Polyhedron Models Custom Built
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!
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.
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.
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.
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:
You are visitor number—
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??
Anyway, please add 54337 to the above figure for a more accurate count.
Visitors to
this site since January 23, 2000 redesign is—
Text
and photos at this website ©1997 George
Olshevsky.
