kaleido/README Version 3.10 (93/11/22)
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Uniform Polyhedra - Computation and 3D Display
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Uniform polyhedra, whose faces are regular and vertices equivalent, have
been studied since antiquity. Best known are the 5 Platonic solids and the 13
Archimedean solids. We then have 2 infinite families of uniform prisms and anti
prisms. Allowing for star faces and vertices, we have the 4 Kepler-Poinsot
regular star polyhedra, and a row of 53 nonconvex uniform polyhedra discovered
in the 1880's and the 1930's. The complete set appeared in print for the first
time in 1953, in a paper by Coxeter, Longuet-Higgins and Miller. Wenninger's
1971 book "Polyhedron Models" contains photos and building instructions for
cardboard models of the 75 uniform polyhedra.
In the paper "Uniform Solution for Uniform Polyhedra", published in
Geometriae Dedicata, 47 (1993), 57-110, we propose a uniform approach to an
arbitrary precision solution of uniform polyhedra and their duals, given a
simple formula which describes the method of generation of each polyhedron by
successive reflections in a trihedral kaleidoscope. The theory is complemented
by 8 tables and 160 computer generated figures. A postscript version of the
paper, along with C programs implementing the algorithms, are available for
anonymous ftp from gauss.technion.ac.il (IP address 132.68.112.3), from the
directory kaleido.
The program kaleido may be used, without any further programming, to compute
the metrical properties of the polyhedra, such as angles and radii. and their
combinatorial properties, such as the Euler characteristic and the covering
density. Furthermore, the program is capable of generating Cartesian coordinates
for the vertices and faces, which are then used to display a rotating wire-frame
images of the polyhedra, with depth simulated by edge brightness, and to
generate a pic file which can be included in any TeX or troff manuscript. The
computational features are available on any machine with a decent C compiler.
The graphic features are currently available for Unix machines with X Windows
or LucasFilm graphics, UNIX V/386 machines, and MS-DOS machines, but may be
extended quite easily to other graphic environments. The source code is
carefully broken into small logical units, so it may be used by an experienced
programmer in any environment which requires a precise computation of polyhedra,
such as a computer modeling software.
The source code may be found in kaleido/src, and the documentation in
kaleido/doc. In addition, we provide in kaleido several subdirectories
which include executable code for common platforms, e.g., x-msdos, x-ix386,
x-sparc, etc. Each subdirectory has a CONTENTS file, for further information.
To fetch the software, in a compressed tar format, use the ftp command
ftp> get kaleido.tar.Z
or to fetch a single subdirectory, use the commands
ftp> cd kaleido
ftp> get src.tar.Z
etc. These commands use the ftp features of automatic archiving and compression.
More details about the ftp site are obtainable by executing
% telnet gauss.technion.ac.il 4096
on the shell prompt.
The help of the following persons is acknowledged with many thanks:
Nadav Har'El
Mark Phillips
Jim Buddenhagen
David W. Sanderson
John Firth
Roman Maeder
Comments and bug reports will be greatly appreciated. Please send them
to the author:
Dr. Zvi Har'El III VVVVVVVVVVVVV Z Z
Department of Mathematics, II VV ZZ ZZ
Technion - Israel Institute of Technology, II VV ZZ ZZ
Haifa 32000, Israel. II VV ZZZ
E-Mail: rl@gauss.technion.ac.il VV ZZ
Phone: +972-4-294094 (Although it may be Greek to you, VV ZZ
FAX: +972-4-324654 it is still Hebrew!!) VVVVVVVVVVVVVVV ZZZZZZZZZZZZZ