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Added December 12, 2004: Six more prints (#21–26)!

You are visitor number— —as of November 22, 2004,
when this website went public and the visitor count stood at 50.
On February 14, 2006, the annoying
AOL Phantom Counter Resetter struck again, as it has done at least once
to most of my other websites, and the count was reset to zero.
So add 8282 to the above figure for a more accurate count.

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 February 16, 2006 and is currently showcasing prints #1–26.

Nuts About Nets!

Tetrahedron net partly unfolded 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.

Cube net partly unfolded 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.

Octahedron net partly unfolded 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.

Dodecahedron net partly unfolded 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.

Icosahedron net partly folded 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.)

Prints of nets for sale

SO, here I offer for sale original signed (in the margin) computer-generated prints of polychora nets that I found to be particularly striking or interesting during the course of my ongoing research. I have a number of these framed on my wall, where they make great conversation pieces and snazzy decorations. I use Great Stella software to produce these pictures. Most of these nets have never been seen anywhere before, and many could (if four-space were real) fold up to make polychora not previously described in the mathematical literature The prints make great presents for Christmas, Father’s Day, and so forth, especially for your math-inclined friends and relatives. If you were going to spend some bucks for, say, a SpongeBob SquarePants doll or something like that, why not spring for one of these prints instead?. They’re decorative and even educational. Proceeds from these sales will buy more time for me to work in geometry and to prepare manuscripts for publication on these and other geometric explorations that I have undertaken.

Each print is available in two versions: normal size, which fits on an 8.5"x11" page, and large size, which fits on an 11"x17" page. These are printed on glossy photoprint stock at high resolution in bright colors. The pictures below are low-resolution GIFs to save website space and decrease download time, and although they give you a good idea of what the figure and the colors look like, the print you purchase will be larger, clearer, and less “jaggy.” Normal size prints will look good in a portrait-size frame on your desk (for example), while large prints are suitable for wall mounting.

Shipping costs are included in the print price. Unless otherwise indicated at the PayPal buttons, all normal size prints are $25.00, and all large size prints are $40.00.

How to Order Prints

If you use PayPal, simply click on the Buy Now buttons for the prints you want, and PayPal will take you through the automatic payment process. If you do not use PayPal, you can still order whatever prints you like by mail. Simply write down the numbers of the prints you want, together with their sizes—normal or large—and any other information that specifies the print you want, total up the amounts, and send a check or money order (cash is OK, but there’s a theft risk sending cash by mail) payable to me, in U.S. funds drawn on a U.S. bank. Be sure to include where you want the prints sent, of course!

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.

Polychora Montage

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 3rd 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 #1 Normal Size $25.00
Print #1 Large Size $40.00

120-Cell Montage
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 #2 Normal Size with lettering $25.00
Print #2 Normal Size with no lettering $25.00
Print #2 Large Size with lettering $40.00
Print #2 Large Size with no lettering $40.00

120-Cell Tower Net
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 #3 Normal Size $25.00
Print #3 Large Size $40.00

600-Cell Fancy Net
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 #4 Normal Size $25.00
Print #4 Large Size $40.00

600-Cell Fancy Net
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 #5 Normal Size $25.00
Print #5 Large Size $40.00

53-gonal Duoprism Net
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 pxqsquares. 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 #6 Normal Size $25.00
Print #6 Large Size $40.00

Rectified Hecatonicosachoron Net
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 #7 Normal Size $25.00
Print #7 Large Size $40.00

Omnitruncated Hexacosihecatonicosachoron
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 #8 Normal Size $25.00
Print #8 Large Size $40.00

Hexacosihecatonicosachoron Net (Zoom)
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 #9 Normal Size $25.00
Print #9 Large Size $40.00

Grand Antiprism Net
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 20th 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 #10 Normal Size $25.00
Print #10 Large Size $40.00

Snub Icositetrachoron Net
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 #11 Normal Size $25.00
Print #11 Large Size $40.00

Rectified Hexacosichoron Exploded
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 #12 Normal Size $25.00
Print #12 Large Size $40.00

Modular Rox Dual Net
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 #13 Normal Size $25.00
Print #13 Large Size $40.00

Fancy Rox Dual Net
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 #14 Normal Size $25.00
Print #14 Large Size $40.00

Grand Antiprism Dual Net
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 #15 Normal Size $25.00
Print #15 Large Size $40.00

Snub Icositetrachoron Dual Net
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 #16 Normal Size $25.00
Print #16 Large Size $40.00

 Tridiminished-icosahedral Tetracontaoctachoron
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 #17 Normal Size $25.00
Print #17 Large Size $40.00

Decadiminished Rectified
Hecatonicosachoron Net
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 #18 Normal Size $25.00
Print #18 Large Size $40.00

Enneagonal Bicingulum Net
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 #19 Normal Size $25.00
Print #19 Large Size $40.00

Hatching Net
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 #20 Normal Size $25.00
Print #20 Large Size $40.00

Icositetrachoron Modular Tower Net
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 #21 Normal Size $25.00
Print #21 Large Size $40.00

Icositetrachoron Hatching Net
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 #22 Normal Size $25.00
Print #22 Large Size $40.00

Decachoron Siamese 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 #23 Normal Size $25.00
Print #23 Large Size $40.00

Tetracontaoctachoron Tower Net
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 #24 Normal Size $25.00
Print #24 Large Size $40.00

Snub-Cuboctahedral Prism Net
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 #25 Normal Size $25.00
Print #25 Large Size $40.00

Rectified Hexacosichoron
Modular Tower Net
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.

Print #26 Normal Size $25.00
Print #26 Large Size $40.00

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.