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.

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.

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:

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.

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

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.

Cut, score, and fold as shown above.

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.

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.