October 3, 2007 8:24 PM Subscribe

The Johnson Solids are a set of 92 semi-regular polyhedra, all of which are uniquely named and numbered. Except for the familiar square pyramid they all have exotic names like the Hebesphenomegacorona. A Hebesphenomegacorona in space. Number 26, the Gyrobifastigium, is unique in that if copies of itself are properly stacked together they will leave no gaps, thus making it the only space filling Johnson Solid.

posted by Tube (28 comments total) 16 users marked this as a favorite

posted by Tube (28 comments total) 16 users marked this as a favorite

Too bad it was wrong. Planetary orbits are not circular

.

posted by Brian B. at 8:39 PM on October 3, 2007

I see your Johnson and raise you a Wenninger. (It only sounds dirty.)

Thanks; this is neat stuff. Time to make more models.

posted by phooky at 8:51 PM on October 3, 2007

Thanks; this is neat stuff. Time to make more models.

posted by phooky at 8:51 PM on October 3, 2007

So, can someone very patient explain to me why there are so few space filling polyhedra? The Gyrobifastigium consists of two triangular prisms. Couldn't there be any number of similar conflations?

posted by roll truck roll at 8:54 PM on October 3, 2007

posted by roll truck roll at 8:54 PM on October 3, 2007

"jonson," solid.

posted by humannaire at 8:59 PM on October 3, 2007

posted by humannaire at 8:59 PM on October 3, 2007

Nice post, but with a name like Hebesphenomegacorona I was expecting something much, much more metal. I mean as in mathematicians should have gone insane from contemplating the complexity, or something.

posted by TheOnlyCoolTim at 9:01 PM on October 3, 2007

posted by TheOnlyCoolTim at 9:01 PM on October 3, 2007

Looking at the Gyrobifastigium it's a pleasant exercise to visualize stacking a space with them.

Nice.

posted by jouke at 9:04 PM on October 3, 2007

Nice.

posted by jouke at 9:04 PM on October 3, 2007

If you'd like to play around with polyhedra, get yourself a free copy of Poly.

posted by hydrophonic at 9:12 PM on October 3, 2007 [1 favorite]

posted by hydrophonic at 9:12 PM on October 3, 2007 [1 favorite]

Unfortunately, we live in a square, square world, so stacked gyrobifastigiums would leave little triangles next to the floor and walls. And they'd slide out. I'll stick to rubbermaids.

posted by anthill at 9:13 PM on October 3, 2007

posted by anthill at 9:13 PM on October 3, 2007

roll truck roll: Well, there's only one space-filling Johnson polyhedron because the definition of a Johnson polyhedron insists that it is convex but rules out the obvious space-filling convex polyhedrons (prisms, mostly). It's also already hard to fill space with (semi-, quasi-, double-latte-)regular polyhedra because of all the constraints: the angles between faces around an edge need to be able to add up to 2π; the solid angles need to be able to add up to 4π, etc. etc. Since there's a finite number of Johnson polyhedra and the definition rules out the easy stuff, it's not too surprising that there's only one that fills space. Did that make sense? *(Scratches own head)*

posted by phooky at 9:18 PM on October 3, 2007 [1 favorite]

posted by phooky at 9:18 PM on October 3, 2007 [1 favorite]

A soccer ball pattern, made of pentagons and hexagons, seems to fit the definition of a Johnson solid as I understand it, but I don't see any such shape in the list. What am I missing?

posted by scottreynen at 9:37 PM on October 3, 2007

posted by scottreynen at 9:37 PM on October 3, 2007

Very true, though a model of stacked Gyrobifastigiums I made with toothpicks and blobs of wax suggests that it might make an interesting "jungle gym".

posted by Tube at 9:40 PM on October 3, 2007

scottreynen: A soccer ball pattern is an Archimedian solid because each vertex has the same number of equilateral faces around it.

As someone once said, "If you think the shape of space doesn't constraint you, try making a fifth Platonic solid."

posted by lupus_yonderboy at 9:50 PM on October 3, 2007

As someone once said, "If you think the shape of space doesn't constraint you, try making a fifth Platonic solid."

posted by lupus_yonderboy at 9:50 PM on October 3, 2007

All puns aside, Robert Webb's affordable Stella Windows apps generate nets to construct colorful paper Johnson Solids and other polyhedra. See his gallery — including models by Father Magnus Wenninger — and model-making tips.

posted by cenoxo at 9:53 PM on October 3, 2007

posted by cenoxo at 9:53 PM on October 3, 2007

I'm glad the space program is going so well we can afford to send astronauts up there to goof around.

posted by TheOnlyCoolTim at 10:11 PM on October 3, 2007

posted by TheOnlyCoolTim at 10:11 PM on October 3, 2007

"What's a hebesphenomegacorona doing onboard the ISS? We're not sure."

Neither am I. Secretly I was hoping that by "a hebesphenomegacorona in space" you meant that astronauts had spotted one of these things floating around out the window. The universe is large enough it seems like there has to be at least one wayward celestial hebesphenomegacorona out there, waiting to be discovered.

posted by tepidmonkey at 1:06 AM on October 4, 2007

Neither am I. Secretly I was hoping that by "a hebesphenomegacorona in space" you meant that astronauts had spotted one of these things floating around out the window. The universe is large enough it seems like there has to be at least one wayward celestial hebesphenomegacorona out there, waiting to be discovered.

posted by tepidmonkey at 1:06 AM on October 4, 2007

SNUB SQUARE ANTIPRISM

SNUB SQUARE ANTIPRISM

SNUB SQUARE ANTIPRISM

SNUB SQUARE ANTIPRISM

SNUB SQUARE ANTIPRISM

SNUB SQUARE ANTIPRISM

SNUB SQUARE ANTIPRISM

SNUB SQUARE ANTIPRISM

posted by RTQP at 1:29 AM on October 4, 2007 [1 favorite]

SNUB SQUARE ANTIPRISM

SNUB SQUARE ANTIPRISM

SNUB SQUARE ANTIPRISM

SNUB SQUARE ANTIPRISM

SNUB SQUARE ANTIPRISM

SNUB SQUARE ANTIPRISM

SNUB SQUARE ANTIPRISM

posted by RTQP at 1:29 AM on October 4, 2007 [1 favorite]

These give me the horn.

posted by Henry C. Mabuse at 2:10 AM on October 4, 2007

posted by Henry C. Mabuse at 2:10 AM on October 4, 2007

Got this for my first wedding anniversary, and been messing with it quite a bit. Highly recommemded:

A Constellation of Origami Polyhedra

posted by Roger Dodger at 4:37 AM on October 4, 2007 [1 favorite]

A Constellation of Origami Polyhedra

posted by Roger Dodger at 4:37 AM on October 4, 2007 [1 favorite]

The story I've heard about how it was proved that there are only 92 Johnson polyhedra is that some Russian mathematician, when he was trying to prove that there are only 92, divided up the problem into a whole bunch of small chunks and farmed out the pieces (redundantly) to a bunch of Russian schoolchildren. Sort of like a student-powered brute force solution.

Norman Johnson is still alive and being a productive mathematician. He still attends conferences, and he's been working for a very long time on producing the definitive work on Uniform Polyhedra. This summer, he said that he's been encouraged by the publisher to break it up into a couple of parts, so maybe we'll see some of it sooner than later (which would be very cool, if you're interested in Uniform Polyhedra).

posted by leahwrenn at 5:54 AM on October 4, 2007

Norman Johnson is still alive and being a productive mathematician. He still attends conferences, and he's been working for a very long time on producing the definitive work on Uniform Polyhedra. This summer, he said that he's been encouraged by the publisher to break it up into a couple of parts, so maybe we'll see some of it sooner than later (which would be very cool, if you're interested in Uniform Polyhedra).

posted by leahwrenn at 5:54 AM on October 4, 2007

Elongated birotunda describes me perfectly.

Very cool post, Tube.

posted by Cat Pie Hurts at 6:11 AM on October 4, 2007

Very cool post, Tube.

posted by Cat Pie Hurts at 6:11 AM on October 4, 2007

This is going to make my Magnetix fiddling so much more mathematically impressive.

posted by Rock Steady at 6:17 AM on October 4, 2007

posted by Rock Steady at 6:17 AM on October 4, 2007

I think I still have some old Ikoso kits somewhere. I hadn't really explored any of the Johnson solids with those, but I had built all of the Platonic solids and some of the smaller Archimedean solids with those.

When built with kits like this, which have edges of fixed length, but flexible (or at least somewhat flexible) vertices, solids consisting entirely of triangular faces are rigid, while others are not. So I decided to make some of those other figures stable by adding pyramids to square and pentagonal faces. (I had some ideas for more complex structures which could stabilize even larger faces, but I never got around to actually trying them out.) In order to maintain the convex hull, I made the added pyramids inverted - on the inside of the solid.

Got some very neat structures that way. Adding inverted pyramids to a cube, the point of the pyramid is just a little bit beyond the center of the cube, so the six inverted pyramids all intersected each other. Adding them to the Platonic dodecahedron, it turns out you get multiple faces from different pyramids in the same plane, making a very pretty structure, with some internal five-pointed, planar stars! Meanwhile, the distance from any vertex of a cuboctahedron to its center is exactly the same as the length of an edge (the edges make up four interesecting regular hexagons in planes passing through the center), so stabilizing that entails adding just a single extra connector at the center, and connecting every vertex to it!

Another neat group is the Catalan solids. Unlike the Platonic, Archimedean, or Johnson solids, their faces are not regular polygons, and their edges are not necessarily all the same length, but they do make fair dice (along with Platonic solids, disphenoids bipyramids, and trapezohedra), unlike Archimedean or Johnson solids. In fact, gamers may be familiar with the rhombic triacontahedron which is used for a d30.

posted by DevilsAdvocate at 9:16 AM on October 4, 2007

When built with kits like this, which have edges of fixed length, but flexible (or at least somewhat flexible) vertices, solids consisting entirely of triangular faces are rigid, while others are not. So I decided to make some of those other figures stable by adding pyramids to square and pentagonal faces. (I had some ideas for more complex structures which could stabilize even larger faces, but I never got around to actually trying them out.) In order to maintain the convex hull, I made the added pyramids inverted - on the inside of the solid.

Got some very neat structures that way. Adding inverted pyramids to a cube, the point of the pyramid is just a little bit beyond the center of the cube, so the six inverted pyramids all intersected each other. Adding them to the Platonic dodecahedron, it turns out you get multiple faces from different pyramids in the same plane, making a very pretty structure, with some internal five-pointed, planar stars! Meanwhile, the distance from any vertex of a cuboctahedron to its center is exactly the same as the length of an edge (the edges make up four interesecting regular hexagons in planes passing through the center), so stabilizing that entails adding just a single extra connector at the center, and connecting every vertex to it!

Another neat group is the Catalan solids. Unlike the Platonic, Archimedean, or Johnson solids, their faces are not regular polygons, and their edges are not necessarily all the same length, but they do make fair dice (along with Platonic solids, disphenoids bipyramids, and trapezohedra), unlike Archimedean or Johnson solids. In fact, gamers may be familiar with the rhombic triacontahedron which is used for a d30.

posted by DevilsAdvocate at 9:16 AM on October 4, 2007

A new friend of mine told me recently that he made this incredible Modular Pie-cosahedron for Thanksgiving last year. Now if that's not a perfect solid...

posted by croquette at 9:58 AM on October 4, 2007

posted by croquette at 9:58 AM on October 4, 2007

My apartment has lots of weirdly angled walls. The mater bathroom has 17 corners. Your typical gyrobifastigium might be a little less disturbed by this than I am.

posted by Foosnark at 2:55 PM on October 4, 2007

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v.cool, thanks!

posted by kliuless at 8:31 PM on October 3, 2007 [2 favorites]