Origami polyhedra

[mmcirvin]

Here's a page on origami polyhedron modules. The one I recently learned how to work with is the very simple Sonobe module, which my coworker Bill Barnert showed us how to produce one day at lunch (Bill had a big pile of leftover handbills for a Lions Club carnival that we used for the pieces).

With the exception of the "Epcot ball", which has plane tesselations on the large faces, the shapes on that second page are all cumulations of Platonic solids with triangular faces, where the pyramid used for the cumulation is a cube-corner. You need one module for every edge in the uncumulated polyhedron, so the little cube (which is really a cumulated tetrahedron) uses 6, the octahedron uses 12, and the icosahedron 30. The icosahedron is an interesting toy, since you can push some of the vertices inward to make it undergo startling transformations, including one form that looks like a saucer-shaped cluster of cubes.

I figured out that you can also make flat square faces by linking together four of the modules instead of three (though they pull apart easily, so shapes that use them hold together best if they've also got pyramidal faces). You can actually, I think, make cubes of at least three different sizes: that little one that is a cumulated tetrahedron, a bigger one made from 12 modules with the flat square faces, and an even bigger one from 24 modules that is a cuboctahedron with the triangular faces cumulated. (Even bigger flat-faced cubes would require square tesselations on the faces and would probably be extremely flimsy.)

You can go beyond that by using other non-Platonic polyhedra as a basis. A simple, strange-looking one I did was the gyrobifastigium, which I've known about ever since Kibo taught me that it's fun to say "gyrobifastigium". What I really want to make someday is the 60-module Sonobe version of the snub cube, which I suspect would be awesome, kind of like the aftermath of six cubes violently colliding in hyperspace.

It's also interesting to think about the minimal nontrivial shapes made from these Sonobes. You can hook two of them together to make a flat (but thick) square, and three to make a sort of tiny, squat triangular dipyramid that is the cumulation of a flat triangle (or, I suppose, really two flat triangles face to face).

Comments

[ronebofh.livejournal.com]

Mathworld's polyhedron pages are one of the least suckful uses of Java in existence.

Also, there is no gyrobifastigium user on LiveJournal.

[paracelsvs.livejournal.com]

They still use Java, though, which doesn't work on this LUNUX MACHENE for whatever reason that I am too lazy to work out.

If I wasn't so lazy, I would try making a Javascript-and-<canvas> version of the polyhedra viewer.

[doctroid.livejournal.com]

Modular origami polyhedra... fun stuff. I made a Christmas tree topper once, using wrapping paper. I have this book which is the bible of the field, apparently.

What I find really remarkable are this book and this one, which show how to make all the Platonic polyhedra and quite a few others without modular techniques -- using a single square sheet of paper, no cuts.

I have the first book and I've found even some of the ones marked as simpler to be very challenging. I'm sure folding the dodecahedron qualifies you for a black belt in origami. And designing the model in the first place probably puts Montroll squarely in the legendary category.

The king of Lineland speaks

[mmcirvin.livejournal.com]

I think the little flat square can be seen as the cumulation of a line segment.

[mmcirvin.livejournal.com]

Also, I think Lavavej is right about needing 270 pieces for the Epcot Ball. It's one per edge. The 12 pentagonal clusters have 5 internal and 5 external edges each for a total of 120, then each of the 20 hexagonal clusters has 6 internal edges for another 120, and the hexagons meet each other in 30 additional edges corresponding to the edges of an icosahedron for a total of 270.

[mmcirvin.livejournal.com]

Many pretty variants of the Sonobe cube (http://lhs1701.freewebpages.org/sonobe_unit.html), and instructions for some of these and many other things (http://home.comcast.net/~meenaks/diagrams/).

[mmcirvin.livejournal.com]

If the Web server is being stupid on that first link, try refreshing.

[paracelsvs.livejournal.com]

Man, I want to do the Magic Rose Cube. But the INTERNET has eaten the pages!

[paracelsvs.livejournal.com]

UPDATE: Internet defeated, Magic Rose Cube made!

It's a cube! And it's a rose!

[mmcirvin.livejournal.com]

The different-looking variations on that page mostly, I think, have to do with the precise nature of the fold used to make the pocket into which the tab on an adjoining piece goes. I get the impression that the corners on Sonobe's original module didn't have the second "needlenose paper airplane" fold described on the Epcot Ball page; that would make tabs and pockets that are less complex looking but maybe slightly harder for beginners to put together. Making those corner folds in other directions or combinations could produce interesting two-tone effects with origami paper that has two different sides.

[paracelsvs.livejournal.com]

I just made an octahedron. I want to make an icosahedron, but the main problem is that I am lazy.

[paracelsvs.livejournal.com]

UPDATE: I made an icosahedron to pass the time while watching the Eurovision Song Contest (Which made NO SENSE WHATSOEVER), and it's totally awesome! The alternate configurations are neat.

Do keep us updated if you ever make the snub cube, and if it's worth it.

[mmcirvin.livejournal.com]

While you are folding all the pieces, you can enliven the process by figuring out what crazy things you can put together from the number of pieces you currently have (or just one or two more).

[mmcirvin.livejournal.com]

Rosa Sanchez has a much more elaborate cumulated snub cube made from a more complex module on this page (http://home.earthlink.net/~rosa_sanchez/digital/page11.html).

[mmcirvin.livejournal.com]

...Also, the bird made from basic Sonobe modules on that page is great.

[mmcirvin.livejournal.com]

...also, holy crap look at that (http://home.earthlink.net/~rosa_sanchez/digital/page13.html).

[paracelsvs.livejournal.com]

Now where do you get the folds for that spinning top? Google has never heard of Manpei Arei, not even on that page.

[mmcirvin.livejournal.com]

I don't know!

[mmcirvin.livejournal.com]

The Lewis Simon orb on that page is identical to a commercial assemble-it-yourself Christmas ornament that I had many years ago.

[paracelsvs.livejournal.com]

I really want the orb now.

[mmcirvin.livejournal.com]

I don't think it's hard to construct. It's basically a cube woven out of three bands connected into loops, with strategically located semicircular creases. The hard part is just the little bit of geometry to figure out how large everything has to be.

[mmcirvin.livejournal.com]

I finished the snub cube, but don't have any good photos of it yet. It's large but not that spectacular; the large size and square faces make it less structurally sound than the icosahedron, and it tends to deform under its own weight. But I think you could make a more attractive one by folding the units better, and maybe using three or six colors of paper to accentuate the (nearly) cubic symmetry.

[paracelsvs.livejournal.com]

Also, it occured to me that if you make the final creases on the Sonobe module in the opposite directions, you the pyramids to face inwards instead.

Furthermore, it would then be theoretically possible to construct the usual shapes INSIDE-OUT, giving a them a much different look, with no visible pockets and tabs, and more solid colours. The trick, of course, is to get the final folds in place. I'm pretty sure somebody could do it, but not me.

[paracelsvs.livejournal.com]

Update: I just used this trick to make a cube with solid-colour faces out of six reverse Sonobe module. As expected, the final pocket is hard, but sort of possible. However, the cube does not hold together nearly as well as the normal version.

[mmcirvin.livejournal.com]

I think you can do that more easily by using something more like Sonobe's original module, which lacked the second "paper airplane" fold for the initial corner.

[paracelsvs.livejournal.com]

I tried re-folding the modules, and it was maybe marginally easier, but it still doesn't hold together very well. I think it works best with very well-folded modules.

[paracelsvs.livejournal.com]

Took some photos with the horrible phone camera, and posted them here:

http://magnesium.net/~dag/kibo/res/206.html

Features the usual octahedron and icosahdron, the inverse cube, and the Magic Rose Cube.

[mmcirvin.livejournal.com]

I learned from here (http://www.geocities.com/SoHo/Studios/8012/creation/sonobe.html) that the 3-module dipyramid is called "Toshie Takahama's Jewel" (Takahama is sometimes credited as co-inventor of the Sonobe module).

[mmcirvin.livejournal.com]

...though the PDF there also erroneously refers to these cumulated polyhedra as "stellated", an error also made by the Mathematica polyhedra library (and corrected by Mathworld). Some cumulations are also stellations (e.g. the stellated dodecahedron is both), but not all cumulations are stellations and not all stellations are cumulations.

[mmcirvin.livejournal.com]

and I think it's wrong about the number of units you need for the spiked pentakis dodecahedron (it would be 90, not 60, right?)