kaleido/README Version 3.10 (93/11/22) ~~~~~~~~~~~~~~ ~~~~~~~ ~~~ ~~~~~~~~~~ Uniform Polyhedra - Computation and 3D Display ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 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