight graduated
    shadings.

    For either EGA or VGA, the user may select not to use color for solid
    surface rendering, but to instead use variable pattern fill, which just
    uses the primary foreground color.  Variable pattern fill is always used
    on monochrome systems.


    3.3  Geometry Manipulations

    Central to the operation of the HYPERGEO program is the ability to
    interact with the geometry as it is being displayed and to have the
    picture on the screen respond to the user's directions.  It is just this
    ability to manually manipulate the hyperobject in all four dimensions of
    hyperspace that creates a sense of "hands-on" control and establishes an
    intuitive feel for the extradimensional shape of the hyperobject.

    In light of the two-stage process required to capture an image of a
    four-dimensional geometrical object on a two-dimensional computer
    display screen (first, 4-D to 3-D reduction, and second, 3-D to 2-D
    projection), HYPERGEO provides interactive functions to manipulate the
    geometry both in its original hyperspace configuration (in 4-space) and
    in its reduced three-dimensional representation (in 3-space).  In


        HYPERGEO    Version 1.1                             Page 3-7


    addition, the program allows the user to control a number of parameters
    that determine how the final 2-D picture is generated.

    A word or two is needed here on the subject of coordinate systems.  The
    HYPERGEO program uses the letters X, Y, Z, and W to label the four
    coordinates in hyperspace.  Wherever ordering is significant (as, for
    example, in the specification of coordinates within a geometry
    definition file), the order is (X,Y,Z,W).  As is conventional with
    computer graphics, three-dimensional coordinate systems are
    "left-handed"; thus, the positive Z-axis points away from the viewer and
    into the screen.


    3.3.1  Rotations

    The primary method of examining an object is to turn it about to view it
    from many different angles.  In this manner, it becomes possible to
    discover the overall nature of something which - for whatever reason -
    is not immediately accessible in its entirety to our perceptions.  In
    the HYPERGEO program, rotations in all possible directions are available
    through interactive commands, and it is possible to position the
    geometry in whatever orientation is desired.

    Rotations can be performed either on the 4-D geometry before it is
    reduced to a 3-D image, or on the 3-D geometry of the reduced image
    itself.  The effect of the two classes of rotation is different, not
    only because of how they are handled by the HYPERGEO program, but also
    because of fundamental differences between the structure of 3-space and
    that of 4-space.

    In our familiar 3-space, rotations are performed about an "axis" which
    is a straight line in space.  The rotation leaves all points on the axis
    unchanged while all other points in 3-space move in circles that are
    centered on the axis and that lie in a plane perpendicular to the axis.
    However, in 4-space, there are an infinite number of lines that pass
    through a given point on a plane and that are perpendicular to the
    plane.  Thus when a hyperobject is rotated so that the points in a
    particular plane move in circles about a fixed point on the plane, it is
    not correct to say that the rotation takes place around an axis.  It is
    most meaningful to our three-dimensional minds to think of 4-D rotations
    as being "on" or "of" a particular plane rather than being "around"
    anything.  (Actually, rotations in hyperspace take place around a plane
    in the sense that for any given direction of hyperspatial rotation there
    is a plane on which no points move, namely, the plane determined by any
    two distinct lines perpendicular to the rotating plane and passing
    through the rotational centerpoint on it.  This is virtually impossible
    to picture for our 3-D brains, so it is simpler to refer to such
    rotations in terms of the plane which is being rotated.)

    Any particular spatial orientation can be reached by performing an
    appropriate sequence of rotations of the planes determined by pair-wise


        HYPERGEO    Version 1.1                             Page 3-8


    combinations of the space's coordinate lines.  In our 3-space there are
    three such planes: the XY-plane, the XZ-plane, and the YZ-plane.  In
    hyperspace, which has four orthogonal coordinate axes, there are six
    planes that form the basis for all rotations: XY, XZ, XW, YZ, YW, and
    ZW.  (In 3-space the rotations are normally referred to by the name of
    the axis around which the rotation occurs rather than the plane of the
    rotation.)  HYPERGEO has interactive commands for performing rotations
    of all nine planes.  For all HYPERGEO rotations, the coordinate system's
    origin is always the centerpoint; thus the origin is never moved by any
    rotation.  (Note: the configuration file accepts a parameter for
    repositioning the origin within the original geometry; this in turn has
    the effect of repositioning the center of all rotations to any desired
    point in hyperspace.)

    All rotations in HYPERGEO are in incremental steps of the current Turn
    Angle.  The value of the Turn Angle can be set by the user in the
    configuration file, and it can also be modified interactively at any
    time.  Its current setting in degrees is displayed in the menu area.
    The Turn Angle can be set to a large number of degrees (say, 30 or 45)
    for rapid rotations of the object, and then reduced to a smaller value
    for fine positioning.  Interactive commands exist for rotating the
    object one incremental step, through either a positive or negative Turn
    Angle, within any one of the three 3-D and six 4-D base rotational
    planes.  Also, there are commands for automating any one of the possible
    modes of rotation; in these automated modes HYPERGEO continues to
    perform incremental rotations in the specified direction as rapidly as
    the computer's processing speed will allow.

    The HYPERGEO program provides two features to assist the user in keeping
    track of the orientation of the active object in hyperspace as it
    undergoes various rotations.  First, there is the ability to display the
    projected image of all four unit vectors emanating from the origin along
    the four hyperspace coordinate axes.  This provides good visual feedback
    of how the object is turning in hyperspace.  Second, the menu area
    contains data fields that display the angular displacement of each
    coordinate axis from its initial, unrotated position.  These
    displacement angles can vary from 0 degrees (when the axis is coincident
    with its initial alignment) to 180 degrees (indicating that it has been
    rotated to point in the completely opposite direction).

    It is important to understand how four-dimensional rotations of the
    hyperobject in hyperspace differ from three-dimensional rotations of the
    reduced image in normal 3-space.  The critical point is that rotations
    in 4-space affect the shape and structure of the reduced 3-D image and
    do not necessarily produce directly equivalent rotations of that 3-D
    image.  Rotations in 3-space, on the other hand, are performed after the
    4-D to 3-D reduction, and they always directly rotate the 3-D image;
    they never alter its shape.

    To clarify this, consider what is going on as the hyperobject is turned
    in hyperspace.  If the program is using 4-D to 3-D intersection mode,


        HYPERGEO    Version 1.1                             Page 3-9


    the slicing hyperplane will pass through varying cross-sections of the
    object, and these cross-sections will be three-dimensional polyhedra of
    varying topologies.  The 3-D polyhedra themselves will not necessarily
    rotate.  For the projection mode of 4-D to 3-D reduction, rotations of
    the hyperobject will have the effect of altering the pattern of the
    network of connected edges that is projected into 3-space, but the
    motion of the network will not be any simple three-dimensional rotation.

    This distinction - that 4-D rotations alter the shape and structure of
    the reduced 3-D image, while 3-D rotations rotate the image - while
    generally true, is, alas, not the whole story.  Four-dimensional
    rotations behave differently depending on whether or not the plane of
    the rotation contains the W-axis.  In HYPERGEO, 4-D to 3-D intersections
    always occur along a hyperplane that is defined by a constant W value,
    and 4-D to 3-D projections always are performed onto a hyperplane that
    is similarly defined by a constant W value.  For this reason, rotations
    that occur on planes that do not contain the W-axis do not alter the
    shape of the resulting 3-D image (because they do not modify the
    W-coordinate of any points in the hyperobject). Instead, they behave
    like 3-D rotations of the corresponding plane (for example, a 4-D
    rotation of the XY-plane has the same effect as a 3-D rotation around
    the Z-axis).  The net outcome of all this is that the 4-D rotations of
    the XY-, XZ-, and YZ-planes look like simple 3-D rotations and have the
    same effect on the appearance of the object's display in HYPERGEO as 3-D
    rotations around the Z-, Y-, and X-axes, respectively.  The other three
    4-D rotations (that is, of the XW-, YW-, and ZW-planes) do alter the 3-D
    image's shape and do not resemble any simple 3-D rotations.

    One final point is needed to complete this discussion of rotations.
    Although 4-D rotations of the XY-, XZ-, and YZ-planes look the same as
    3-D rotations around the Z-, Y-, and X-axes, they are not treated the
    same by the HYPERGEO program.  The 4-D rotations permanently alter the
    orientation of the hyperobject in 4-space as that orientation is
    calculated and remembered by the program; they are cumulative in the
    sense that each subsequent 4-D rotation takes as its initial orientation
    the one arrived at by the previous 4-D rotation.  3-D rotations, on the
    other hand, do not permanently modify the four-dimensional orientation
    of the hyperobject; rather, they serve as temporary repositionings of
    the current 3-D image in 3-space.  For this reason, 3-D rotations are
    most useful as a means of examining the shape of the current 3-D image
    expressly without altering the 4-D orientation of the hyperobject that
    produced that image.  A 4-D rotation performed following a sequence of
    3-D rotations reverts back to the most recent 4-D orientation for its
    starting position since that is the permanent orientation remembered by
    HYPERGEO.  The transient 3-D rotations of the reduced image are lost and
    forgotten.  (It is worth noting that although the 4-D rotations may look
    like 3-D rotations, they involve the additional computational effort of
    updating the permanent 4-D database and thus are noticeably slower.)

    This functional distinction between 4-D and 3-D rotations is reflected
    in the way HYPERGEO reports the hyperobject's orientation.  In


        HYPERGEO    Version 1.1                             Page 3-10


    particular, the axis displacement angles (displayed in the menu area)
    and the axis vectors (displayed in the primary graphics area) are only
    affected by 4-D rotations.  During temporary, 3-D rotations, they remain
    unchanged and continue to indicate the permanent 4-D orientation from
    which the next 4-D rotation will proceed.


    3.3.2  Translation of the Intersecting Hyperplane

    For the intersection mode of 4-D to 3-D reductions, the program
    calculates the three-dimensional intersection of the hyperobject with a
    hyperplane that is defined by the equation, w = k, that is, having a
    constant W coordinate value.  (In four dimensions, a linear equation of
    the four coordinate variables, Ax + By + Cz + Dw = E, defines a
    hyperplane.  A hyperplane can perhaps best be envisioned as a normal
    three-dimensional space that is situated in hyperspace and has zero
    extent into the fourth dimension.  Our universe at any given instant can
    be considered a hyperplane in four-dimensional space-time.)  The shape
    of this intersection can be varied by rotating the hyperobject; this
    approach has been discussed in the above section on rotations.  However,
    the intersection can also be modified by moving the hyperplane through
    hyperspace, in particular, by moving it along the W-axis by altering the
    value for 'k' in the hyperplane's equation.  The HYPERGEO program
    provides interactive commands for performing such a translation of the
    intersecting hyperplane.  (It should be emphasized that the position of
    the intersecting hyperplane only has an effect in the intersection mode
    of 4-D geometry reduction; it is irrelevant when using projection mode.)

    Two parameters relate to the interactive translation of the intersecting
    hyperplane: the current W value of the hyperplane's equation, and the
    amount by which that value will be incremented (or decremented) for each
    translation of the hyperplane.  Both of these values are displayed in
    the HYPERGEO menu area: the current W value of the intersecting
    hyperplane is called the W-INTER parameter, and the incremental step
    value is called DELTA-W.  An interactive command exists that increments
    the W-INTER value by the value of DELTA-W and updates the HYPERGEO
    display accordingly.  Another command serves to decrement the W value
    and thus move the intersecting hyperplane in the reverse direction.  In
    addition, both parameters may be explicitly set to any desired numeric
    value using other interactive functions.  The initial values of the
    W-INTER and DELTA-W parameters may also be set in the HYPERGEO
    configuration file.  If W-INTER is not specified by the user in the
    configuration file, its default value is calculated as the average of
    the minimum and maximum W values for the active geometrical object thus
    centering the intersecting hyperplane within the geometry.

    Translation of the intersecting hyperplane is most revealing when it
    proceeds from one edge or face of the hyperobject, through its entire
    body, and on to the opposite side.  A feature exists which automates
    just such a traversal of the hyperplane.  It continuously increments the
    W value until the body of the hyperobject is exited.  The direction of


        HYPERGEO    Version 1.1                             Page 3-11


    translation is then reversed and the W value is continuously decremented
    until the opposite side of the hyperobject is reached.  The direction of
    translation continues to be reversed at each extreme so that the
    hyperplane passes back and forth (in hyperspace) through the entire body
    of the hyperobject.  Performing such passes upon varying orientations of
    the hyperobject in hyperspace provides an excellent method for acquiring
    a sense of its four-dimensional shape.  (Note: the effective speed of
    the hyperplane's motion along the W-axis can be adjusted by setting the
    DELTA-W value appropriately.)


    3.3.3  View Control

    The final step in the processing chain that leads from the specification
    of the geometry of a four-dimensional hyperobject to the depiction of an
    image of that object on the computer screen is the generation of the
    two-dimensional graphical output.  As described above in a previous
    section of this document, the program uses a number of variations of a
    basic 3-D to 2-D linear projection to perform this step.  The user has
    access to several interactive commands with which to adjust a number of
    parameters that affect the appearance of the final display.  Paramount
    among these is the output display mode.

    As outlined earlier, there are five possible projection modes in which
    the final graphical output can be presented: orthographic, perspective,
    stereoscopic anaglyph, hidden-line, and solid.  The selection of a
    particular mode is one of the parameters which the user can modify; any
    one of the five modes can be invoked at any time (with some exceptions -
    see below).  A data field in the menu area shows which mode is in
    effect.  Whenever the display mode is changed, the program immediately
    redraws the representation of the current geometry using the method of
    the new mode.  The basic orientation of the hyperobject remains fixed
    across any such switch in display mode.

    The exceptions referred to in the preceding paragraph are as follows:
    first, the stereoscopic anaglyph mode cannot be invoked on monochrome
    systems (since the anaglyph requires red and blue); and second, the
    hidden-line and sold display modes can only be selected if the 4-D to
    3-D reduction step is using 4-D intersection, not 4-D projection (since
    4-D projections do not produce a 3-D geometry which is a solid 3-D
    polyhedron amenable to hidden-line analysis).

    Two numeric parameters are key in the calculations which generate the
    graphics seen on the screen.  First, the eye-to-screen distance is used
    to calculate all forms of 3-D to 2-D projection (except orthographic, in
    which the eye-to-screen distance is effectively infinite).  The
    eye-to-screen distance is shown as a data item in the menu area.  It can
    be adjusted up or down interactively by the user.  Decreasing the
    eye-to-screen distance has the effect of enhancing the perceived degree
    of perspective in the picture, and increasing the distance reduces the
    perspective.  The eye-to-screen distance can be adjusted in increments


        HYPERGEO    Version 1.1                             Page 3-12


    or decrements equal to 10% of its current value, or it can be set to an
    explicit numeric value.  The program will allow the eye-to-screen
    distance to be reduced to a very small value; however, it should never
    be made so small that the viewpoint effectively enters the object being
    displayed as this results in distorted and meaningless graphics.

    The second key display parameter is the scale factor which is the ratio
    of geometry units to screen inches.  This value determines the apparent
    size of the object as finally drawn on the screen.  The scale ratio is
    displayed as a data field in the menu area.  It can be adjusted up or
    down in increments of 10% of the current value, or it can be assigned an
    explicit numeric value.  When a new geometry file is read in, and if
    there is no overriding information in the configuration file, the
    program automatically sets the scale ratio so the image of the object
    fits comfortably within the primary graphics area allowing plenty of
    room for rotations.

    Both the eye-to-screen distance and the display scale ratio can be
    specified in the configuration file.


    3.4  Special 3-D Geometry Definition

    The designed function of the HYPERGEO program is the display and
    examination of the properties of four-dimensional geometrical objects.
    The program's input is normally expected to be a geometry definition
    file that specifies four coordinate values for each vertex point.  All
    of the preceding discussion has been in terms of this basic scenario.
    However, the program will also accept geometry definitions of
    three-dimensional objects and will display them with all features
    available that relate to the 3-D level of processing.  When the input
    geometry is three-dimensional, all of the program's commands that
    normally perform operations in four dimensions are deactivated; if the
    user selects a keyboard command that only operates on four-dimensional
    geometries, a message window appears with an explanation that the
    command is not applicable to the current 3-D object, and the command is
    ignored.  In special 3-D geometry mode, the section of the mouse control
    pad area that contains the selection boxes for 4-D rotations is shaded
    over with a stippled pattern, and those mouse functions cannot be
    selected.

    A 3-D geometry definition file has the same format as a 4-D file (see
    Appendix A) with two exceptions: first, the value of the "number of
    dimensions" field is 3 instead of 4; and second, each point has only
    three coordinates, not four.

    When operating upon a 3-D geometry definition, the program treats all
    3-D rotations as permanent modifications to the object's database.  The
    displayed axis angle displacements and the unit axis vectors are changed
    upon 3-D rotations.  (Recall that for 4-D geometries these items remain
    fixed during 3-D rotations and are only changed when a 4-D rotation is
    performed.)


        HYPERGEO    Version 1.1                             Page 4-1




    Section 4.    Inner Workings


    4.1  Units

    The coordinates of the geometry definition are considered to be in
    so-called "geometry units" which are not assumed to have any particular
    relation to any real physical unit.  Regardless of the size of the
    coordinate values, HYPERGEO will scale the display to fit within the
    visible screen area.  Some of the parameters that can be specified by
    the user are expressed in geometry units and must be entered that way.
    Other parameters, in particular, those relating to the projection of the
    final 2-D image on the computer's screen, are expressed in real-world
    units.  The program uses inches for all real-world dimensions, and the
    display scale is expressed as a ratio of geometry units to inches (the
    inches here refer to distances on the face of the computer's display
    screen).

    In the detailed reference material contained in the appendixes to this
    document and in all program prompts and messages it is always indicated
    what units a particular data value is expressed in.


    4.2  Verification of the Geometry Definition

    When HYPERGEO reads in a geometry definition file, either 4-D or 3-D, it
    analyzes the topology of the described object.  The number of vertexes,
    edges, faces, and (for 4-D) hyperfaces is calculated, and various
    aspects of the input file and its derived topology are checked for
    validity:

        *  The number of dimensions must be either 3 or 4.

        *  Every vertex must have at least as many edges connected to it as
           the number of dimensions.

        *  All edge connections must be consistently specified between both
           connected vertexes, that is, if vertex #1 is in the connection
           list of vertex #2, vertex #2 must be in vertex #1's list, also.

        *  The numbers of vertexes, edges, faces, and (for 4-D) hyperfaces
           must agree with the theoretical values as given by Euler's
           Formula or its hypergeometric version.

    If any of these checks fails, an error message is displayed, and when
    the message window is dismissed, the program exits back to DOS.


        HYPERGEO    Version 1.1                             Page 4-2



    4.3  Calculating 4-D to 3-D Projections

    To project the original 4-D geometry onto 3-space, an imaginary
    viewscreen (which is a three-dimensional hyperplane) and an imaginary
    viewpoint must be positioned in hyperspace.  The HYPERGEO program uses
    the so-called "radius" of the hyperobject to determine where the
    viewscreen and viewpoint are located.  The radius of the hyperobject is
    the maximum four-dimensional distance from the origin of all of the
    object's vertex points.  The viewscreen is positioned at a distance
    equal to the hyperobject's radius along the negative W-axis and normal
    (or "hypernormal") to that axis.  The viewpoint is located at a distance
    equal to 1.5 times the radius along the negative W-axis.

    This arrangement produces properly sized images with a reasonable
    measure of 4-D "hyperperspective" if the configuration of vertex points
    in the geometry definition is more or less symmetrical in hyperspace
    about the origin.  Ideally, the geometry specification should be
    perfectly symmetrical, with all vertex points lying on the surface of a
    hypersphere centered on the origin.  This is possible with the regular
    four-dimensional "polytopes".  (A polytope is the extra-dimensional
    equivalent of a 3-D polyhedron; a regular four-dimensional polytope has
    hyperfaces that are all identical polyhedra and that all adjoin one
    another across identical polygonal faces at identical angles.)


        HYPERGEO    Version 1.1                             Page 5-1




    Section 5.    Restrictions and Limitations


    5.1  Geometry Definition File

    The database maintained by the HYPERGEO program is dynamically allocated
    and expands to fit the size of the geometry being handled so there are
    no precise and absolute capacity limits on how big and complicated an
    object may be.  The ultimate barrier is the 640K limit of available
    memory under DOS.  Although I have not done the actual experiment, my
    guess is that the two largest regular four-dimensional polytopes, with
    120 and 600 hyperfaces respectively (and as many as 1200 edges and
    faces), would far overwhelm HYPERGEO and DOS with their memory
    allocation requirements.  (In any event, if, through some miracle, those
    geometries could be made to fit in memory, the computations involved in
    their topological analysis and 3-D image generation would be so
    prolonged as to reduce the program's responsiveness to a state that
    could only be described as "glacier-like".)

    For more reasonably sized geometries, the most important restriction is
    that all 4-D hyperobjects must be "simple" and externally convex.
    (Simple means the object must be solid without any holes passing through
    it.)  For 3-D geometries, the requirement that the object be externally
    convex is removed, but it must still be a simple polyhedron, and all its
    faces must be simple polygons without internal edges.

    As mentioned above, HYPERGEO performs some checking on the topological
    correctness of the input geometry.  However, to be truthful, this
    checking is not very comprehensive; it can certainly fail to detect some
    errors in the input, and when it does find errors, the messages issued
    are not as helpful as could be wished in pinpointing the problem.  I am
    sure that it would be possible to get the program to misbehave quite
    seriously, including hanging or crashing, by feeding it pernicious input
    data.  Therefore, until a future version of the program possesses
    improvements in this regard, the user must take some pains to insure
    that all new geometry definition files are created carefully and
    accurately.  If the program crashes upon startup, consider that to be a
    very loud, broadband error message, and check the geometry file closely.
    (However, HYPERGEO is not totally deficient in this regard, and it will
    issue error messages when certain problems are detected.  A geometry
    definition file with only a handful of conventional errors will be
    handled adequately.)


        HYPERGEO    Version 1.1                             Page 5-2



    5.2  Mathematical Precision

    All calculations in HYPERGEO are performed using double precision
    floating-point arithmetic.  The double precision word length is 64 bits
    which provides about fifteen decimal digits of accuracy.  Any
    floating-point values given to the program (such as coordinates in the
    geometry definition file or scale ratio in the configuration file) may
    usefully contain up to fifteen digits.


        HYPERGEO    Version 1.1                             Page 6-1




    Section 6.    Supplied Geometry Definition Files


    The HYPERGEO program comes with several "ready-to-run" geometry
    definition files, including both 4-D and 3-D objects.  These files
    include four of the six regular four-dimensional polytopes and all five
    of the three-dimensional regular polyhedra.  (As indicated above, the
    remaining two regular 4-D polytopes are almost certainly too big and
    complex to fit in HYPERGEO under DOS; they also are too far beyond my
    powers of hypergeometric comprehension to capture in a HYPERGEO input
    file.)

    All of the supplied geometry definition files have the file extension
    .GEO, and the root portion of the file name corresponds to the name of
    the geometric object contained in the file (with some abbreviations as
    required by DOS file name length limits).  The regular four-dimensional
    polytopes are in the files:

        File Name       Vertexes    Edges       Faces       Hyperfaces
       ------------     --------    -----       -----       ----------

          5CELL.GEO         5          10          10           5

       HYPRCUBE.GEO        16          32          24           8

         16CELL.GEO         8          24          32          16

         24CELL.GEO        24          96          96          24


    Note that the polytope names are of the form 'N-cell' where N is the
    number of hyperfaces.  Also, the hypercube, technically and to be
    consistent, could be called the 8-cell (just as the cube could be called
    a 'hexahedron' and the square a 'tetragon').  The two 4-D regular
    polytopes that aren't included here are the 120-cell and the 600-cell.
    (Note: The 24-cell can be quite slow to read in and to display using
    hidden-line and solid modes.)


        HYPERGEO    Version 1.1                             Page 6-2



    For three dimensions, all of the regular polyhedra names except "cube"
    are of the form '<prefix>hedron' where <prefix> is the Greek numeral
    indicating the number of faces: tetrahedron, octahedron, dodecahedron,
    and icosahedron.  The HYPERGEO geometry definition file names are formed
    from just the prefix without the base 'hedron':

        File Name       Vertexes        Edges           Faces
       ------------     --------        -----           -----

          TETRA.GEO         4               6               4

           CUBE.GEO         8              12               6

           OCTA.GEO         6              12               8

         DODECA.GEO        20              30              12

          ICOSA.GEO        12              30              20


    In addition to the regular polytopes and polyhedra, the following
    geometry definition files are also included:

        File Name     Dimensions    Geometrical Object
       ------------   ----------    -----------------------------------

        BLOCKS.GEO        3         A three-dimensional cross formed by
                                    stacking cubes along each of the three
                                    coordinate dimensions.

        STELLA.GEO        3         The "stella octangula", an interesting
                                    polyhedron formed by merging two 3-D
                                    tetrahedra.

            HG.GEO        3         A shape that forms the HYPERGEO logo by
                                    presenting the outline of the letter 'H'
                                    when viewed from two of the three
                                    orthogonal directions and the outline of
                                    the letter 'G' when viewed from the
                                    third.  (Note: This geometry can be
                                    quite slow to read in, and is very slow
                                    for hidden-line and solid display modes
                                    - although the results are attractive.)


        HYPERGEO    Version 1.1                             Page 6-3



       HYTETRA.GEO        4         These four objects are "hyperprisms"
        HYOCTA.GEO        4         derived from the 3-D regular polyhedra
      HYDODECA.GEO        4         contained in their names.  A hyperprism
       HYICOSA.GEO        4         is formed by positioning two 3-D objects
                                    in hyperspace at a given separation
                                    along the W-axis and connecting each of
                                    the pairs of corresponding points
                                    between the two objects.  The resulting
                                    hyperobject has two hyperfaces that are
                                    the same shape as the base 3-D object
                                    plus an additional number of hyperfaces
                                    equal to the number of faces in the base
                                    3-D object.  These additional hyperfaces
                                    are 3-D prisms whose end faces are the
                                    same polygonal shape as the faces of the
                                    base 3-D object.  Thus, for example, the
                                    HYTETRA.GEO hyperobject is based on the
                                    3-D tetrahedron; it has two hyperfaces
                                    that are tetrahedra and four hyperfaces
                                    that are prisms with triangular end
                                    faces.  (Note: Technically the hypercube
                                    is itself a hyperprism  - one based on
                                    the cube - but since it is also one of
                                    the regular 4-D polytopes, it is not
                                    included in this series.)


        HYPERGEO    Version 1.1                             Page A-1




    Appendix A.    Geometry Definition File


    This appendix describes the format of the geometry definition file.
    This file is the primary input to the HYPERGEO program.  It specifies
    the topology and dimensions of the geometrical object to be displayed.
    In particular, the file contains the coordinate values for all vertex
    points of the object plus connectivity information that indicates which
    vertexes are joined by edges.

    The geometry definition file is an ordinary text file that can be edited
    using any available text editor.  It is read by the HYPERGEO program a
    line at a time; each line comprises one complete data record, and there
    is no way to continue data across multiple lines.  The maximum length of
    a line is 255 characters.

    The geometry definition file can contain notes and comments that are not
    part of the data read by the program, but serve as human-readable
    annotation.  This is indicated by preceding any such comments with the
    character '!'.  HYPERGEO ignores any '!' character plus whatever follows
    it on the same line.  If the first character on a line is a '!', the
    entire line is skipped by the program.  If the '!' character appears
    following some data on a line, that data is read by the program, but
    everything after the '!' is ignored.

    The geometry definition file can contain any number of blank lines,
    which are ignored.  A line which contains only comment text is
    considered by the program to be a blank line.

    Every geometry definition file must begin with two particular data
    lines.  The first contains a single integer value which is the number of
    dimensions of the geometric object; this number must be either 3 or 4.
    The second line contains a single integer that is the number of vertex
    points in the geometry.  Following these two lines, there must be one
    line with point data for each vertex point.

    Each point data line contains the coordinate and connectivity
    information for a single vertex point.  Its format is:

        n:  x.x y.y z.z [w.w]   p1  p2  p3 [... pN]

    where 'n' is the sequence number of the point data line; 'x.x', 'y.y',
    'z.z', and 'w.w' are the X, Y, Z, and W coordinates of the point; and
    'p1' through 'pN' are the sequence numbers of all other points connected
    to this point by edges.

    Each point's sequence number serves to identify the point within the
    file.  It is the number used in the connection lists, and it is also the
    number used to refer to the point in any error messages that may be
    issued.  All point sequence numbers are integers, and they must be in


        HYPERGEO    Version 1.1                             Page A-2


    ascending order within the file; the first point data line must have
    sequence number 1, and the last point data line must have a sequence
    number equal to the total number of vertex points in the object.

    The coordinate values are floating-point numbers and may contain an
    optional decimal point and fractional digits.  They may also contain a
    decimal exponent in standard scientific notation format.  The coordinate
    values may be in any units desired; the program always scales the final
    display to fit the screen.  The W coordinate must be specified for a
    four-dimensional geometry and must not be specified for a
    three-dimensional one.

    The list of connected points contains the sequence numbers of all points
    connected to the point described by the given data line.  Connection
    information must be symmetrical, that is, if point A is in point B's
    connection list, point B must also be in point A's connection list.  It
    is an error for a point to be in its own connection list.  Unlike the
    coordinate values, the program has no way of knowing how many
    connections should exist for each point and hence cannot directly report
    an error if there are too few or too many.  Therefore, it is incumbent
    upon the user to construct these lists with some care.

    Except for the ':' that must follow the sequence number, no punctuation
    is used to separate the data values, just one or more spaces.

    Following all point data, the geometry definition file may optionally
    contain configuration information.  If it appears, it must consist of
    statements that are identical in format to the primary configuration
    file (see Appendix B).  Configuration parameters specified in a geometry
    definition file override corresponding parameters set in the primary
    configuration file, but they remain in effect only while that geometry
    definition file is active in the HYPERGEO program.

    Note:  There is no formal delimiter that marks the end of the geometry
    data and the start of configuration data in a geometry definition file.
    The program expects to find a number of point data lines equal to the
    total number of vertex points as specified on the second data line of
    the file.  Any non-comment lines found in the file after that number of
    point data lines have been read are assumed to be configuration data.
    This can lead to some very long and confusing error reports if the
    number of point data lines is not correct.


        HYPERGEO    Version 1.1                             Page B-1




    Appendix B.    Configuration File


    This appendix describes the format of the HYPERGEO configuration file.
    This file allows the user to set a number of parameters that control
    various aspects of the program's performance.  The file is read once
    upon start-up of the program, and is reread each time a new geometry
    definition file is entered.

    In addition to the configuration file, similar configuration information
    may also be contained in any geometry definition file.  The format is
    the same in both places, but the parameter specifications in the geometry
    definition file override those in the primary configuration file;
    however, they are in effect only while that geometry file is active in
    the program (see Appendix A).

    A command line argument exists that allows the user to specify the name
    of the particular primary configuration file to use.  This permits
    multiple configuration files to exist and to be selected as desired for
    different executions of HYPERGEO.  If no configuration file name is
    specified via a command line argument, the program tries to open one
    called HYPERGEO.CFG.  It is not an error if this default configuration
    file cannot be opened, but it is a fatal error if a configuration file
    whose name is specified by the command line argument cannot be opened.

    The configuration file is an ordinary text file that can be edited using
    any available text editor.  It is read by the HYPERGEO program a line at
    a time; each line comprises one complete data record, and there is no
    way to continue data across multiple lines.  The maximum length of a
    line is 127 characters.

    The configuration file can contain notes and comments that are not part
    of the data read by the program, but serve as human-readable annotation.
    This is indicated by preceding any such comments with the character '!'.
    HYPERGEO ignores any '!' character plus whatever follows it on the same
    line.  If the first character on a line is a '!', the entire line is
    skipped by the program.  If the '!' character appears following some
    data on a line, that data is read by the program, but everything after
    the '!' is ignored.

    The configuration file can contain any number of blank lines, which are
    ignored.  A line which contains only comment text is considered by the
    program to be a blank line.

    Each data line in the configuration file contains a single parameter
    definition.  The format of each line is simply the key word identifying
    the parameter followed by whatever data values the particular parameter
    requires.  No punctuation should be used; all parameters and values
    should be separated by one or more spaces.


        HYPERGEO    Version 1.1                             Page B-2


    The case of letters used in the configuration file does not matter, so
    lower or upper case (or both) can be used as desired.  However, the
    program is absolutely unforgiving about spelling; all parameter key
    words must be spelled exactly as they are shown in this document - no
    abbreviations, no hyphens instead of underscores, no near misses.

    Most parameters take a single data value, either a number or another key
    word.  Some parameters take a sequence of several numeric values, and
    some take no value at all.  Numeric values that are inherently integral
    must be entered as pure integers.  Where floating-point values are
    appropriate, the data item may include a decimal point and fractional
    digits as well as an exponent in scientific notation.

    Errors encountered by the program while trying to read a configuration
    file (or configuration data in a geometry definition file) are reported
    in a message window.  All configuration file errors are fatal and
    terminate the program after the error message window is dismissed.

    The following is a list of all parameters that may be specified in the
    HYPERGEO configuration file or a geometry definition file:


    Parameter:  COLOR
                COLOR_BACKGROUND
                COLOR_VECTOR
                COLOR_MENU
                COLOR_MESSAGE_BACKGROUND
                COLOR_MESSAGE_TEXT
                COLOR_HELP_BACKGROUND
                COLOR_HELP_TEXT

    Value:      A color name chosen from the sixteen standard EGA colors.
                The possible colors are:

                    BLACK
                    BLUE
                    GREEN
                    CYAN
                    RED
                    MAGENTA
                    BROWN
                    LIGHTGRAY
                    DARKGRAY
                    LIGHTBLUE
                    LIGHTGREEN
                    LIGHTCYAN
                    LIGHTRED
                    LIGHTMAGENTA
                    YELLOW
                    WHITE


        HYPERGEO    Version 1.1                             Page B-3


    Use:        These eight parameters set the colors used for each of the
                component parts of the HYPERGEO display:

                    COLOR - Primary geometry graphics plus the transient
                        text values in the menu area

                    COLOR_BACKGROUND - Screen background

                    COLOR_VECTOR - Unit axis vectors

                    COLOR_MENU - Outlines and fixed labels in menu area plus
                        the mouse control pad; also the geometry file name

                    COLOR_MESSAGE_BACKGROUND - Background color for all
                        message and prompting windows

                    COLOR_MESSAGE_TEXT - Text color for all message and
                        prompting windows

                    COLOR_HELP_BACKGROUND - Background color of the on-line
                        help screen

                    COLOR_HELP_TEXT - Text color of the on-line help screen

                The screen background color should be a dark color (BLACK or
                DARKGRAY) if the stereoscopic anaglyph mode of display is to
                be used.

    Default:    COLOR                           WHITE
                COLOR_BACKGROUND                DARKGRAY
                COLOR_VECTOR                    CYAN
                COLOR_MENU                      YELLOW
                COLOR_MESSAGE_BACKGROUND        BROWN
                COLOR_MESSAGE_TEXT              WHITE
                COLOR_HELP_BACKGROUND           LIGHTGRAY
                COLOR_HELP_TEXT                 BLACK


        HYPERGEO    Version 1.1                             Page B-4



    Parameter:  COLOR_SOLID

    Value:      A red/green/blue color specification in the form of three
                integers each in the range 0 to 63.  The integers should be
                entered without punctuation, separated by spaces.

    Use:        On VGA color systems, defines the colors to be used for
                surface rendering in solid display mode.

                For solid surface rendering the program uses eight shades of
                graduated intensity based on the same color.  The shades
                range from light to dark and are used to represent the
                varying degrees of illumination received by the different
                faces of the object from a theoretical light source.

                The color specified by the COLOR_SOLID parameter defines the
                lightest of the eight shades; the program creates the
                remaining seven to be increasingly dark variations.
                Therefore, the color specified should be fairly light, that
                is, have relatively high values for the red, green, and blue
                color components.

                Note: This parameter is only meaningful on systems with VGA
                color graphics.

    Default:    53  63  43

                This defines a light, slightly grayish green.  It produces a
                spectrum of eight shades that range through medium
                gray-greens down to a dark, evergreen green.


    Parameter:  DELTA_W

    Value:      numeric value (floating-point)

    Use:        Specifies the incremental step amount used to modify the W
                intersection value during execution of the HYPERGEO command
                for translation of the intersection hyperplane (Page Up/Page
                Down - see the section on Interactive Commands below).  This
                value is in the units of the geometry definition.

                Because this parameter is in units that pertain to a
                particular geometry definition, it is most useful in a
                geometry definition file; however, it may be specified in
                the primary configuration file.

    Default:    0.05 times the span of the geometrical object.  The "span"
                is the greatest extent of the object in any one of its three
                or four dimensions.


        HYPERGEO    Version 1.1                             Page B-5



    Parameter:  DISPLAY_INFO

    Value:      One of:     OFF
                            ON

    Use:        Specifies whether or not the HYPERGEO information window
                should be displayed automatically at program start up and
                whenever a new geometry definition file is entered.

    Default:    ON


    Parameter:  DISPLAY_MENU
                DISPLAY_NAME
                DISPLAY_VECTOR

    Value:      One of:     OFF
                            ON

    Use:        Specifies whether a particular component of the HYPERGEO
                display will be drawn.

                DISPLAY_MENU controls the display of the menu area which
                occupies the right edge of the screen.  If display of the
                menu area is specified OFF, the entire screen is used for
                the primary geometry graphics (in this state, the mouse
                functions are not available).

                DISPLAY_NAME controls the display of the name of the
                geometry definition file; if ON, the name will be displayed
                at the upper, left corner of the screen.

                DISPLAY_VECTOR controls the display of the coordinate axis
                vectors.  The axis vectors are displayed as dotted lines in
                the primary graphics; each vector is labelled with the name
                of its axis, X, Y, Z, or W.

                Note: Regardless of the DISPLAY_VECTOR setting, the axis
                vectors are not displayed in hidden-line or solid display
                mode.

    Default:    DISPLAY_MENU            ON
                DISPLAY_NAME            ON
                DISPLAY_VECTOR          OFF


        HYPERGEO    Version 1.1                             Page B-6



    Parameter:  DISPLAY_MODE

    Value:      One of:     ANAGLYPH
                            HIDDEN_LINE
                            ORTHOGRAPHIC
                            PERSPECTIVE
                            SOLID

    Use:        Specifies the display mode used to draw the primary geometry
                graphics.

                Note: The HIDDEN_LINE and SOLID modes cannot be used with
                4-D to 3-D projections (only intersections).  If they are
                specified together with the PROJECTION parameter (see
                below), a warning message will be issued, and the program
                will revert to PERSPECTIVE mode.

    Default:    PERSPECTIVE


    Parameter:  EPSILON

    Value:      numeric value (floating-point)

    Use:        Specifies a small value to be used as an effective zero in
                all floating-point comparisons required by the program's
                calculations.

                This value is unitless; all calculations are performed
                internally on normalized coordinates.

    Default:    1.0E-8

                Note: The performance of some of the algorithms used in
                hidden-line and solid surface displays is very sensitive to
                this value.  The default was set after much experimentation
                and appears to provide reliable behavior almost all of the
                time.  The user fiddles with this value at his own risk.


        HYPERGEO    Version 1.1                             Page B-7



    Parameter:  EYE_DISTANCE

    Value:      numeric value (floating-point)

    Use:        Specifies the viewing distance, that is, the distance
                between the eye and the display screen.  This value is in
                inches.  It determines the degree of perceived perspective
                in the various display modes.  For greatest realism, it
                should be set to accurately reflect the true eye-to-screen
                distance for the user's normal viewing position.

                Decreasing the EYE_DISTANCE enhances the perspective effect,
                which can be useful.  However, the EYE_DISTANCE should never
                be reduced to the point that the viewpoint appears to enter
                the object; this cause the graphics to become distorted and
                meaningless.

    Default:    30.0


    Parameter:  EYE_SEPARATION

    Value:      numeric value (floating-point)

    Use:        Specifies the distance in inches between the viewer's eyes.
                This value is used in the perspective calculations required
                for the separate left- and right-eye images of the
                stereoscopic anaglyph display.

    Default:    2.5

                Note: This default value should be entirely acceptable for
                all users save those with close relatives residing in remote
                upper reaches of the Himalayas.


        HYPERGEO    Version 1.1                             Page B-8



    Parameter:  INTERSECTION

    Value:      None

    Use:        Causes HYPERGEO to use the intersection mode of 4-D to 3-D
                reduction (rather than projection mode).

                The INTERSECTION parameter is meaningless for
                three-dimensional geometries.

                INTERSECTION and PROJECTION (see below) are mutually
                exclusive alternatives; if both appear in configuration file
                data, the final one encountered will take precedence.

    Default:    By default HYPERGEO uses intersection mode for 4-D to 3-D
                reductions.


    Parameter:  ORIGIN

    Value:      Coordinate values as a sequence of three (for 3-D) or four
                (for 4-D) floating-point numbers.  Values should be entered
                without delimiting punctuation, using only spaces as
                separation.

    Use:        Permits translating the geometry along any (or all) of the
                coordinate axes.  The coordinate values specified define the
                new origin which is to be used by HYPERGEO as a centerpoint
                for all rotations.

                The ORIGIN parameter may only be used in configuration data
                appearing within a geometry definition file; it may not be
                used in the primary configuration file.  This is because the
                validity of the origin coordinates - in terms of their
                number and the units they are in - depends on the associated
                geometry definition.  Also, the translation is applied
                dynamically to the geometry point definitions as they are
                read in.

                The ORIGIN coordinate values are in the same units as the
                original vertex point coordinates in the geometry definition
                file.

    Default:    The origin is assumed to be (0,0,0) for 3-D and (0,0,0,0)
                for 4-D geometries.


        HYPERGEO    Version 1.1                             Page B-9



    Parameter:  PROJECTION

    Value:      None

    Use:        Causes HYPERGEO to use the projection mode of 4-D to 3-D
                reduction (rather than intersection mode).

                The PROJECTION parameter is meaningless for
                three-dimensional geometries.

                PROJECTION and INTERSECTION (see above) are mutually
                exclusive alternatives; if both appear in configuration file
                data, the final one encountered will take precedence.

    Default:    By default HYPERGEO uses intersection mode for 4-D to 3-D
                reductions.


    Parameter:  RATIO

    Value:      numeric value (floating-point)

    Use:        Specifies the scale ratio at which the display is drawn.
                The units of this value are: geometry units per screen inch.

                Because this parameter is in units that pertain to a
                particular geometry definition, it is most useful in a
                geometry definition file; however, it may be specified in
                the primary configuration file.

    Default:    Autoscale: the display is scaled to fit comfortably within
                the primary graphics area allowing ample room for rotations.


        HYPERGEO    Version 1.1                             Page B-10



    Parameter:  SCREEN_HEIGHT
                SCREEN_WIDTH

    Value:      numeric value (floating-point)

    Use:        These two parameters specify the vertical and horizontal
                size in inches of the actual area on the display screen in
                which graphics are drawn.  They can best be determined for a
                particular monitor by simply using a ruler to measure the
                height and width of the displayed area on the face of the
                screen.

                The precise accuracy of these parameters isn't critical, but
                they are useful to insure that the program's display scale
                is consistent between the horizontal and vertical
                directions (so that squares look square and not rectangular,
                for example).

    Default:    SCREEN_HEIGHT                   6.0
                SCREEN_WIDTH                    8.0

                Note: These default values are about right for the typical
                12-inch monitor.


        HYPERGEO    Version 1.1                             Page B-11



    Parameter:  SOLID_FILL

    Value:      One of:     COLOR
                            PATTERN

    Use:        Specifies whether to use color or pattern for surface
                rendering in solid display mode.

                Solid surface rendering that uses color displays the faces
                of the object with varying levels of color brightness to
                indicate how they are illuminated by a theoretical light
                source.  If pattern is used instead, the same effect is
                achieved by varying the density of fill (displayed with a
                single color) used to render the object's solid faces.

                If color is used, the COLOR_SOLID shades (see above) are
                used for VGA graphics, and a set of eight pre-programmed
                colors are used for EGA graphics.

                If pattern is used, it is displayed using the color
                specified for the primary geometry graphics.

                Note: This parameter is only meaningful on systems with
                color graphics capability (EGA or VGA); on monochrome
                systems, PATTERN is always used.

    Default:    COLOR (for EGA and VGA color systems)
                PATTERN (for monochrome systems)


    Parameter:  TURN

    Value:      numeric value (floating-point)

    Use:        Specifies the incremental step angle used by all interactive
                rotation commands.  This value is specified in degrees.

    Default:    5.0


        HYPERGEO    Version 1.1                             Page B-12



    Parameter:  VECTOR_LENGTH

    Value:      numeric value (floating-point)

    Use:        Specifies the length in screen inches of the unit axis
                vectors which may optionally be displayed as part of the
                primary geometry graphics (see DISPLAY_VECTOR above).  The
                VECTOR_LENGTH is the length an axis vector appears on the
                screen when it is situated parallel to the XY-plane.  As the
                object is rotated, the various axis vectors will appear to
                rotate with it and become visibly shorter as they assume
                different oblique orientations.

    Default:    1.0


    Parameter:  W_INTER

    Value:      One of:     CENTERED
                            numeric value (floating-point)

    Use:        Specifies the W-coordinate value at which a four-dimensional
                hyperobject will be "sliced" when using 4-D to 3-D
                intersection mode (that is, the value 'k' in the equation,
                w = k, of the intersecting hyperplane).  If a numeric value
                is specified, it should be in the same units as the geometry
                definition.

                When CENTERED is specified, the program automatically
                calculates the W value to use as the average of the minimum
                and maximum of all W coordinate values in the geometry
                definition.  This slices the hyperobject more or less down
                the middle.

                Because this parameter is in units that pertain to a
                particular geometry definition, it is most useful in a
                geometry definition file; however, it may be specified in
                the primary configuration file.

    Default:    CENTERED


        HYPERGEO    Version 1.1                             Page B-13



    The following is an example of a HYPERGEO configuration file:

        !   Example HYPERGEO Configuration File
        !

        PROJECTION          ! change from default Intersection

        Color_Background    Black
        Color_Vector        LightMagenta        ! very vivid

        COLOR_SOLID         63 53 43    ! flesh tone - produces darker
                                        ! shades that range through cocoa
                                        ! on down to dark reddish-brown

        display_vector      on
        vector_length       2.0         ! up from default 1 inch

        TURN                45          ! big angle for fast rotations

        DISPLAY_MODE        anaglyph    ! start up in 3-D graphics mode


        HYPERGEO    Version 1.1                             Page C-1




    Appendix C.    Command Line Arguments


    The HYPERGEO command line consists of the name of the program's
    executable file, HYPERGEO.EXE, followed optionally by additional
    parameters that are passed to the program.  This additional text is used
    to transmit control specifications to the HYPERGEO program. Parameters
    passed to a program in this manner are called command line arguments.

    Note: Normally the .EXE extension is not typed when the program is being
    executed since DOS knows it without being told.

    The HYPERGEO program accepts a number of command line arguments that can
    be used to control the behavior of the program.  Many of the command
    line arguments perform a function identical to parameters that can be
    specified in the HYPERGEO configuration file.  Whenever a command line
    argument and a configuration file parameter are in conflict, the command
    line argument takes precedence.  Furthermore, the command line
    specifications will continue to override configuration parameters that
    might be contained in any new geometry definition files that are read
    into the program interactively.

    Each command line argument consists of a single key letter that is
    entered on the HYPERGEO command line.  The key letters should be typed
    on the command line with a preceding hyphen, for example,

        HYPERGEO  -a

    Multiple arguments can either be strung together following a single
    hyphen or typed separately preceded by individual hyphens; thus,

        HYPERGEO  -aIv

    and

        HYPERGEO  -a  -I  -v

    are equivalent.

    Several arguments are flags indicating that a file name follows them
    immediately on the command line.  For these arguments, there must be no
    other argument letters following them before the file name.  There may,
    however, be one or more (or no) spaces between the flag letter and the
    file name.

    In all instances, the case (lower or upper) of the key letter is
    significant.  A number of arguments use the two cases of the same letter
    to mean opposite settings of a single parameter.  All arguments that use
    only one case of a letter require it to be lower-case; it will be
    rejected as an error if typed in upper-case.


        HYPERGEO    Version 1.1                             Page C-2


    (Confession: The standard notation of using a hyphen to prefix command
    line argument letters is styled after the Unix convention.  HYPERGEO
    follows this usage, and hyphens will always work.  However, to be
    candid, the arguments will also be handled correctly if preceded by a
    slash ('/', NOT a back-slash) or by nothing at all.  The only possible
    source of confusion is with file names appearing in the command line,
    but as long as all file names are always correctly and immediately
    preceded by their flag argument, the hyphen (or slash) prefix is
    superfluous and can be dispensed with.)

    Most arguments perform an independent function and can be used in any
    desired combination with other arguments.  However, five of the
    arguments serve to set the value of a single parameter, the 3-D graphic
    display mode, and they are mutually exclusive.  These arguments are -a,
    -h, -o, -p, and -s;  if more than one of them appears in a HYPERGEO
    command line, the one typed last takes effect.

    The following is a list of all command line arguments for the HYPERGEO
    program:

    Argument:   -a

    Use:        Selects the stereoscopic anaglyph mode of display.

                This argument is mutually exclusive with -h, -o, -p, and -s.

    Default:    Perspective display mode.


    Argument:   -b

    Use:        Instructs HYPERGEO to create all graphics in black and white
                even if the graphics display device supports color.  This
                can be useful when a monochrome monitor is connected to a
                color graphics controller card.

                With black and white graphics the stereoscopic anaglyph mode
                is unavailable, and solid surface fill always uses patterns.

    Default:    Color if graphics device supports it, else monochrome.


        HYPERGEO    Version 1.1                             Page C-3



    Argument:   -c

    Use:        Specifies the name of a HYPERGEO configuration file to use.
                The argument letter must be followed on the command line by
                the file name, with no other arguments intervening.  The
                file name can contain full DOS path information; if there is
                no path specified, it is assumed to exist in the current
                directory.  If the file name specification has no extension,
                it is assumed to have the extension .CFG.

                If this argument is used and the specified file cannot be
                opened, it is a fatal error and HYPERGEO terminates.  It is
                not an error, however, if the default configuration file
                cannot be opened.

    Default:    HYPERGEO.CFG in current directory


    Argument:   -f

    Use:        Specifies the name of a HYPERGEO geometry definition file to
                use.  The argument letter must be followed on the command
                line by the file name, with no other arguments intervening.
                The file name can contain full DOS path information; if
                there is no path specified, it is assumed to exist in the
                current directory.  If the file name specification has no
                extension, it is assumed to have the extension .GEO.

                If either the geometry file specified by this argument or
                the default geometry file cannot be opened, it is a fatal
                error and HYPERGEO terminates.

    Default:    HYPRCUBE.GEO in current directory

                This is the standard four-dimensional hypercube.


    Argument:   -g/-G

    Use:        Selects the method for reducing the 4-D geometry to a 3-D
                image for display.  The lower-case argument, -g, selects
                intersection mode, and the upper-case argument, -G, selects
                projection mode.

    Default:    Intersection mode


        HYPERGEO    Version 1.1                             Page C-4



    Argument:   -h

    Use:        Selects the hidden-line mode of 3-D display.

                This argument is mutually exclusive with -a, -o, -p, and -s.

                Also, hidden-line display mode cannot be used with 4-D
                projections.  If this argument is used together with the -G
                argument, a warning message is issued and the program starts
                up in perspective display mode.

    Default:    Perspective display mode.


    Argument:   -i/-I

    Use:        Specifies whether or not to display the special HYPERGEO
                information window automatically upon program start up and
                whenever a new geometry definition file is read.  The
                lower-case argument, -i, selects automatic display, and the
                upper-case argument, -I, deactivates it.

    Default:    The information window is automatically displayed.


    Argument:   -m/-M

    Use:        Specifies whether or not to display the menu area along the
                right edge of the screen.  The menu area lists a number of
                parameters that describe the current display and that can be
                modified interactively within the HYPERGEO program.  It also
                contains a mouse control pad area which allows the mouse to
                be used for controlling rotations of the geometry.  If the
                menu area is not displayed, the primary geometry graphics
                use the entire screen area; the mouse may not be used
                without the display of the menu area.

                The lower-case argument, -m, selects display of the menu
                area, while the upper-case argument, -M, causes the menu not
                to be displayed.

    Default:    The menu area is displayed.


        HYPERGEO    Version 1.1                             Page C-5



    Argument:   -n/-N

    Use:        Specifies whether or not to display the name of the geometry
                definition file in the upper, left corner of the graphics
                display.

                The lower-case argument, -n, selects display of the file
                name, while the upper-case argument, -N, causes the name not
                to be displayed.

    Default:    The geometry definition file name is displayed.


    Argument:   -o

    Use:        Selects the orthographic mode of 3-D display.

                This argument is mutually exclusive with -a, -h, -p, and -s.

    Default:    Perspective display mode is used.


    Argument:   -p

    Use:        Selects perspective display mode.

                This argument is mutually exclusive with -a, -h, -o, and -s.

    Default:    Perspective display mode.


    Argument:   -s

    Use:        Selects solid display mode.

                This argument is mutually exclusive with -a, -h, -o, and -p.

                Also, solid display mode cannot be used with 4-D
                projections.  If this argument is used together with the -G
                argument, a warning message is issued and the program starts
                up in perspective display mode.

    Default:    Perspective display mode.


        HYPERGEO    Version 1.1                             Page C-6



    Argument:   -v/-V

    Use:        Specifies whether or not to display the unit axis vectors
                along with the primary geometry graphics.

                The lower-case argument, -v, selects display of the axis
                vectors, while the upper-case argument, -V, causes the axis
                vectors not to be displayed.

                Note: Regardless of the setting of this option, the axis
                vectors are never displayed in hidden-line or solid display
                mode.

    Default:    The unit axis vectors are not displayed.


    Argument:   -?

    Use:        Prints to the screen a description of the appropriate syntax
                and usage of these command line arguments and terminates the
                HYPERGEO program.

    Default:    No syntax description is printed.



    The following is an example of a HYPERGEO command line:

        HYPERGEO  -f cube  -cc:\configs\special.dat  -Gav  -I

    This command invokes the following command line argument functions:

        *   Use the geometry definition file CUBE.GEO located in the current
            directory

        *   Use the configuration file SPECIAL.DAT located in the directory
            C:\CONFIGS\

        *   Use projection mode for reducing the 4-D geometry to a 3-D image

        *   Use stereoscopic anaglyph display mode for the 3-D image

        *   Display the unit axis vectors

        *   Deactivate the automatic display of the HYPERGEO information
            window


        HYPERGEO    Version 1.1                             Page D-1




    Appendix D.    Interactive Commands


    When the HYPERGEO program is executed, it accepts any command line
    arguments, reads in the appropriate configuration file (if any) and
    geometry definition file, and generates the initial display of the
    geometric object.  From that point on, the program remains responsive to
    a set of interactive commands which enable the user to manipulate the
    orientation of the object and to modify the various parameters that
    control the manner in which the display is generated.  Many of these
    interactive commands affect the same parameters that can be initially
    set through the configuration file and command line arguments.

    All interactive commands are invoked using either the keyboard or the
    mouse.  The mouse (if the system has one) provides an alternative means
    for invoking the commands for rotation of the object; if there is no
    mouse, keyboard commands can be used to perform the same functions.
    (Also, if the menu area is not displayed, the mouse cannot be used, and
    the keyboard equivalents must be similarly resorted to in its stead.)


    D.1  Keyboard Commands and the Modifier Keys

    The keyboard commands consist of either a single keystroke or a single
    keystroke accompanied by the simultaneous pressing of one of the
    standard keyboard modifier keys: Shift, Ctrl, or Alt.  These modifier
    keys are only used with some of the interactive commands, and in all
    cases they have a consistent meaning:

        Shift   - Used with keystrokes that invoke the various rotations; it
                  causes the direction of rotation to be reversed, that is,
                  to be equivalent to a negative turn angle.

                  Note: For all command keystrokes that are letters, if
                  the Shift key has no special meaning for that command, the
                  command may be entered equivalently in either lower or
                  upper case.  Only three letter commands (X, Y, and Z) have
                  special Shift-modified meanings.

        Ctrl    - Used with keystrokes for the rotation commands plus the
                  command for translating the intersecting hyperplane along
                  the W-axis in hyperspace; causes the command action to be
                  repeated continuously as fast as the computer's processing
                  speed will allow.  This "command automation" is stopped by
                  pressing any key.

                  In the case of the hyperplane translation command, the
                  automated motion is automatically reversed in direction
                  each time the intersecting hyperplane reaches an extreme


        HYPERGEO    Version 1.1                             Page D-2


                  of the hyperobject.  This causes the hyperplane to move
                  continuously back and forth through the entire body of the
                  hyperobject.

        Alt     - Used with keystrokes for the rotation commands; has the
                  same "command automation" effect as the Ctrl modifier
                  except the rotation is in the reverse direction.  The
                  continuous, automated rotation is stopped by pressing any
                  key.


    D.2  Mouse Selected Interactive Commands

    When the mouse is used to invoke a rotation command, no special
    modifiers are necessary since the action of the mouse buttons provides
    the equivalent control.  A rotation command can be invoked with the
    mouse (whenever the HYPERGEO menu area is displayed) by moving the mouse
    so the cursor is over the selection box labeled with the coordinate
    letters indicating the desired axis or plane of rotation; this will
    highlight the selection box.  To begin the rotation, press the left or
    right mouse button; the right mouse button causes a rotation in the
    positive direction, and the left mouse button rotates the object in the
    negative direction.  The rotation will continue as long as the mouse
    button remains pressed.  A single incremental rotation can be achieved
    by quickly clicking and releasing the mouse button.

    There are nine rotation command selection boxes in the HYPERGEO menu
    area, divided into two sections: 3-D axial rotations, and 4-D planar
    rotations.  All nine are available for 4-D geometries, but only the
    three 3-D rotations can be selected when the input geometry is
    three-dimensional.  The 4-D rotation selection boxes are labeled with
    the letters of the two coordinates specifying the plane of the rotation;
    thus, selecting the box labeled "XY" initiates a rotation in hyperspace
    of the XY-plane.  The 3-D rotation selection boxes are labeled with the
    single letter of the axis around which the rotation occurs: X, Y, or Z.

    If your system has no mouse or if the menu area isn't displayed (or if
    you simply prefer to use the keyboard), there is an equivalent keyboard
    command for each of the nine mouse rotations:

            Keystroke       Mouse Selection Box Label
            ---------       -------------------------
                X                      X
                Y                      Y
                Z                      Z
               F1                     YZ
               F2                     XW
               F3                     XZ
               F4                     YW
               F5                     XY
               F6                     ZW


        HYPERGEO    Version 1.1                             Page D-3


    The keystrokes F1, F3, and F5 correspond to the YZ, XZ, and XY 4-D
    planar rotations.  Note that these are the three rotations that occupy
    the top row of the six 4-D rotation selection boxes and that they are
    the 4-D rotations that don't involve the W coordinate.  These three
    rotations produce the same visual effect in the geometry display as the
    three 3-D rotations whose selection boxes are directly above them in the
    menu area: X, Y, and Z.

    The keystrokes F2, F4, and F6 correspond to the three 4-D rotations that
    do involve the W coordinate: the XW, YW, and ZW planar rotations.  These
    are the 4-D rotations that occupy the bottom row of selection boxes
    within the menu area.  They do not correspond in effect to any of the
    3-D rotations.


    D.3  Interactive Parameter Specification

    Some of the interactive commands prompt for the input of a value to be
    assigned to a program parameter.  This value is generally either a
    numeric quantity, a file name, or a particular key word.  The input is
    requested from the user via a prompting window which pops up in the
    primary graphics area when the command is executed.  The prompting
    window contains an entry line in which the user's typed response
    is echoed.  After typing in the appropriate value, the user should
    indicate that the entry is complete and correct by pressing <Enter>;
    this will cause the command to accept the typed in value for the given
    parameter.  A prompting window may be cancelled at any time by pressing
    <Esc>; this aborts the command that caused the prompting window to
    appear.

    Prompts that accept character input - either key words or file names -
    are case insensitive; upper or lower case may be used as desired.
    Numeric values may be either integers or floating-point numbers
    depending upon the context of the command.  If a floating-point number
    is appropriate, it may be entered with an optional decimal point and
    fractional digits and may include an optional decimal exponent in
    scientific notation.

    If the user's entry in a prompting window is invalid, an error message
    window will appear.  Like all HYPERGEO message windows, it can be
    dismissed by pressing any key.  All errors resulting from incorrect
    input to interactive parameter prompts are non-fatal; the program will
    continue running, ready for the next interactive command, once the
    message window is dismissed.


        HYPERGEO    Version 1.1                             Page D-4



    D.4  HYPERGEO Interactive Keyboard Commands

    The following is a list of all HYPERGEO interactive commands that can be
    entered from the keyboard while the program is displaying an active
    geometry:


    Keystroke:  ?

    Use:        Displays the on-line help screen.  This screen contains a
                brief description of the interactive commands available
                within the HYPERGEO program (that is, the contents of this
                Appendix).

                The on-line help screen can be dismissed by pressing any
                key.  This returns the program to the graphics state it was
                in prior to the '?' command.


    Keystroke:  A

    Use:        Selects the stereoscopic anaglyph mode of display.

                If the program is operating with monochrome graphics, the
                stereoscopic anaglyph display cannot be used; a message
                window will appear indicating this, and the current display
                mode will not be changed.


        HYPERGEO    Version 1.1                             Page D-5



    Keystroke:  C

    Use:        Accepts a new color specification for drawing the primary
                graphics.

                In most cases, this will be the single EGA color used to
                draw the edge outlines of the geometrical image being
                displayed.  (The primary graphics color is also used for the
                transient data value fields within the HYPERGEO menu area.)
                The command accepts a value which is the name of one of the
                sixteen standard EGA colors:

                    BLACK
                    BLUE
                    GREEN
                    CYAN
                    RED
                    MAGENTA
                    BROWN
                    LIGHTGRAY
                    DARKGRAY
                    LIGHTBLUE
                    LIGHTGREEN
                    LIGHTCYAN
                    LIGHTRED
                    LIGHTMAGENTA
                    YELLOW
                    WHITE

                However, if solid display mode is currently active and the
                system has VGA color capability, the command will accept a
                new specification for the eight graduated shades of color to
                be used for solid surface rendering.  The value to be
                entered is a red/green/blue number triple in the form of
                three integers in the range 0 to 63.  This defines the
                lightest of the eight shades; the program will calculate
                seven additional, darker shades from this to create the full
                range required for realistic surface rendering.


    Keystroke:  D

    Use:        Accepts a new delta value that will be used to increment (or
                decrement) the W-value of the intersecting hyperplane.  This
                delta value is used by the interactive command which
                performs a translation of the intersecting hyperplane on
                geometries being displayed via the 4-D intersection mode
                (see Page Up/Page Down below).

                The value entered is a positive floating-point number.  It
                is in the units of the currently active geometry.


        HYPERGEO    Version 1.1                             Page D-6



    Keystroke:  E

    Use:        Accepts a new value for the eye-to-screen distance.  This
                parameter is used in the generation of all 3-D to 2-D screen
                projections.  Decreasing the eye-to-screen distance enhances
                the amount of apparent perspective in the display.  For
                greatest realism, the eye-to-screen distance should be set
                to correspond to the true distance between the viewer's eyes
                and the screen.

                It is possible to set the eye-to-screen distance to any
                arbitrarily small positive value.  However, it should never
                be made so small that the viewpoint appears to enter the
                object being displayed.  This will result in distorted and
                meaningless graphics.

                The value entered is a positive floating-point number.  It
                is in inches.


    Keystroke:  F

    Use:        Accepts the name of a new geometry definition file.  The
                name may include a full DOS path specification.  If no path
                is specified, the file is assumed to be in the current
                directory.  If no extension is specified, the extension .GEO
                is assumed.


    Keystroke:  G

    Use:        Switches between the two modes of 4-D to 3-D geometry
                reduction: intersection and projection.

                Because hidden-line and solid displays cannot be used with
                4-D to 3-D projections, if either of those two modes is in
                effect when the program switches from 4-D intersection to
                4-D projection, perspective display mode is automatically
                selected.

                The G command is not applicable to three-dimensional
                geometric objects.


    Keystroke:  H

    Use:        Selects hidden-line display mode.

                If the current geometrical object is four-dimensional and
                the 4-D to 3-D reduction mode is projection, the hidden-line
                display cannot be used.


        HYPERGEO    Version 1.1                             Page D-7



    Keystroke:  I

    Use:        Displays the special HYPERGEO Information Window.  This
                window contains a summary of the current object's topology,
                plus a number of related program parameters.  For the
                topology, the window shows:

                    *   number of dimensions
                    *   number of vertex points
                    *   number of edges
                    *   number of faces
                    *   number of hyperfaces (4-D geometries only)
                    *   span (geometry units)
                    *   radius (geometry units)

                The "span" of an object is calculated by finding the minimum
                and maximum extents of the object in each of its three or
                four dimensions; the span is the largest difference between
                the minimum and maximum extents in a single dimension.  The
                "radius" of an object is the greatest distance in
                (hyper)space of any vertex point from the origin.

                Additional parameters shown are:

                    *   screen height and width (inches)
                    *   horizontal and vertical pixels per inch
                    *   free memory (bytes)

                The HYPERGEO Information Window is an ordinary message
                window; it can be dismissed by striking any key.


    Keystroke:  M

    Use:        Toggles on and off the display of the HYPERGEO menu area.

                The menu area must be displayed to enable the use of the
                mouse for command selection.


    Keystroke:  N

    Use:        Toggles on and off the display of the name of the current
                geometry definition file.

                From time to time, interactive manipulations of the current
                object may cause the graphics of the displayed geometry to
                pass over the text of the geometry definition file name and
                partially or wholly erase it.  The display of the file name
                can be restored by toggling it off and then on again using
                two 'N' commands in immediate succession.


        HYPERGEO    Version 1.1                             Page D-8



    Keystroke:  O

    Use:        Selects orthographic display mode.


    Keystroke:  P

    Use:        Selects perspective display mode.


    Keystroke:  Q

    Use:        Quits the HYPERGEO program and returns to DOS.


    Keystroke:  R

    Use:        Accepts a new value for the current display scale ratio.  A
                smaller scale ratio increases the size of the image on the
                screen, while a larger scale ratio decreases it.

                The value entered is a positive floating-point number.  Its
                units are geometry units per screen inch.


    Keystroke:  S

    Use:        Selects solid display mode.

                If the current geometrical object is four-dimensional and
                the 4-D to 3-D reduction mode is projection, the solid
                display mode cannot be used.


    Keystroke:  T

    Use:        Accepts a new value for the turn angle.  The turn angle is
                the basic incremental angle used by each of the rotation
                commands.  A large turn angle is useful for rapid
                manipulation of the geometrical object under study.  Smaller
                angles can be used for fine positioning.

                The value entered is a positive floating-point number.  Its
                units are degrees.


        HYPERGEO    Version 1.1                             Page D-9



    Keystroke:  V

    Use:        Toggles on and off the display of the unit axis vectors.

                Regardless of the setting of this toggle, the unit axis
                vectors are not displayed with hidden-line or solid display
                mode.


    Keystroke:  W

    Use:        Accepts a new value for the W-coordinate of the intersecting
                hyperplane used for 4-D to 3-D intersections.  Repositioning
                the intersecting hyperplane at various W-coordinate values
                provides a means for examining the hypergeometrical
                structure of the current object.

                It is possible (indeed, easy) to assign a value to this
                parameter that causes the intersecting hyperplane to lie
                entirely outside the extents of the current object in
                hyperspace.  This results in an empty display.  The program
                has no feature for "finding" the object in hyperspace in
                such cases, so it behooves the user to remember the extents
                of the object.  (This is no problem if the original geometry
                contains the origin point; then, setting the hyperplane's
                W-coordinate to zero will restore the display.)

                The value entered is a floating-point number.  It is in
                geometry units.

                Note: This command has no apparent effect if the program is
                currently using the projection method of 4-D to 3-D image
                reduction.  However, the next time intersection mode is
                entered, the modified W coordinate value of the intersecting
                hyperplane will be used.


        HYPERGEO    Version 1.1                             Page D-10



    Keystroke:  X
                Y
                Z

    Use:        Executes a three-dimensional rotation of the current 3-D
                image about the particular axis: X, Y, or Z.  The size of
                the rotation is determined by the current turn angle.

                When the active geometry is four-dimensional, all 3-D
                rotations are transient and operate only on the immediate
                3-D image which has been derived from the hyperobject's
                geometry via a 4-D to 3-D reduction using either projection
                or intersection.  In this case, the program does not update
                the display of the unit axis vectors or the values shown in
                the menu's axis angle data fields during 3-D rotations.

                For three-dimensional geometries, however, 3-D rotations are
                permanent; the internal database is modified accordingly,
                and the unit axis vectors and the displayed values for the
                axis angles are updated with each 3-D rotation.

                By themselves, the X, Y, and Z keys produce a single
                incremental rotation in the positive direction.  This effect
                can be altered by combining the keystroke with the
                simultaneous pressing of one of the standard keyboard
                modifier keys (for example, Shift-X is entered by pressing
                and holding down the Shift key, then pressing X):

                    Shift - Reverses the direction of the rotation

                    Ctrl  - Automates continuous rotations in the positive
                            direction.  The rotations are performed as fast
                            as the system's processing speed will allow.  To
                            stop the animation, press any key.

                    Alt   - Automates continuous rotations in the negative
                            direction.  The rotations are performed as fast
                            as the system's processing speed will allow.  To
                            stop the animation, press any key.


        HYPERGEO    Version 1.1                             Page D-11



    Keystroke:  F1
                F2
                F3
                F4
                F5
                F6

    Use:        Executes a four-dimensional rotation of the current
                hyperobject's geometry in hyperspace.  The six function keys
                F1 through F6 correspond to the six possible coordinate
                planes in which 4-D rotation may occur:

                    F1  -   YZ plane
                    F2  -   XW plane
                    F3  -   XZ plane
                    F4  -   YW plane
                    F5  -   XY plane
                    F6  -   ZW plane

                The size of the rotation is determined by the curent turn
                angle.

                After each 4-D rotation, the hyperobject's geometry is
                reduced to a 3-D image using the currently active method
                (intersection or projection), and the 3-D image is converted
                into a 2-D display on the computer's screen using the
                currently active display mode (orthographic, perspective,
                stereoscopic anaglyph, hidden-line, or solid).

                By themselves, the F1 through F6 keys produce a single
                incremental rotation in the positive direction.  This effect
                can be altered by combining the keystroke with the
                simultaneous pressing of one of the standard keyboard
                modifier keys (for example, Shift-F1 is entered by pressing
                and holding down the Shift key, then pressing F1):

                    Shift - Reverses the direction of the rotation

                    Ctrl  - Automates continuous rotations in the positive
                            direction.  The rotations are performed as fast
                            as the system's processing speed will allow.  To
                            stop the animation, press any key.

                    Alt   - Automates continuous rotations in the negative
                            direction.  The rotations are performed as fast
                            as the system's processing speed will allow.  To
                            stop the animation, press any key.


        HYPERGEO    Version 1.1                             Page D-12



    Keystroke:  3
                4

    Use:        Performs continuous automated random rotations.  The
                keystroke '3' continuously executes the three 3-D rotations
                in a randomly determined order, and '4' randomly executes
                the six 4-D rotations.  All rotations are in a positive
                angular direction with a size determined by the current turn
                angle.  To stop the automation, press any key.


    Keystroke:  Home

    Use:        Restores the orientation of the current geometry as it
                existed when the geometry definition file was first read in.
                All axis angles are reset to zero degrees, and (for 4-D
                geometries) the W coordinate of the intersecting hyperplane
                is reset to its original value.


    Keystroke:  End

    Use:        Restores the current geometry to the orientation it was in
                prior to the most recent Home command (see above).
                Parameters reset include the axis angles and (for 4-D
                geometries) the W coordinate value of the intersecting
                hyperplane.

                If no Home command has yet been performed on the current
                geometry, the End command has no effect.


    Keystroke:  +
                -

    Use:        Performs a "zoom" operation on the visible image.  The '+'
                command enlarges the graphical display, and the '-' command
                reduces it.

                These commands operate by changing the display scale ratio.
                The '+' command reduces the scale ratio by 10% of its
                current value, and the '-' command increases it by 10%.


        HYPERGEO    Version 1.1                             Page D-13



    Keystroke:  Insert
                Delete

    Use:        Alters the degree of perceived 3-D perspective for the
                currently displayed image.  These commands operate by
                modifying the eye-to-screen distance.  The Insert command
                increases apparent perspective; it moves the viewpoint in
                towards the object by reducing the eye-to-screen distance by
                10% of its current value.  The Delete command decreases the
                perspective effect; it moves the viewpoint away from the
                object by increasing the eye-to-screen distance by 10%.

                The eye-to-screen distance should not be reduced to the
                point that the viewpoint enters the 3-D geometry of the
                displayed image.  This will result in distorted and
                meaningless graphics.


    Keystroke:  Page Up
                Page Down

    Use:        Performs a translation of the intersecting hyperplane.  The
                hyperplane is moved along the W-axis in hyperspace.  The
                size of the translation is determined by the value of the
                DELTA_W parameter.  The Page Up command increments the W
                coordinate of the intersecting hyperplane by this delta
                amount, and the Page Down command decrements it.

                Note: This command has no apparent effect if the program is
                currently using the projection method of 4-D to 3-D image
                reduction.  However, the next time intersection mode is
                entered, the modified W coordinate value of the intersecting
                hyperplane will be used.

                By themselves, the Page Up and Page Down commands perform a
                single incremental translation of the intersecting
                hyperplane.  When combined with the simultaneous pressing of
                the Ctrl key, these keystrokes initiate the continuously
                automated motion of the hyperplane through the body of the
                current four-dimensional geometry.  (For example, to enter
                Ctrl-Page Up, first press and hold down the Ctrl key, then
                press Page Up.)  In automated mode, the direction of the
                hyperplane's motion is reversed each time it reaches an
                extreme edge of the hyperobject, that is, the point at which
                it ceases to intersect the hyperobject in hyperspace.  Thus,
                the hyperplane, continues to move back and forth through the
                entire body of the hyperobject.  The automated motion is
                stopped by pressing any key.


        HYPERGEO    Version 1.1                             Page D-14



    Keystroke:  Arrow Keys: Left, Right, Up, Down

    Use:        These keys move the position of the current graphical
                display on the screen.

                When a new geometry is read in, HYPERGEO automatically
                centers the display within the primary graphics area.  The
                arrow keys may be used to reposition it as required by
                subsequent rotations or other interactive manipulations.


        HYPERGEO    Version 1.1                             Page E-1




    Appendix E.    Running HYPERGEO under Microsoft Windows


    The program file supplied with HYPERGEO (HYPERGEO.EXE) is a standard DOS
    executable; it is not a special Windows program.  However, it can be run
    under Windows using that system's ability to execute non-Windows DOS
    programs.  Two files are supplied with HYPERGEO to assist in adding the
    program to the Windows Program Manager: an icon file (HYPERGEO.ICO), and
    a Program Information File (HYPERGEO.PIF).

    The procedure to follow to add a new program to a particular program
    group within the Program Manager is described in the Windows User's
    Guide in the section "Adding Program Items and Documents to a Group" in
    the Program Manager chapter (page 88 for Windows 3.0).  Within the
    Program Item Properties dialog box that appears, the name of the
    HYPERGEO PIF file (including its DOS path) should be entered in the
    Command Line text box.  You should also supply the name of the icon file
    by choosing the "Change Icon..." option from the Program Item Properties
    dialog box.  Enter the icon file name (HYPERGEO.ICO) along with its full
    DOS path in the File Name text box within the Select Icon dialog box.

    One final step may be necessary to enable the running of HYPERGEO in
    Windows.  The supplied PIF file has an entry which specifies the
    program's start-up directory.  This initially contains the path
    specification C:\HYPERGEO.  If you have installed HYPERGEO in a
    different directory, you will have to change this entry in the PIF to
    reflect the actual directory in which the program executable
    HYPERGEO.EXE exists.  To modify a PIF you use the Windows PIF Editor.
    This is described fully in the Section "Working with PIFs and the PIF
    Editor" (page 444 for Windows 3.0) and subsequent sections in the
    chapter "More About Applications" in the Windows User's Guide.

    The PIF Editor also lets you specify any optional command line
    parameters you wish to pass to HYPERGEO each time it is executed.

    The PIF file as supplied will cause HYPERGEO to run as a full-screen DOS
    program.


        HYPERGEO    Version 1.1                             Page F-1




    Appendix F.    Registration Form


                            Stuart Esten
                            33 Banks Ave.
                            Lexington MA  02173
                            USA


                HYPERGEO Customer Registration Form


    Name and Address:


        ____________________________________________________________


        ____________________________________________________________


        ____________________________________________________________


        ____________________________________________________________


    Diskette size:      5-1/4" ( )          3-1/2" ( )



    HYPERGEO -  includes latest version of program
                and one pair of plastic 3-D glasses:

                       _____ copies at $20 each     =   _____________


    Additional 3-D glasses: ______ pairs at $2 each =   _____________


    Massachusetts residents add 5% state sales tax  =   _____________


    Shipment outside of U.S. and Canada add $5      =   _____________



                                        Total       =   _____________



    For domestic orders payment should be via check payable to Stuart Esten.
    Foreign orders should use an International Money Order denominated in
    dollars.  For either domestic or foreign orders cash is acceptable.


    (Please use the back of this form for any comments, suggestions, or
    criticisms you have concerning the HYPERGEO program.)