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1992-11-13
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This geomview external module lets you fly through the tesselation of
hyperbolic space by a right-angled regular dodecahedron which appeared
in the mathematical animation "Not Knot" produced by the Geometry
Center. You can either pick a pre-computed flight path or fly around
interactively.
All 30 edges of the dodecahedron are white except the three pairs of
edges colored green, blue and red corresponding to the three loops of
the Borromean rings. Every face of the dodecahedron has exactly one
non-white edge, so we can color the face by this color.
All flight paths begin and end at the center of a green face. There
are three other green faces: one adjacent to this one, at right
angles along the green beam; and a pair which border the other green
beam, on the other side of the dodecahedron.
The light blue "Direct" path is the simplest to understand:
we go straight through to the green face directly opposite from the
original face.
The yellow "Quarter Turn" path, which goes to the adjacent green face,
simply circles around the green axis which the two faces share.
The "Full Loop" path is also yellow: it repeats this quarter turn four
times so that we start and finish in the same place. The three other
paths just jump back to the starting place when they reach the end.
The magenta "Equidistant" path, which goes to the other green face
which doesn't border the original face, is the most interesting. It
follows a so-called equidistant curve: in this case, one that is
equidistant to the red axis that connects the two green faces in
question. This curve is like a parallel line in Euclidean space: it
stays a constant distant from the red axis, but it's not a geodesic in
hyperbolic space.
In the small 3D diagram window, you can use the left mouse button to
spin around a dodecahedron with colored coded flight paths as
mentioned above. It's easier to see what's going on in the Euclidean
diagram, while the hyperbolic version is more similar to what you see
in the flythrough.
You can either choose one of four flight paths through the tesselation
or stop the automatic flight by hitting the "Stop" button and fly
around yourself. For interactive flight, hit the "Cam Fly" button on
the geomview Tools panel: then dragging the mouse with the middle
button down moves you forwards or backwards, and dragging with the
left button down is like turning your head. When you hit "Go", the
automatic flight will continue.
You can choose one of four tesselation levels: level 0 is a single
dodecahedron, level 1 adds a layer of 12 dodecahedra (one for each
face of the original dodecahedron), level 2 tesselates two layers
deep, and level 3 has three layers. The more layers you have the
slower the update rate: level 3 is glacially slow, but each frame
looks pretty impressive. You can change the size of the dodecahedra
with the "Scale Dodecahedra" slider: at 1.0 they fit together exactly.
The "Steps" buttons control the smoothness of the flight path: you can
set the number of steps to 10 (jerky but fast), 20, 40, or 80 (smooth
but slow).
Authors:
Charlie Gunn (geometry and flight paths) gunn@geom.umn.edu
Tamara Munzner (interactive interface) munzner@geom.umn.edu
Stuart Levy (3D diagram) levy@geom.umn.edu
Copyright (c) 1992
The Geometry Center
1300 South Second Street
Minneapolis, MN 55454
email: software@geom.umn.edu
Available free via anonymous ftp from geom.umn.edu
You can redistribute and/or modify this program according to the terms
of the the GNU Emacs Public License.