Uniform Polyhedra - Computation and 3D Display

Kaleido’s README file

    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 ftp.math.technion.ac.il, from the directory kaleido, or, through the Word-Wide Web, from the URL: http://www.math.technion.ac.il/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-linux, x-sparc, etc. Each subdirectory has a README 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.

    If you use a WWW browser to access our archive, we provide appropriate hooks to get single files or whole directories. More information about the Technion Mathematics Department in general and its FTP archive in particular may be obtained from the URL: http://www.math.technion.ac.il/.

    As an illustration to kaleido’s possibilities, kaleido may produce virtual reality models (using VRML97) of the polyhedra, which can be viewed using any current VRML browser. Each polyhedron face is solid, colored according to its valency using a simple color coding. To overcome the limitations of the browsers, faces bounded by self-intersecting polygons are broken down to simple sub-faces by insertion of auxiliary vertices. Ray traced pictures of the models, and the VRML files themselves, may be viewed in the URL: http://www.math.technion.ac.il/~rl/kaleido/.

    The help of the following persons is acknowledged with many thanks:
    Roman Maeder has ported the kaleido algorithms to Matehmatica. His programs are available from the Mathematica Information Center, URL: http://library.wolfram.com/infocenter/Articles/2254/. The polyhedra he generated may be viewed through the WWW in his Uniform Polyhedra page, URL: http://www.mathconsult.ch/showroom/unipoly/.

    Comments and bug reports will be greatly appreciated. Please send them to the author:
Dr. Zvi Har'El     mailto:rl@math.technion.ac.il     Department of Mathematics
tel:+972-54-227607 icq:179294841     Technion - Israel Institute of Technology
fax:+972-4-8293388 http://www.math.technion.ac.il/~rl      Haifa 32000, ISRAEL
"If you can't say somethin' nice, don't say nothin' at all." -- Thumper (1942)

Copyright © Zvi Har’El
$Date: 2006/11/14 10:15:47 $