Fold Me!

Last year, Jiangmei Wu and I worked on some infinite polyhedra that can be folded into two different planes. Today, you get the chance to make your own (finite version of it). This is a simple craft that, time and energy permitting, will be featured at a fundraiser for the WonderLab here in Bloomington. You will need 3 (7 for the large version) sheets of card stock, scissors, a ruler and craft knife for scoring, and plenty of tape. A cup of intellectually satisfying tea will help, as always. 

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Begin by downloading the template, print the first three pages onto card stock, and cut the shapes out as above.  Lightly score the shapes along the dashed and dot-dashed line, and valley and mountain fold along them.  Note that there are lines that switch between mountain and valley folds, but all folds are easy to do.

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The letters come into play next. Tape the edges with the same letters together. Begin with the smaller yellow shape, and complete the two halves of the larger blue shapes, but keep them separate for a moment, like so:

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Stick the yellow piece into one of the blue halves, this time matching the digits. Complete the generation 2 fractal by taping the second blue half to the yellow generation 1 fractal and the other blue half.

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This object can be squeezed together in two different planes. Ideal for people who can’t keep their hands to themselves. 

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The next 2 pages of the template repeat the first three without the markings, if you’d like to build a cleaner model. You then need two printouts of page 5. The last page allows you to add on and build the generation 3 fractal. You need 4 printouts. 

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Cut, score, and fold as shown above.

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Again, tape edges together as before. There are no letters here, but the pattern is the same as before. Finally, wiggle the generation 2 fractal into the new orange frame, as you did before with the yellow piece into the blue piece.

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Here is how they now grow in our backyard. If anybody is willing to make a  generation 4 or higher versions of this, please send images.

All these polyhedra have as boundary  just a simple closed curve. Topologists will enjoy figuring out the genus.

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Hidden Simplicity (Maybe-Ferns 5)

Mathematicians like to do things a little differently. An excellent example was the Mathematische Arbeitstagung, a yearly event held in Bonn, where the (mathematical) audience was asked to publicly suggest speakers.

Hirzebruch

Friedrich Hirzebruch would write the suggested names on the board (he sometimes misheard…), and then create a list of speakers on the fly. Sometimes they ended up with unexpected results. One year, Michael Barnsley was suggested, who had been working on a new fractal image compression method.

Nonfern4

His talk was exciting for us graduate students, because we for once could understand something. The idea was to use special types of iterated function systems: Take a few linear maps that are all contractions, and use them to map a subset  of the plane to the union of the images of that set under all the linear maps. This becomes a contraction of the space of closed subsets of the plane to itself with respect to the Hausdorff distance, and hence has a fixed point, which is again a subset of the plane.

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It turns out that these subsets are highly complicated fractals, encoded just by a few numbers. For instance, all images on this page (except for the photo of Hirzebruch at the top) were made with just two linear maps, requiring 12 decimal numbers.

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Barnsley claimed that he could reverse engineer this: Start with an image, and find a small collection of linear maps that would produce the given image very accurately. If true, this would revolutionize image compression.  We went home and tried it out on our Atari ST computers and the likes. All we could produce were ferns, twigs, and leaves.

Nonfern1

Paul Bourke has a nice web site where he explains how one can design some simple fractals, and has also some very impressive images of ferns using four and more linear maps. Below are the two simple maps used to create the polypodiopsida psychedelica above.

Formula

Carpets (Foldables 4)

The last (for now) example in this series of bifoldable designs is a woven carpet. Will create a doubly periodic polyhedron that consists of the Miura tubes below (which are almost 50 years old!).MiuraTube

We begin with a corner type we call Double L

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Four copies of it (using reflecions) can be combined into a translationa fundamental piece like so:

MiuraWeave 2

The tubes (of double length) emerge when we replicate this piece several times in both directions:

MiuraWeave 3

Above is its most symmetric state. This carpet does not need to be rolled, it can be squeezed in both of its translational directions, as below:MiuraWeave 4

So you can push this Miura Carpet to any of the four sides of a room.

Dos Equis (Foldables 3)

Whenever you show a mathematician two examples, s(he) wants to know them all. So, after the introductory examples of Butterfly and Fractal it’s time to make something more complicated. Jiangmei and I started by classifying all possible vertex types that can occur when you build polyhedra using only translations of four of the six types of faces of the rhombic dodecahedron (and make sure they attach to each other as they do it there). We found 14 different ones, and a particularly intriguing one is what we called the X:

TripleX1The central vertex has valency 8, and we were wondering whether we could use it to build a triply periodic bifoldable polyhedron. It is easy to combine two such Xs to a Double X:

TripleX2

One can then put a second such Double X (with the order of the Xs switched) in front. Note that these are still polyhedra. Below are two deformation states of these quadruple Xs. We see that they are quite different.

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So far, the construction can be periodically continued up/down and forward/backward. It is also possible to extend to the left/right, and there are in fact two such possibilities, allowing for infinite variations, because one has this choice for every left/right extension. They are indicated by the arrays below. 

TripleX3carrowIf you don’t have the time to build your own model, here again is a movie showing the unfolding/folding of a rotating Dos Equis.

 

The Fractal (Foldables 2)

The second bifoldable object Jiangmei showed me was this:

Fractal 1

You can find a movie showing how this folds together in two ways here. To understand how and why this works, let’s first look at a simple saddle:

Saddle

This is a polyhedron with a non-planar 8-gon as boundary. Its faces are precisely the four types of faces that are allowed in our polyhedra: All others have to be parallel to these four. The four edges that meet at the center of this saddle constitute the star I talked about the last time. Again, all edges that can occur must be parallel to one of these four. One can fold the saddle by moving the upwards pointing star edges further up (or down), and the downwards pointing edges further down (or up), thereby keeping the faces congruent. This works locally everywhere and therefore allows a global folding of anything built that way. Fractal 0

For instance, the hollow rhombic dodecahedron above can be bi-folded. Now note that this piece is also a polyhedron with boundary. In fact, its boundary is exactly the same octagon as the boundary of the saddle. 

Observe also that at the center of this piece we have a vertex in saddle form. This suggests to subdivide all rhombi into four smaller rhombi, remove the saddle an the middle vertex of the doubled hollow dodecahedron, and replace it by a copy of the standard hollow dodecahedron. This gives you Jiangmei’s fractal. Repeating this is now easy. Below is the generation 2 fractal (animation):

Fractal 2

And, just for fun, the generation 10 fractal:

 Fractal 10 colorgradient

You can see it being bifolded here. So far, the two completely folded states of our polyhedra looked very much the same. We will see next week that this doesn’t need to be the case.

The Butterfly (Foldables 1)

We all know that cardboard cubes are rigid, which is why we get our packages in boxes. We also all know that if we remove two opposite faces from a cube, we can fold it together. This started to interest me when I noticed that the polyhedral approximation of the Schwarz P surface is surprisingly flexible. This summer, I showed this to our local Origami and Paper Folding expert, Jiangmei Wu from our School of Art and Design, and she became interested. A few days later she came with a paper model that looked like this:

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She called it a simple variation of the polyhedral P-surface. Hmm. This is a triply periodic polyhedral surfaces tiled with rhombi. To understand it, we build it out of smaller units (which we called butterflies):

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The really cool thing about it is that it can be folded together in two different ways, like so:

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You can find an animation showing the continuous deformation here. We stared at this for a (long) while, until we realized that this has to do with rhombic dodecahedra. The structures up above are composed of the rhomboids from last week that tile a rhombic dodecahedron. The latter has, as the name hints, 12 faces, which occur in opposite pairs. Like the cube, it is rigid per se, but becomes foldable if we remove two pairs of parallel faces, leaving us with four faces to use, which are distinguished by color up above.Fractal0b

Above you can see the four hollow parallelepipeds (which we called hollowpeds). The almost trivial but nevertheless mind bending realization is that everything you build out of these hollowpeds becomes a structure foldable in two different ways. Next week I’ll show Jiangmei’s second model, a foldable fractal… If you can’t wait, check out this.

 

 

 

A Double Figure 8

Recently, a local artist had an intriguing question. Suppose you have a hook in the ceiling (who hasn’t?), and  two spot lights in front of the hook, slightly to the left and to the right. Suppose also that you have drawn two curves on the back wall. Can you bend a wire and suspend it from the hook so that the two projections match the drawings?

 

Sketch

I first thought: Yes, this means we just have to determine the intersection of two cones, so this is possible but maybe tricky.

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After playing around with it a little I realized that this is simpler than I thought: Of both curves have the same height, this is essentially always possible, and even completely explicit. In fact, this is almost as simple as using two perpendicular parallel projections.

For instance, below you see a single red wire that has two figure 8 curves as projections.

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Then of course one wants to play with it and rotate the wire.

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Clearly, there are two more rotational positions where one of the projections is again a figure 8, the one above and the one below.

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Now we need to find somebody who can accurately bend wires for us.