Polyhedron Models Custom Built

Something New logo Just added November 22, 2004: A website where you can view and even purchase beautiful prints of interesting polychora nets. Go to Nuts About Nets!

Fra Luca

Above: Painting of Fra Luca Pacioli and pupil, by Jacopo de Barbari (1440/50?–1516), perhaps the most famous Renaissance painting with a geometric theme. Currently at the Museo di Capodimonte in Naples, Italy, it was executed in 1495, and besides the figures of Renaissance geometer Pacioli and an unnamed mathematics student, it shows a beautiful glass model rhombicuboctahedron suspended by a string and a paper or wooden model regular dodecahedron on the bench.
YMMETRIC FIGURES in solid geometry have fascinated people since ancient times. Early Egyptians played with icosahedral dice, and the five regular solids were well known to the mysterious Pythagoreans of classical Greece, who named such figures “polyhedra.” Other symmetric solids were discovered by Archimedes. Johannes Kepler discovered the first star solids, and geometers of the 19th and 20th centuries found many more. The aesthetic value of such objects was not lost on artists of the Renaissance and the Enlightenment, who used them in their works of art and architecture. Today, a small group of dedicated model-makers continues this gentle, age-old art, producing figures of striking intricacy and beauty.

I’ve been making polyhedron models since I was a grade-school student back in 1958 (my technique has improved somewhat since those days!), for the sheer enjoyment of contemplating the finished figures. For more than a decade and a half, I lacked the time to pursue this little avocation, but a few years ago my interest in polyhedra (and figures in higher-dimensional spaces) was rekindled, and I went on something of a model-making binge. Now colorful polyhedra decorate our living room, dining room, and bedrooms, to the point where we really can’t accommodate many more. Needing an outlet to satisfy my model-building urge, I decided to see whether there was any interest by the outside world in this minor art form.
Quasitruncated Great Stellated Dodecahedron

Above: Photo of my model of a quasitruncated great stellated dodecahedron: a nonconvex uniform polyhedron with 90 edges whose faces are 12 regular decagrams (10-pointed stars; yellow) and 20 equilateral triangles (red), two decagrams and one triangle meeting at each of 60 vertices (corners or points). Its diameter is approximately 35 cm. (Note that the faces intersect one another; those portions of the faces that pass into the polyhedron’s interior cannot be seen. This polyhedron is sometimes called a stellated truncated great dodecahedron, and even a great stellated truncated dodecahedron! Its Wythoff symbol is 2 3 | 5/3.)

As of January 23, 2000, I have redesigned this website so that a visitor no longer need wait for more than a dozen JPG pictures to download. I broke the single large home page up into several smaller ones, each comprising some of the original text and several pictures from the previous version. I also added some general remarks on the geometry of polyhedra and the craft of polyhedron model-making, and an atlas of Mathematica-generated pictures of the nine regular polyhedra. Link to these pages in the following order to view this website fully and to see more photographs of polyhedron models:

Page 2: What Are Polyhedra? This page displays some more polyhedron models and introduces a working definition of a polyhedron. Use the chart of Greek Numerical Prefixes at the bottom of this page to create formalized names for all kinds of polyhedra.

Page 3: The Regular Polyhedra. This page has a photo of my set of nine regular polyhedron models and describes how to build each one. See an atlas of the regular polyhedra and a table of their various numerical properties (dihedral angles, symmetry groups, circumradii, and so forth). With 10,000+ words and more than a dozen pictures, this page takes a bit of time to download.

Page 4: Specifications and Prices. Still more photos of models here, along with descriptions of the materials and methods I use in my craft. Find out how much I would charge to custom-build a polyhedron for you.
Great Ditrigonary Dodekicosidodecahedron

Above: Photo of my model of a great ditrigonary dodekicosidodecahedron: a nonconvex uniform polyhedron with 120 edges whose faces are 12 regular decagrams (olive), 20 equilateral triangles (yellow), and 12 regular pentagons (green), two decagrams, one triangle, and one pentagon meeting at each of 60 vertices. Its Wythoff symbol is 3 5 | 5/3. The decagrams are particularly evident, since they’re entirely on the polyhedron’s exterior. Its diameter is approximately 32 cm. Here I’ve manually blacked out the background for effect.
I presently correspond by mail and e-mail with a number of other polyhedron model-makers, including the master of us all, Magnus Wenninger (see Fr. Magnus Wenninger’s Home Page).

To order one or more polyhedron models, to discuss sizes and color schemes, or simply to correspond about geometric topics, just e-mail me at Polycell@aol.com. Let’s hear from you about which polyhedra you’re interested in, and what you think about this website.

Naturally, this website is perpetually under construction. I’ll be adding more links and pictures shortly, and every so often I’ll add a new page: Coming soon! A new page about modeling isohedra.

This website last updated 12/06/06.

Name and Location:

George Olshevsky
Post Office Box 161015
San Diego, California 92176–1015

Business Description:

Writing, editing, publishing about dinosaurs; professional indexing; and polyhedron model-making.

Dextro and Laevo Five Tetrahedra

Above: Photo of my models of a matched dextro-laevo pair of the compound of five tetrahedra in a dodecahedron. The five tetrahedra in each figure are colored cream, beige, yellow, orange, and violet, and are arranged so that either model is a reflection of the other. Five tetrahedra have a total of 20 vertices, which in these figures are located at the corners of a regular dodecahedron. Alike except for being mirror images, the models have a diameter of about 30 cm.
Here are a few links to other interesting polyhedron websites:

For information about and pictures of all the uniform polyhedra, go to Roman Maeder’s Uniform Polyhedra or Steven Dutch’s Uniform Polyhedra.

For another very handsome collection of all the uniform polyhedra, see Vladimir Bulatov’s Polyhedra Collection.

This website has polyhedron software and an interesting polyhedron poster that can be ordered: Pedagogical Polyhedra.

For pictures of the “stellated” icosahedra (what I call the 59 icosahedral aggregates), see Roman Maeder’s Stellated Icosahedra.

And for what has to be the absolutely most thorough website on all kinds of polyhedra, go to Virtual Reality Polyhedra (George Hart), but for best results there, you will need a VRML viewer. George Hart has almost 1,000 polyhedra on display at this website.

Daniel Green’s models of portions of infinite regular polyhedra may be viewed at his Infinite regular polyhedra website.

For a table of all the convex uniform polytopes in four dimensions, see my website Four Dimensional Figures. All the vertex figures of these polytopes, called polychora, are illustrated there. I recently acquired an early version of Mathematica, and I’ve used its three-dimensional graphics features to make some pictures of polyhedra. One of these days, I’ll adapt it for four-dimensional display.

Mathematical references on the subject of polyhedra and their geometry are cited at this website: Polyhedra References.

Geometric-art objects for sale appear at Christopher Guest’s Sacred Shapes website: Sacred Shapes.

My graduate-school adviser at the University of Toronto was the famous geometer H. S. M. Coxeter, who appears in a photo at that website. The photo shows him examining a large model of the retrosnub ditrigonary icosidodecahedron, or yog-sothoth, which Bruce L. Chilton constructed from templates I calculated with a computer program and plotted using a CalComp 30" drum plotter. Displayed at the Shaping Space conference at Smith College in 1984, this was the first accurate (to the last facet) model of this polyhedron ever built, and it is the most complicated polyhedron I’ve ever worked on. It’s the model featured in the article in 21st Century Science & Technology I mention at the Specifications and Prices page. Surrounding it are numerous smaller polyhedron models built by Chilton, one of the world’s premier polyhedron model-makers and my mentor in this hobby.

I used the same CalComp plotter to draw plane projections of the edge-skeletons of all the regular four-dimensional star polytopes, some of which Professor Coxeter used in his book Regular Complex Polytopes. You can see one of them, the great stellated hecatonicosachoron, at its website, which can also be accessed from the H.S.M. Coxeter home page. Another Coxeter website is at GCS - Donald Coxeter -- Mathematician. Sadly, Professor Coxeter passed away March 31, 2003 at the age of 96.

You’ll find pictures of all kinds of polyhedra, as well as an enormous amount of other math and science material, at the website operated by Eric W. Weisstein.

Take a look at some more virtual polyhedron models at Tom Gettys’s place.

Lots of links to other interesting geometric websites may be found by going through The Geometry Junkyard.

Take a short trip into spaces of dimension higher than three on Professor Tom Banchoff’s home page.

And finally, here is the link to the website where you can find out about my dinosaur publications, if you’re so inclined: Dinogeorge’s Home Page.
This page has received a Links2Go Award as a polyhedra resource. To see what this is all about, try these links:

Key Resource

Links2Go Key Resource

Polyhedra Topic

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The insidious AOL Phantom Counter Resetter finally hit
this website, as it has all my other websites,
on April 13, 2006, and again on September 13, 2006, and again on December 6, 2006.
This is driving me nuts! What is the matter with the server??
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Text and photos at this website ©1997 George Olshevsky.