when this website went public and the visitor count stood at 50.

On

to most of my other websites, and the count was reset to zero.

So add

By coincidence, November 22 happens to be my late mother’s birthday,

so I’d like to dedicate this website to her memory,

for unflinchingly encouraging my interests in mathematics and the natural sciences.

Happy 90th birthday, mom. Mnogoye leto!

This website was last updated

**Welcome to Nuts About Nets!** During the past few months I’ve been using a
program called Great Stella to
picture some pretty polyhedral configurations called **nets**. When limned with suitable
color schemes against contrasty (usually black) backgrounds, they make eye-catching *objets
d’art*. Nets of the kind I’ve been working with are seldom encountered in the
mathematical literature because historically they’ve been difficult to draw by hand, but
with Great Stella software we can now actually construct and examine them in three dimensions
and thereby gain insight into the appearance and properties of a host of interesting
four-dimensional polytopes. If you Google search something like “nets of
four-dimensional polytopes,” you’ll turn up very little information, and nothing like
what is on display here. This is the first appearance anywhere for most of these figures, products
of an ongoing research project I’ve called the Dinogeorge Net Factory, Dinogeorge being one of my
Internet aliases. They’re so novel, and print so beautifully with an inkjet printer on
high-gloss photo paper, that I decided to offer for sale color prints, suitable for framing, of some
of the
nicer nets I’ve constructed (details below).After all, astronomers produce prints of spiral
galaxies and interstellar wonders, and naturalists produce prints of butterflies, flowers, and other
beautiful organisms, so how about a mathematician offering prints of geometrical objects?
People should know that such things exist within the realm of human knowledge! Geometers will
surely find these figures intriguing, but persons with little or no formal mathematical training
may also enjoy them simply for their peculiar charm and aesthetic appeal. In time I’ll have
other kinds of polytope-geometric prints for sale here too, which is why I’m calling this
the **Polyto-Prints** website.

**What are nets?** Well, a common high-school geometry exercise for students is to
model three-dimensional solids, or *polyhedra*, by cutting patterns of polygons out of
paper or cardboard, folding them up, and gluing them together. A pattern of polygons suitable for
this kind of cutting and pasting is called a *planar net*, and it is believed that every convex
polyhedron has at least one planar net that includes all its faces, so that it can be folded and glued
together from one piece. (This, incidentally, has yet to be proved mathematically, but most
geometers believe it to be true, since nobody has yet produced a convex polyhedron that cannot
be unfolded into a planar net. Most convex polyhedra have lots and lots of different
nets—the more faces, the more nets.) Polyhedra usually modeled by high-school geometry
students are the five convex regular polyhedra known from ancient times, namely, the
tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron—the so-called
*Platonic solids*. Most any craft book on polyhedron modeling as a matter of course
displays nets of all kinds of polyhedra, often in color and with instructions on cutting, folding,
and pasting. A Google search for “polyhedron nets” yields thousands of websites
where one may
find pictures of the nets of numerous interesting polyhedra. This part of this website is decorated
with pictures of the nets of the five regular convex polyhedra partly unfolded so you can get an
idea of the relationship of planar nets to their corresponding polyhedra. To see pictures and
models of the convex regular polyhedra and many others, please feel free to visit my Polyhedron Models webite.

**The net concept** generalizes readily to convex polytopes of *n* dimensions, for
*n* greater than or equal to 2: the net of a convex *n*-dimensional polytope is simply
an assemblage of (*n*–1)-dimensional polytopes in
(*n*–1)-dimensional space connected along some of their facets. Of course, the net of
a convex polygon is nothing more than a string of line segments (a *linear net*), so the case
*n*=2 may be regarded as trivial. For *n*>4, nets are difficult to visualize and
display, and the technology to picture them is generally unavailable. But nets of convex
four-dimensional polytopes, or *polychora*, are three-dimensional assemblages of
polyhedra that are easily grasped visually (once their pictures have been prepared)—at least
the less complicated ones—and may readily be understood. If you were a four-dimensional
polychoron-model maker, these are the configurations of polyhedra you would cut out of
three-dimensional paper and fold up and glue together to make convex polychoron models. We
can call them *solid nets*.

**How would one “fold up” a solid net?** Well, if you look at the pictures of
partly folded
planar nets surrounding this text, you’ll see that a fold occurs along an edge joining two
faces.
One face at the edge rotates through some angle relative to the other face, thereby moving it out
of the plane of that face out into three-space. Any parts of the net attached to the face rotate along
with it. When the angle of rotation is exactly the *dihedral angle* between the two faces,
and if like rotations are performed along all the join edges simultaneously, the polyhedron
*closes*. That is, the faces all come into contact along their *free edges*, where they
can be “glued together.” Assuming that the net has been properly constructed, we
are left with a
closed polyhedron that has no free edges, in which each face adjoins exactly one other face along
each of its edges. In regular polyhedra all the dihedral angles are equal (that’s part of what
being
“regular” is all about), but in semiregular and irregular polyhedra the dihedral angles
need not be
the same. By no means does every assemblage of faces fold up into a closed polyhedron; the
faces must be properly arranged and have the correct face angles and edge lengths for the
operation to succeed.

In folding a solid net, everything gets scaled up one dimension. Instead of polygons connected by
edges, we have polyhedra, called *cells*, connected by faces. Folds take place at faces
rather than at edges (rotations in four-space have planes as axes, all whose points remain fixed
during the rotation, just as rotations in three-space have lines as axes). When one cell is rotated
relative to its adjoining cell at their common face, this forces the cells into two different
three-spaces, called *realms*, that intersect in the plane of the face at the fold (which
belongs to both cells). As the cells move out of the realm of the solid net through four-space,
their free faces approach one another, meeting when the various angles of the rotations equal the
various dihedral angles between the cells. The exact criteria for a solid net to fold up into a
convex polychoron are even more stringent than for a planar net to fold up into a closed
polyhedron, but nevertheless it can be done. As far as we know, real space is three-dimensional,
so any excursions like this into four-space unfortunately can take place solely in the mind.
Folding a solid net into a four-dimensional figure, called a *polychoron*, is entirely a
mental exercise. You squint your eyes, Zen out, and imagine the cells of the net coming together,
the cracks between them getting narrower and narrower, and cells from other places in the net
slowly being brought to their contact points. The pictures at this website will, I hope, assist the
viewer in performing this mental exercise and make it a bit more real for him or her than it has
been before. Not to mention that they make *great decorations* for desktop or wall! (And
I’m working on calendars and sets of postcards, too.)

Each print is available in two versions:

Shipping costs are included in the print price. Unless otherwise indicated at the PayPal buttons, all normal size prints are

If you use PayPal, simply click on the

George Olshevsky

P.O. Box 161015

San Diego, CA 92176–1015 USA

If you email your order to me at Polycell@aol.com when you send your payment, I’ll pull your prints the day I receive the email. There is a few days’ curing time required before I can ship a fresh print, to allow the ink to set properly.

**Print #1: Montage of nets of the six convex regular polychora**

To recapitulate some of the preceding discussion, a regular polyhedron is a polyhedron all of
whose faces are regular polygons, the same number of which meet at each corner. Of all the
infinitely many different kinds of polyhedra, only nine are regular: five are convex and four have
interpenetrating faces. The five convex regular polyhedra, whose partly unfolded nets are
illustrated in the text above, are also called the Platonic solids, because they were already known
to Plato and other classical Greek philosophers more than 1700 years ago. The concept of a
regular polyhedron extends easily to polytopes in *n* dimensions, and in particular in
four-space a **regular polychoron** is one whose cells are all regular polyhedra, meeting the
same way at each corner. There are just 16 such figures, six convex and ten with interpenetrating
cells. We can call the six convex regular polychora the **Platonic polychora**, extending the
“Platonic” concept up one dimension. In all spaces of dimension greater than four,
there are only three regular polytopes, all convex, so there is no need to distinguish the convex
ones by calling them Platonic. The definitive work on the regular polytopes is H.S.M.
Coxeter’s *Regular Polytopes*, still available in its
3^{rd} edition from Dover Books. Coxeter was my advisor when I
was a math grad student at the University of Toronto in the late 1960s.

Print #1 is, as far as I know, the first time that nets of all six Platonic polychora have appeared in
the same picture. The large size print in particular makes a striking wall display. If you were a
four-dimensional high-school geometry student, one of your exercises could well be to build
models of the set of six. The first four would probably not be particularly difficult, assuming you
had the kinds of tricky tools that would allow you to cut triangles, squares, and other polygons
out of three-dimensional paper. But the hecatonicosachoron, with its 120 dodecahedra, and the
hexacosichoron, with its 600 tetrahedra, would be fairly time-consuming efforts. The
pentachoron (with five tetrahedra as cells, hence its name pentachoron, which is Greekish for
“five-cell”), tesseract (with eight cubes as cells), and hexadecachoron (with 16
tetrahedra as cells) are perhaps too simple to stand alone as prints (I’ll be happy to make
prints of them if there is some demand, of course). But the icositetrachoron (with 24 octahedra as
cells), hecatonicosachoron, and hexacosichoron are available below as prints on their own.

**Print #2: Four views of the hecatonicosachoron net**

The hecatonicosachoron from Print #1 is the star of this picture, with its net viewed from the
front, back, and side and obliquely. Along with the hexacosichoron, it is one of the two Big
Kahunas of the set of convex regular polychora. The six colors used here define twelve girdles of
ten dodecahedral cells: If the net is folded up in four-space, the helices of ten cells of the same
color come together front to back and “straighten out” into rings of dodecahedra that
surround the center of the figure. This pattern was discovered by one of H.S.M. Coxeter’s
geometry students, John R. Wilker, way back when we were both geometry grad students at the
University of Toronto in the late 1960s. Later, in the late 1990s, I used this pattern to construct
the small swirlprism. The coloring is chiral and comes in “left-handed” and
“right-handed” versions. To get one version, fold the dodecahedra up; to get the
other version, fold the dodecahedra down. **This print is also available without the
lettering.**

**Print #3: Hecatonicosachoron net in side view**

The hecatonicosachoron from Print #1 is also featured in this print, with its net in side view
filling the whole picture. The coloring is as for Prints #1 and #2. You can really see what’s
going
on in this net! In the “netting trade,” we call this kind of net a “modular tower
net with spiral
staircase.” It is made of ten identical modules of twelve dodecahedra joined into a
“tower.” This
arrangement forces one set of ten dodecahedra (colored white) to spiral helically around the
tower formed by the other 110. The spine of the tower is also made of white dodecahedra, and
when the net folds up in four-space, the two white girdles circle the figure in absolutely
perpendicular “polar girdles.” Indeed, each of the other five colors defines a
congruent pair of
polar girdles located elsewhere on the hecatonicosachoron. The nets of the tesseract and the
icositetrachoron in Print #1 are also modular tower nets, but there is no spiral staircase for that
particular icositetrachoron net (six modules of four octahedra). When a tower net folds up, the
top of the tower joins up with the bottom—it bends around in four-space to do so—and
the levels of the tower all join together. The spine of the tower usually becomes a girdle around
the center of the polychoron, and if there is a spiral staircase, its stairs usually join up sideways to
become another girdle absolutely perpendicular, or “polar,” to the spine-girdle.

**Print #4: Hexacosichoron net in side view**

This print features the hexacosichoron net from Print #1 from a somewhat different angle, one
that perhaps better emphasizes its helical structure. It’s a modular tower net with a spiral
staircase in which the modules that make up the tower are essentially the same as the modules
that make up the staircase. Because tetrahedra have no parallel faces, the tower cannot rise
perfectly straight, so instead it acquires an organic helicity that interacts with the staircase
and gives the net a double-helix feel. When a tower does this, particularly with large
stairs in the staircase, I call it a “fancy” net. There are 300 red tetrahedra and 300
light yellow
tetrahedra in this net, and when it folds up in four-space the two differently colored regions of the
hexacosichoron are congruent. The “surface” that separates the two regions is
topologically a
torus, or doughnut, known as a “spherical torus”—a two-dimensional manifold
embedded
in the three-dimensional manifold of the glome, or “hypersphere,” on which the
corners of the
hexacosichoron all lie.

**Print #5: Hexacosichoron net in side view**

Here is a different kind of fancy hexacosichoron net. As in Print #4, it’s a fancy modular
tower
net with a spiral staircase, but the “stairs” are smaller units of only 15 tetrahedra.
The outer parts
of the stairs in the Print #4 net have been implanted onto the tower. There are 300 turquoise
tetrahedra and 300 beige tetrahedra in this net, and as with the Print #4 net, when this net folds
up in four-space the two differently colored regions of the hexacosichoron are congruent. In a
hexacosichoron 20 tetrahedra meet at each corner, where they form an icosahedral vertex figure.
That’s why the stairs of this particular net so closely resemble icosahedra. The 15
tetrahedra of
each unit, along with five tetrahedra from an adjacent unit that stick into the unit’s back
pocket,
create the icosahedral corner at the unit’s center. The ten units, totaling 150
tetrahedra—fully a quarter of the hexacosichoron’s cells—girdle the
hexacosichoron.
Remarkably, one can find three such independent girdles of 150 tetrahedra in a hexacosichoron
(and someday I will construct a hexacosichoron net that displays them), but the final set of 150
tetrahedra doesn’t make this kind of chain. Rather, those final 150 tetrahedra are spread
like
“glue” in among the three chains and hold them together. The 150 tetrahedra in the
spine of this
net form a second girdle polar to the one made up from the stairs.

**Print #6: Pentecontatrigonal duoprism net**

Who would have thought that a mere duoprism could yield such a striking net? This style of net
was discovered by my polytope-net pal Andrew Weimholt, so I call it a Weimholt net after him.
A duoprism in four-space is a **uniform** polychoron (that is, all its corners are alike and
all its cells are uniform polyhedra) whose cells are all prisms arranged around its center in a pair
of polar girdles. There are infinitely many duoprisms. Given integers *p* and *q*, a
*pq*-gonal duoprism has a girdle of *p* *q*-gonal prisms and a polar girdle of
*q* *p*-gonal prisms, mutually “entangled,” adjoining along their
squares. The
surface of squares is topologically a spherical torus—essentially, a rolled up array of
*p*x*q*squares. When *p*=*q*, we simply call it a *p*-gonal
duoprism. In the case of this net, *p*=*q*=53, so it is the net of a pentecontatrigonal
duoprism. It has 5620 external faces, mainly the tiny squares along the rims of the 53-gonal
prisms. Weimholt nets look best for *p*-gonal duoprisms with *p* in the
20–60 range, so that the cells are quite pancake-shaped. A Weimholt net starkly
emphasizes the two girdles of a duoprism!

**Print #7: Icosidodecahedral hexacosihecatonicosachoron net**

Nonregular uniform polytopes are constructed by locating their corners at various symmetric
places on the regular polytopes, such as the midpoints of edges and the centers of faces. In
particular, one kind of uniform polychoron may be constructed by locating its corners at the
midpoints of the 1200 edges of a hecatonicosachoron. This converts each of the 120 dodecahedra
into icosidodecahedra, and adds 600 tetrahedra where the corners of the hecatonicosachoron
previously were. (Four dodecahedra come together in a tetrahedral vertex figure at each of the
600 corners of a hecatonicosachoron, and this is where the 600 tetrahedra come from.) Because
the three-dimensional counterpart of this polychoron is called an icosidodecahedron—it has
the twelve pentagons of a dodecahedron and the 20 triangles of an icosahedron—I called the
four-dimensional version an “icosidodecahedral hexacosihecatonicosachoron.” This
enormously
cumbersome name, which tells us the polychoron has 600 tetrahedra and 120 icosidodecahedra as
cells, is not used very often. Norman W. Johnson has called it a “rectified
hecatonicosachoron”
or a “rectified 120-cell,” where the “rectification” operation means
“locate the corners at the
midpoints of the edges of” the figure being rectified; and Jonathan Bowers has shortened
this to
**rahi**. It’s a lot easier to call it a “rahi” than any of the other names!
This print shows an
oblique view of a rahi modular tower net with spiral staircase. Each of the ten levels (modules) of
the tower comprises twelve icosidodecahedra and 60 tetrahedra, the twelfth icosidodecahedron
being one stair of the spiral staircase. The icosidodecahedra all have teal pentagons and orange
triangles, and the tetrahedra are orange all over. When folding the net, the teal faces always
attach to teal faces, and the orange faces of the tetrahedra always attach to the orange faces of the
icosidodecahedra.

**Print #8: Great diprismatohexacosihecatonicosachoron net**

This is a side view of a modular tower net with spiral staircase of the Big Kahuna of the convex
uniform polychora, the one with 14,400 corners that is the four-dimensional analogue of the great
rhombicosidodecahedron. Its net has 28,802 free faces (not coincidentally twice the number of
corners plus 2), the greatest number of faces I have ever worked with in a single Great Stella
figure. Its cells are 120 great rhombicosidodecahedra, 600 truncated octahedra, 720 decagonal
prisms, and 1200 hexagonal prisms—and all these cells of course went into the construction
of the net in this print. The net provides a really good idea how the cells all fit together in the
polychoron, and Print #9 (below) is a zoom into the same net that fills the picture with its
polygons. For a summary of the properties of all the convex uniform polychora, visit my Uniform Polychora website. I have
constructed Great Stella nets for all 64 convex uniform polychora (which include the six regular
polychora but not, of course, the infinite sets of duoprisms and antiprismatic prisms) as part of
the Dinogeorge Net Factory project. I will offer prints of all of them if a demand arises, maybe
even a special bulk deal for the set of 64!

**Print #9: Great diprismatohexacosihecatonicosachoron net (zoom)**

Continuing the description of this polychoron from Print #8 above, the net has five colors: teal
for the decagons, red for the hexagons, and white, light green, and light blue for the squares.
Three colors are used for the squares because they have three different situations in the
polychoron: The white squares join the great rhombicosidodecahedra to the hexagonal prisms,
the light blue squares join the truncated octahedra to the decagonal prisms, and the light green
squares join the decagonal prisms to the hexagonal prisms. Of course, the red hexagons join the
truncated octahedra to the hexagonal prisms, and the teal decagons join the great
rhombicosidodecahedra to the decagonal prisms. Whew. The spaces between adjacent
nonadjoining cells in the net are very thin, so one need fold a cell up only a little bit before it
joins its neighbor. Because there are two different kinds of prisms among the cells of this
polychoron, I called it the “great diprismatohexacosihecatonicosachoron,” which
Jonathan
Bowers shortened to **gidpixhi** (pronounced gid-PICK-she). Norman W. Johnson’s
name
for it is “omnitruncated hexacosihecatonicosachoron.”

**Print #10: Grand antiprism net**

The grand antiprism was the last of the “Archimedean” uniform polychora to be
discovered.
Whereas most of them were found by Thorold Gossett and Alicia Boole Stott in the early years of
the 20^{th} century, the grand antiprism was found only in the
mid-1960s, by John Horton Conway and Michael Guy, in a computer search for convex uniform
polychora. The grand antiprism has 100 of the 120 corners of a hexacosichoron, as well as 300 of
its tetrahedral cells. The other 300 tetrahedra are replaced by two polar girdles of ten pentagonal
antiprisms, the two girdles showing up quite well in the net in this print. The 20 antiprisms are
colored light blue, 200 of the tetrahedra are colored dark blue, and the other 100 tetrahedra are
colored orange. The blue tetrahedra share a face with an antiprism, whereas the orange tetrahedra
do not. This is another modular tower net with a particularly evident spiral staircase.

**Print #11: Snub icositetrachoron net**

Thorold Gosset discovered this convex uniform polychoron when he searched for all the convex
semiregular polytopes in *n* dimensions whose cells are regular polytopes, publishing his
results in the year 1900. It is best described as a hexacosichoron from which 24 icosahedral
pyramids are removed, leaving icosahedra in place of icosahedral assemblages of 20 tetrahedra.
When the 24 pyramids are removed, they take 480 of the 600 cells of the hexacosichoron with
them, leaving 120 tetrahedra filling the spaces between the 24 icosahedra. In this tower net with
spiral staircase, the icosahedra are colored white, and the 96 tetrahedra that share a face with an
icosahedra are colored red. The 24 tetrahedra that share their faces only with other tetrahedra are
colored dark blue, but not too many of them can be seen in the net. The stairs of the spiral
staircase are assemblies of four red tetrahedra surrounding a fifth blue tetrahedron, whose four
red neighbors keep it from being seen (to us three-dimensional observers, anyway:
four-dimensional model makers would have no trouble seeing it from outside our three-space).
The 24
icosahedra lie in the cell realms of the 24 octahedra of the icositetrachoron, which is why this
figure is called a “snub icositetrachoron.” The 24 blue tetrahedra also lie in the cell
realms of an
icositetrachoron, one positioned “dually” to the icositetrachoron of the icosahedra.

**Print #12: Icosahedral hexacosihecatonicosachoron exploded net**

Whereas print #7 features the icosidodecahedral hexacosihecatonicosachoron, this net features
the *other* four-dimensional analogue of the icosidodecahedron, the icosahedral
hexacosihecatonicosachoron. Norman W. Johnson’s name for this figure is
“rectified
hexacosichoron,” which Jonathan Bowers has shortened to **rox**. The corners lie at
the
midpoints of the 720 edges of a hexacosichoron, which converts the hexacosichoron’s 600
tetrahedra to octahedra and adds 120 icosahedra where the hexacosichoron’s corners were.
In this
print, the icosahedra are colored fuchsia and the octahedra have fuchsia and beige faces, four of
each. When the net folds up, the octahedra adjoin icosahedra along their fuchsia faces and each
other along their beige faces. This is an example of an exploded net, in which one cell serves as
the center and all the other cells are arranged outward as symmetrically as possible around it.
After all 120 icosahedra are placed in the net, only one threefold symmetry axis remains of its
initally icosahedral symmetry, and in this picture we look almost straight down that axis. The
icosahedron at the center of the picture is the cell opposite the net’s central icosahedron in
the
rox.

**Print #13: Pentagonal-bipyramidal heptacosiicosachoron net (modular)**

As with polyhedra in three-space, most polychora have *duals*, which, roughly speaking,
are polychora whose “element numbers” are reversed. In three-space, for example,
the
dodecahedron is the dual of the icosahedron (and vice versa): The dodecahedron has twelve
faces, 30 edges, and 20 corners, while the icosahedron has twelve corners, 30 edges, and 20
faces. And in four-space the hecatonicosachoron is the dual of the hexacosichoron: The
hecatonicosachoron has 120 dodecahedral cells, 720 pentagonal faces, 1200 edges (three
dodecahedra surround an edge), and 600 tetrahedral corners, whereas the hexacosichoron has 120
icosahedral corners, 720 edges (five tetrahedra surround an edge), 1200 triangular faces, and 600
tetrahedral cells. Cells dualize into corners, edges into faces, and so on. Regular polytopes
dualize into other regular polytopes, but if a polytope is not regular, neither is its dual. For
example, the 13 semiregular polyhedra dualize into the set of 13 Catalan polyhedra.
Great Stella has a terrific dualizing function built into it that works fine for polyhedra but is less
useful when one is working with polychora. Nevertheless, one can still coax it into producing
nets for a few of the duals of the semiregular polychora (which have not previously been studied
much at all). Print #13 displays a four-color modular net with spiral staircase of the dual of the
rox from print #12, so it can be considered one of the “Catalan polychora.” Indeed,
it is one of
the two four-dimensional analogues of the rhombic triacontahedron, the dual of the
icosidodecahedron. The stairs of the spiral staircase are the cells colored green. Just as all 720
corners of the rox are alike (because it is uniform), so the 720 cells of its dual are all alike: each
is a pentagonal bipyramid of a particular height. Since the vertex figure of the rox is an
Archimedean pentagonal prism, the cell of the rox dual may be constructed in Great Stella by
dualizing that prism. Once you have the cell, it is then a matter of putting 720 of them together
into the net (continued under print #14). I formally call this polychoron, the rox dual, a
**pentagonal-bipyramidal heptacosiicosachoron**, because its cells are 720 identical
pentagonal bipyramids.

**Print #14: Pentagonal-bipyramidal heptacosiicosachoron net (fancy)**

I could not make up my mind which rox dual net to offer on this website, so I ended up using
both the modular one in print #13 and the fancy one depicted here. Three hundred sixty of the
cells are in the blue tower, and the other 360 are in or near the red stairs. When the net is folded
up, both regions become congruent halves of the polychoron. The rox dual’s cells are 720
pentagonal bipyramids, and these kinds of cells are rather difficult to fit together into a net when
there are hundreds of them because there is not much space between neighboring cells. The
bipyramids enjoy “crashing,” which is two net cells interpenetrating so that a model
maker would
be unable to cut them out separately in building a model. All my nets are crash-free as far as my
polytope-net pal Andrew Weimholt and I can tell, and removing crashed cells
(“decrashing”)
sometimes requires a major restructure of a considerable portion of a net. This is why some of
the red cells in the depicted net have migrated from the stairs onto the blue ones in the tower. I
have built a number of other nets of this polychoron. They’re all quite pretty, and in time
I’m sure
I’ll be offering prints of them for sale.

**Print #15: Sphenodecahedral hectochoron net**

One dual begets another. The grand antiprism can be vertex-inscribed in a hexacosichoron, so its
vertex figure is inscribable in an icosahedron, and so the cell of its dual is a partial stellation of
the regular dodecahedron. Indeed, two pentagonal faces of the dodecahedron are removed, six
others are extended in their planes into two trapezoids and four kite-shaped quadrilaterals
(“phylloids”), and the other four faces remain unchanged. The resulting heretofore
unnamed
ten-faced cell looks like a wedge or an axe blade, so we can christen it the
**sphenodecahedron**. There are 100 of them in the grand antiprism dual, corresponding to
the 100 corners of the grand antiprism, so the polychoron itself can be called a
**sphenodecahedral hectochoron**. (I use the prefix “hecto-” rather than
“hecaton-” only
when there are exactly 100 facets.) This print shows one of the many wonderful shapes a net of
the sphenodecahedral hectochoron can assume; I’ve built 13 different ones already. The
pentagons are fuchsia, the outer trapezoids are beige and their inner counterparts are red, and the
phylloids are turquoise. Free faces of like colors adjoin when the net folds up in four-space. The
net is a modular helicoidal tower comprising ten identical modules of ten cells each.

**Print #16: Othenneahedral enenecontahexachoron net**

Since a snub icositetrachoron is nothing more than a hexacosichoron with 24 icosahedral
pyramids symmetrically removed, leaving 24 icosahedra at the “stumps, ” its dual is
a hecatonicosachoron with 24 dodecahedral pyramids augmented onto 24 of its dodecahedra.
This removes the underlying 24 dodecahedra, and since the twelve lateral cells of the
dodecahedral pyramids lie in the realms of the removed dodecahedron's twelve neighbors, the 96
remaining dodecahedra become strange-looking nine-faced polyhedra: dodecahedra with tall
pentagonal pyramids erected on three of their faces. Like the grand antiprism, the snub
icositetrachoron has a vertex figure vertex-inscribable in an icosahedron, because the snub
icositetrachoron is itself vertex-inscribable in a hexacosichoron. So each cell of the snub
icositetrachoron dual is again a partial stellation of the dodecahedron: Three of the dodecahedral
faces are removed, three extend into phylloids as with the sphenodecahedron (the cell of the dual
of the grand antiprism), and the remaining six extend into obtuse golden triangles. None of the
original pentagons remains unchanged. In the picture of the net, the phylloids are red and the
obtuse golden triangles are turquoise or beige. When folded up, the red faces adjoin red faces, but
the turquoise triangles always adjoin the beige ones, and vice versa. Each cell may also be
thought of as a triangular bipyramid that has had three of its faces that meet at a corner pushed
apart by a set of three phylloids, so we can christen it the **othenneahedron**, from the
Greek “otheo,” meaning “push.” There are 96 cells in the snub
icositetrachoron dual, so we can call it an **othenneahedral enenecontahexachoron**.The
net is a modular helicoidal tower comprising six identical triangularly symmetrical modules of 16
cells each.

**Print #17: Bi-icositetrachorically diminished hexacosichoron net**

A Johnson polyhedron,
named after Norman W. Johnson, who originally proposed and solved the problem of finding and
enumerating them, is a convex polyhedron whose faces are all regular polygons but which is not
a uniform polyhedron. Examples include the square pyramid, whose faces are one square and
four equilateral triangles; the triangular cupola, whose faces are four equilateral triangles, three
squares, and one hexagon (it is half of a cuboctahedron); and the tridiminished icosahedron,
whose faces are five equilateral triangles and three pentagons (it is an icosahedron with three
pentagonal pyramids removed, hence its name). Johnson found a total of 92 of them back in
1966; in 1969 Viktor Abramovich Zalgaller proved the list was complete, and in 1971 Martin
Berman published planar nets for building them all. They have become quite well known in the
world of geometry; a Google search for “Johnson polyhedra” finds some 25,000
websites.

In four-space, we can define a **Johnson polychoron** as any polychoron whose faces
(two-dimensional elements) are all regular polygons that is not already uniform. This definition
extends with no modification to polytopes in *n*-dimensional space: Having regular faces
forces the facets of a **Johnson polytope** to be Johnson polytopes or uniform polytopes of
one less dimension. So, how many Johnson polychora are there? Alas, there are trillions, as well
as at least one infinite set. But this need not prevent us from building solid nets of some of the
more peculiar or symmetric ones. All 92 Johnson polyhedra are available in Great Stella, so we
can potentially build solid nets of *all* the trillions of Johnson polychora with it.

The net in this print is the first solid net I built with Great Stella. The figure was discovered by
Andrew Weimholt, and it has the interesting properties that its corners are all alike and its cells
are all alike, making it a *noble* Johnson polychoron (as defined by Branko
Grünbaum; interestingly, Noble Johnson is the actor who played the tribal chief in the
1933 movie *King Kong*). Noble Johnson polychora are rare; and indeed there are no
noble Johnson polyhedra at all. Each of its 48 cells is a tridiminished icosahedron. Perhaps the
easiest way to construct it is simply to remove all 24 assemblies of five tetrahedra from the snub
icositetrachoron. When you do this, you also have to take three pentagonal pyramids off the snub
icositetrachoron’s icosahedra when you leave the tridiminished icosahedral
“stump” at each location. The chip that you remove is itself another Johnson
polychoron, a tridiminished-icosahedral pyramid. It’s a neat coincidence that the stump
cell has the same shape as the cells that are left behind. I used Andrew’s name for it in the
title; Jonathan Bowers has already shortened it to **bidex**. This print shows a modular
tower net with a spiral staircase whose stair is one of the cells, six modules of eight cells. The
pentagons are white, the triangles are light yellow, red, and blue. Yellow triangles always attach
to yellow triangles, but the blue ones attach to red ones and vice versa when the net folds up. The
cells form eight girdles of six, connected red triangle to blue, around the polychoron.

**Print #18: Truncated-dodecahedrally decadiminished rectified hecatonicosachoron net**

Here is a Johnson polychoron net that reminds me of some kind of *Star Trek* spaceship.
To construct the polychoron, begin with a rectified hecatonicosachoron (rahi: see print #7).
Remove ten “icosidodecahedral-truncated-dodecahedral rotundae” (yes, another
kind of Johnson polychoron) from the rahi, leaving two polar girdles of five truncated
dodecahedra carved out of the rahi. This operation removes 20 icosidodecahedra (two girdles of
ten) outright and cuts all 100 remaining icosidodecahedra in half, turning them into the Johnson
polyhedra known as pentagonal rotundae. It also removes 400 of the 600 rahi tetrahedra, 40 per
rotunda. For want of a better name, we can call the Johnson polychoron that remains a
**truncated-dodecahedrally decadiminished rectified hecatonicosachoron**. If you want an
even longer name, substitute “icosidodecahedral hexacosihecatonicosachoron” for
“rectified hecatonicosachoron” in the preceding. To summarize, the cells of this
figure are 10 truncated dodecahedra (in dark red), 100 pentagonal rotundae (in light green), and
200 tetrahedra (in light yellow). The net is a modular tower net with a very obvious spiral
staircase of five truncated dodecahedra.

**Print #19: Enneagonal bicingulum net**

Not all Johnson polychora are chips and pieces of the uniform polychora; many are quite
independent of them. Andrew Weimholt discovered an infinite series of prismlike Johnson
polychora, of which a net of the *p*=9 member appears here. The idea is to make a girdle
of three polyhedra from two *p*-gonal antiprisms and one *p*-gonal prism, and then
fill the gaps with a second girdle of *p* alternating tetrahedra and *p* square
pyramids. There being just two girdles in the figure, a good name for it is *p*-gonal
bicingulum, and there is one for every *p* greater than 2. The *p*=3 member
coincidentally happens to be a **diminished dispentachoron** (a very simple Johnson
polychoron of only nine cells: one triangular prism, two octahedra, three tetrahedra, and three
square pyramids), which is a dispentachoron from which a corner triangular-prismatic pyramid
has been removed. You can lengthen the prism-antiprism girdle of three to four with a second
prism, thereby generating the *p*-gonal antiprismatic prisms, which are well known
uniform polychora. But inserting any more prisms or antiprisms into the little girdle yields no
further series of Johnson polychora.

**Print #20: Whatchamacallit hatching net**

The polychoron whose net appears here is a hexacosichoron from which a whole slew of
different kinds of Johnson-polychoric chips have been symmetrically removed. There is almost
no point in figuring out what to call such figures, since there is a nearly endless number of
combinations of ways to dechip (that is, diminish) a hexacosichoron. I call the type of net in this
picture a “hatching net,” because some of the cells are spread apart to reveal more
cells that they would otherwise enclose, something like the shell of an egg cracking away to
reveal the fledgling within. The Johnson polychoron that remains after dechipping comprises two
dodecahedra (in fuchsia), twelve pentagonal pyramids (in dark purple), twelve gyroelongated
pentagonal pyramids (or gyepips: each is an icosahedron with one pentagonal pyramid removed;
in orange; **gyepip** is Jonathan Bowers’s acronym for this Johnson polyhedron),
and 160 tetrahedra (20 colored light blue, 20 light yellow, 60 dark blue, and 60 yellow-orange).
The net resembles an artificial satellite with its solar panels spread.

**Print #21: Icositetrachoron modular tower net**

I’ve added this net (and the next) by popular demand: A recent poll taken at a
four-dimensional forum website named the 24-cell as the “favorite polytope” by a
considerable margin. The net is fairly simple—just 24 octahedra—and this one is quite
attractive. Here the octahedra are painted in four colors, each color representing a girdle of six
octahedra. When the net folds up in four-space, the girdles all meet end to end and circle the
center of the figure. Also featured in Print #1, this is a tower net made up of six modules, each
module comprising four octahedra, one of each different color. It knocks together in seconds in
Great Stella!

**Print #22: Icositetrachoron hatching net**

Like the preceding net, this one is here by popular demand. I colored the cells in the minimum of
three colors—red, white, and blue, of course. When the net folds up, one cell of each of the
three colors appears at every edge. Each color appears in eight cells, and the eight cells of the
same color touch one another solely at their corners. The cell realms of the eight like-colored
cells belong to a tesseract, so you can see that an icositetrachoron will stellate into a regular
compound of three tesseracts, each tesseract naturally corresponding to one of the three colors.
Instead of having four girdles with six cells the same color, as in the preceding net, this coloring
guarantees that all 16 girdles of six will have two opposite cells the same color, all three colors
appearing twice in each girdle. One such girdle is very obvious in the hatching net.

**Print #23: Decachoron Siamese net**

Two convex uniform polychora besides the six convex regular polychora have only one kind of
cell, which makes them noble polychora: the decachoron (net featured here) and the
tetracontaoctachoron (net featured in next print, #24). There are no noble convex uniform
polyhedra besides the Platonic solids (almost by definition), so here is where four-space really
does have one up on three-space. The cells of the decachoron are ten truncated tetrahedra, each
here having its hexagons colored blue and its triangles colored white. Because truncated
tetrahedra and tetrahedra can fill three-space, a few of the hexagons and triangles in this net are
coplanar. I call this a “Siamese” net because it is made of two identical modules (of
five truncated tetrahedra each) joined along one common face, reminiscent of Siamese twins
unfortunately sharing a common body part. The decachoron is the polychoron common to the
compound of two pentachora in “dual position”—situated concentrically so
that the five corners of one are directly above the five cells of the other.

**Print #24: Tetracontaoctachoron tower net**

Along with the decachoron (in the preceding print), this is one of only two noble convex
nonregular uniform polychora. Its cells are 48 identical truncated cubes, here painted with blue
octagons and orange triangles. As with the decachoron, the tetracontaoctachoron is the
polychoron common to the compound of two icositetrachora in dual position. Here is a modular
tower net of eight cruciform modules of six cells each; one cell becomes the stair of the spiral
staircase around the tower. It is not necessary to position the stairs spirally in this kind of net;
they may appear anywhere along their “vertical channels.” For example, they could
all be placed at the same level around the tower—in the case of the tetracontaoctachoron, at
two adjacent levels. The spiral staircase is merely an aesthetic choice. This net also shows how
the cells fall into six girdles of eight joined octagon to octagon, the stairs forming one girdle, for
example, and the spine of the tower forming the polar girdle. A different tower net with spiral
staircase exists, with six levels of eight cells each, showing the cells arranged in eight girdles of
six joined triangle to triangle. The eight-level net in this print folds up (or down) 45 degrees
between levels to make the tetracontoaoctachoron.

**Print #25: Snub-cuboctahedral prism hatching net**

Often even a lowly prism can have an interesting-looking net. Each uniform polyhedron, convex
or not, is the base of a uniform right prism in four-space. Simply translate the base polyhedron
one edge-length perpendicular to its realm and connect it and its image under translation with a
**mantle** (Branko Grünbaum’s term) of right prisms and other elements. If
the base polyhedron is a cube, the corresponding prism is a tesseract. But if the base is a snub
cuboctahedron, as in this print, the corresponding prism is, naturally enough, a
snub-cuboctahedral prism. To make a net, you can simply take the planar net of the snub
cuboctahedron and make a prism out of it, so that each face in the net becomes a prism itself.
These then become the mantle cells of the four-dimensional prism; the two bases are a snub
cuboctahedron attached to the top of the mantle-prism and another attached underneath, as shown
in this print. There is one additional intricacy: the snub cuboctahedron is chiral; that is, it comes
in left- and right-handed versions. In the snub cuboctahedral prism, one base will be left-handed
and the other will be right-handed, because when the mantle prisms fold up in four-space, the net
on one side of the mantle folds up and the net on the other side folds down. This little nuance is
why I picked the snub-cuboctahedral prism net for display.

**Print #26: Icosahedral hexacosihecatonicosachoron modular tower net**

Print #12 featured this convex uniform polychoron, nicknamed “rox,” as an
exploded net. Here we show the same polychoron in a modular tower net with spiral staircase.
The modules of the tower are not exactly parallel, because the face that joins the levels is not on
a symmetry axis of a module. This gives the whole net a slight helical twist, and it also makes
placing some of the cells very tricky, to avoid crashes (interpenetrating cells) between the levels
that would render the net invalid. The spaces between the cells are very narrow, and the
octahedra especially enjoy getting into each other’s way. I began building the net some
time ago but abandoned the project for weeks because of the crashing problem. I finally
overcame it and produced the handsome structure depicted in four views (top, bottom, and two
side views) in this print. The 120 icosahedra are white, and the 600 octahedra are teal. One
icosahedron survives as the stair of the spiral staircase around this tower. The icosahedra go
around the center of this polychoron in “necklaces” of ten. In girdles, the cells are
connected face to face, but in necklaces neighboring cells touch one another only at their
opposite corners, something like the beads in a real necklace. Each icosahedron belongs to six
necklaces. The ten spiral staircase icosahedra make an obvious necklace after the net folds up,
and the ten icosahedra that lie almost entirely hidden in the spine of the tower make another
necklace polar to the spiral-staircase necklace.

And that’s all I have for now. Keep returning to this website, as I’ll be adding more and more Polyto-Prints for sale, particularly of those exotic Johnson polychora, at least until AOL boots me off for using too much of their website space. Let me thank my wife Andrea for putting up with my hours at the computer working with Great Stella, Andrew Weimholt for vetting all these nets for me, Robert Webb for creating the marvelous tool Great Stella, and Norman W. Johnson and Jonathan Bowers for their enthusiatic support of my geometry work.

Text and picture designs ©2004 George Olshevsky, but the math belongs to everyone.