The Unsorted BBS Collection/thegreatunsorted.tar/thegreatunsorted/texts/term_papers/kep.esy

Text File | 1993-01-28 | 12KB | 244 lines

JOHANNES KEPPLER

Johannes Kepler (1571-1630) was born in Weil, a city near Stuttgart
in Southern Germany. He was educated at the University of Tubingen
with one of the centres of Protestant theology, receiving the
degree master of arts in 1591. He was fortunate enough to study
with Michael Mastlin, the professor of mathematics, who taught him
privately the work of Copernicus, while hardly daring to recognize
it openly in his professorial lectures.  In 1593, at the age of 22,
Kepler prepared a discussion enthusiastically supporting the
Copernican theory of a sun-centred universe; but he never had the
opportunity to present it because the professor in charge of those
academic activities, was so opposed to Copernicanism, that he
refused it to be heard.  Kepler abandoned his intentions of
entering the Lutheran ministry because his views were argumentative
with beliefs that then conquered, and instead accepted position of
"provincial mathematician" in the protestant seminary at Graz, the
capital of the Austrian province of Styria.
Kepler viewed the universe as being governed by geometric
correlations that harmonize to the inscribed and circumscribed
circles of the five regular polygons, or the five platonic solids.
These solids were the cube, tetrahedron, dodecahedron, icosahedron,
and the octahedron. Each circle represented the orbit of one of the
six planets, while the shapes determined the size of the circle.
Danish astronomer Tycho Brahe, the mathematician at the court of
Emperor Rudolph II at Prague, was so impressed with Kepler's work
that in 1600 he invited Kepler to come to Prague as his assistant.
 Brahe died shortly after, and Kepler inherited both his master's
position and his vast and very accurate assembly of astronomical
data on the motion of the planets.
The riddle of the orbit of Mars captivated Keplers attention for
eight years.  Tycho's very accurate measures of the position of
Mars relative to the sun permitted Kepler to test various educated
guesses, but abandon them aside when they proved contradictory with
the movement observed.  It was only after many failures to fit the
findings to a circular orbit that he began to suspect that it must
be some other closed path.
Kepler believed that this path was of an elliptical shape.  Years
of hard work and disappointment finally forced him to the decision
that only an elliptical orbit, with the sun occupying one of the
two foci, satisfied Tycho's data.  This information was presumably
true for all other planets, since the harmony of nature demanded
that all "have similar habits."  This was Keplers celebrated first
law.  Another conclusion he obtained from the astronomical data was
that the speed with which a planet traversed its elliptical orbit
varied in a regular pattern, accelerating with approach to the sun,
and decelerating with departure from the sun.  From this he was
lead to another foundation in "celestial mechanics", Kepler's
second law: The line drawn from the sun to a planet sweeps over
equal areas in equal times.
After ten years of further effort Kepler arrived at a relation, his
third and final prominent law of planetary motion, connecting the
times of revolution of any two planets with their respective
distances from the sun.
Johannes Kepler was concerned to answer two distinct questions:
-Why are there 6 planets? (in Copernicus' theory, there were six
planets, from Mercury to Saturn.)
-Why are there five gaps between them, with the sizes, in which
they are?
 In 1621, a second edition of Kepler's work entitled, "Mysterium
Cosmographicum" was printed.  This explained the gaps between
neighbouring planetary orbits by relations between the
circumspheres and inspheres of the five platonic solids, which
include, as mentioned before, the cube, tetrahedron, octehedron,
icosahedron, and the dodecahedron.
The basis of his theory derived from book XIII of Euclids Elements,
which states that the faces of a regular polyhedron are all regular
polygons of the same shape and they meet in the same way at every
vertex of the solid.  Therefore, the vertices must lie on a sphere,
and all the centres of the faces must lie on another sphere that
will touch the faces of these points.
The two spheres will have the same centre.  The outersphere is the
circumsphere and the inner is the insphere and they are similar to
the circumcircle and incircle of a regular polygon, except that one
set is three dimensional and the other is two dimensional.  The
ratios of these spheres can be easily calculated.  There are only
three ratios for the five solids since two are the same.
The ratios of circumsphere to insphere are:
Tetrahedron ----> 3
Cube + Octehedron---->1.7321
Dodecahedron + Icosahedron---->1.2584
Kepler presented his theory in the following way:
If a cube is inscribed in the orbit of Saturn, then its insphere
will be the orbit of Jupiter; and if a tetrahedron is inscribed in
Jupiter's orbit, its insphere is the orbit of Mars; if a
dodecahedron is inscribed in the orbit of Mars, its insphere will
be the orbit of Earth; if an icosahedron is inscribed in the earths
orbit, its insphere will be the orbit of venus; and finally when an
octahedron is inscribed in Venus' orbit, its insphere will be the
orbit of Mercury.
This explains the sizes of the orbits, and the number of orbits.
Since there are five basic platonic solids, six orbits can be
assembled.  Astonishingly, Kepler's calculated ratios for platonic
solids and observational ratios for planetary orbits are very
similar.

Johannes Kepler  1571-1630

Andrew Likakis

Dave Tjandra

















JOHANNES KEPPLER

Johannes Kepler (1571-1630) was born in Weil, a city near
Stuttgart
in Southern Germany. He was taught at the University of Tubingen
with one of the centres of Protestant theology, getting the
degree master of arts in 1591. He was lucky enough to study with
Michael Mastlin, the professor of mathematics, who taught him
privately the work of Copernicus, while hardly daring to admit it
openly in his professorial lectures.  In 1593, at the age of 22,
Kepler got ready a discussion enthusiastically supporting the
Copernican theory of a sun-centred universe; but he never had the
opportunity to present it because the professor in charge of
those academic activities, was so much against Copernicanism,
that he would not listen to it.  Kepler abandoned his intentions
of entering the Lutheran ministry because his ideas were
argumentative with what people believed then, and instead
accepted position of "provincial mathematician" in the protestant
seminary at Graz, the capital of the Austrian province of Styria.

"Kepler viewed the universe as being governed by geometric
correlations that harmonize to the inscribed and circumscribed
circles of the five regular polygons, or the five platonic
solids.  These solids were the cube, tetrahedron, dodecahedron,
icosahedron, and the octahedron. Each circle represented the
orbit of one of the six planets, while the shapes determined the
size of the circle."   Danish astronomer Tycho Brahe, the
mathematician at the court of Emperor Rudolph II at Prague, was
so amazed with Kepler's work that in 1600 he invited Kepler to
come to Prague as his assistant.   Brahe died shortly after, and
Kepler inherited both his master's position and his large and
very accurate collection of astronomical data on the motion of
the planets.
The "riddle" of the orbit of Mars dazzled Keplers attention for
eight years.  Tycho's very accurate measures of the position of
Mars relative to the sun permitted Kepler to test a lot of ideas,
but abandon them when they proved opposite with the movement
observed.  It was only after many failures to fit the findings to
a circular orbit that he began to wonder if there was some other
closed path.
Kepler believed that this path was an elliptical shape.  Years of
hard work and disappointment finally forced him to the decision
that only an elliptical orbit, with the sun occupying one of the
two foci, satisfied Tycho's data.  This information was probably
true for all other planets, since the harmony of nature demanded
that all "have similar habits."  This was Keplers celebrated
first law.  Another conclusion he obtained from the astronomical
data was that the speed with which a planet covered its
elliptical orbit varied in a regular pattern, accelerating with
approach to the sun, and decelerating with departure from the
sun.  From this he was lead to another foundation in "celestial
mechanics", Kepler's second law: The line drawn from the sun to
a planet sweeps over equal areas in equal times.
After ten years of further effort Kepler arrived at a relation,
his third and final prominent law of planetary motion, connecting
the times of revolution of any two planets with their respective
distances from the sun.
Johannes Kepler was concerned to answer two distinct questions:
-Why are there 6 planets? (in Copernicus' theory, there were six
planets, from Mercury to Saturn.)
-Why are there five gaps between them, with the sizes, in which
they are?
 In 1621, a second edition of Kepler's work entitled, "Mysterium
Cosmographicum" was printed.  This explained the gaps between
nearby planetary orbits by relations between the circumspheres
and inspheres of the five platonic solids, which include, as
mentioned before, the cube, tetrahedron, octehedron, icosahedron,
and the dodecahedron.
"The basis of his theory derived from book XIII of Euclids
Elements, which states that the faces of a regular polyhedron are
all regular polygons of the same shape and they meet in the same
way at every vertex of the solid.  Therefore, the vertices must
lie on a sphere, and all the centres of the faces must lie on
another sphere that will touch the faces of these points.
The two spheres will have the same centre.  The outersphere is
the circumsphere and the inner is the insphere and they are
similar to the circumcircle and incircle of a regular polygon,
except that one set is three dimensional and the other is two
dimensional.  The ratios of these spheres can be easily
calculated."  There are only three ratios for the five solids
since two are the same.
The ratios of circumsphere to insphere are:
Tetrahedron ----> 3
Cube + Octehedron---->1.7321
Dodecahedron + Icosahedron---->1.2584
Kepler presented his theory in the following way:
"If a cube is inscribed in the orbit of Saturn, then its insphere
will be the orbit of Jupiter; and if a tetrahedron is inscribed
in Jupiter's orbit, its insphere is the orbit of Mars; if a
dodecahedron is inscribed in the orbit of Mars, its insphere will
be the orbit of Earth; if an icosahedron is inscribed in the
earths orbit, its insphere will be the orbit of venus; and
finally when an octahedron is inscribed in Venus' orbit, its
insphere will be the orbit of Mercury."
This explains the sizes of the orbits, and the number of orbits.
Since there are five basic platonic solids, six orbits can be
found.  Astonishingly, Kepler's calculated ratios for platonic
solids and observational ratios for planetary orbits are very
similar.
Understanding Johannes Keppler's work was very important back in
his time since his theories and data were like a stepping stone
for mathematicians like Copernicus.

Johannes Kepler  1571-1630
Andrew Likakis
Dave Tjandra