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.

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:


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.


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:

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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:


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):

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And, just for fun, the generation 10 fractal:

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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.