Those who have spent as many hours as I have watching maths YouTube videos may well have come across the idea of a flexagon. Flexagons are origami-like models of 2D shapes, normally made out of strips of paper. As the name suggests, they can be folded (or flexed) to reveal new faces not previously seen. Flexagons hide their complexity in layers of faces, but the 3D analogy of a flexagon—a flexahedron, or sometimes called a tri-kaleidocycle—puts that complexity on show.
As a lover of polyhedra, these 3D flexagons fascinated me and ever since I first heard the word flexahedron, I’ve had ambitions of designing my own. These shapes are so beautiful that I can’t understand why anyone wouldn’t want to!
The first flexahedron I ever saw was the (disgustingly named) infinity cube, which I came across when I was 11. Initially, it appears to comprise of eight small cubes in a $2\times2\times2$ arrangement. On further inspection though, its components are connected together in such a way that twisting them morphs from large cube, to cuboid, and back to cube through a number of different transitions.
An infinity cube
The Yoshimoto cube—designed by Naoki Yoshimoto—builds on the ideas of the infinity cube, but the result is something even more beautiful. Each of the small cubes in the infinity cube is dissected through three of its diagonals, with the entire shape splitting into two parts that each move in the same way as the whole. What’s more, when folded up individually each of these creates a stellated dodecahedron.
One half of a Yoshimoto cube at various points in their flexing journeys
Naoki Yoshimoto, whose first cube was unveiled in his From Cube to Space exhibition, also designed two further flexahedra based on the cube. These both consist of circles of polyhedra connected at their edges, able to form exciting shapes as they cycle and fold. Each of these flexahedra is made of a loop of identical (up to reflection) polyhedral modules joined together in a repetitive way, giving them a pleasing symmetry.
The two halves of a Yoshimoto cube
We can make all these models using nets. For the original Yoshimoto cube, we make eight modules from this net: 
We can work out the relative lengths of the sides—every edge of the module is either an edge of the cube it dissects, or lies along a diagonal of the cube. A quick application of Pythagoras’ theorem tells us that the diagonal is $\sqrt3$ times the length of a side, so the equal sides of each isosceles triangular face are $\sqrt3/2$ times the cube side length.
Before designing my own flexahedron—which I’d decided would be an imitation of a Yoshimoto cube—I had to first consider the necessary qualities. Symmetry seems to be crucial to a good flexahedron: all those that we have encountered so far are made of either identical modules, or equal numbers of mirror-image modules. This both makes the shape feel purposeful and allows it to exploit the quirks of geometry. Another important aspect of all the flexahedra we have discussed is their ability to cycle: we can repeat the same mechanical moves without ever having to reverse them. For me this is vital, as it gives a certain freedom of movement, and contributes to the fluidity of the shape. One of the most impressive things about the original Yoshimoto cube is that it seems as if it achieves this with nothing to spare—we know the modules are arranged in a ring in terms of how they are connected, but there is never a gap between its parts as it folds. Finally, I want there to be something interesting about the shape: maybe it folds into many patterns or maybe it moves in a fascinating way similar to the original Yoshimoto cube.
I wanted the flexahedron to be able to take the form of a `nice’ polyhedron in the same way that the Yoshimoto cube is based around (to nobody’s surprise) a cube. I’ll call this our basis polyhedron. It made sense to begin my construction here, so naturally, I turned my attention to the platonic solids. Yoshimoto had already created a flexahedron with a cube as its basis polyhedron, so I wanted to choose one of the remaining four platonic solids to build my model from. Given the dual of the cube is an octahedron (ie joining the centres of the faces of a cube gives an octahedron and vice versa), I decided that it was the octahedron that seemed most promising.
Considering the symmetries of the basis polyhedron is important, as it suggests a natural dissection, namely cutting the shape along every plane of symmetry. This, however, generally creates more modules than is ideal for the final flexahedron. When a flexahedron has more than an arbitrary limit of, say twenty modules, the final shape resembles more of a floppy loop than anything else. Dissecting a cube or octahedron along its planes of symmetry splits each into 48 parts—far more than is ideal! We therefore seek structure amongst this dissection.
Splitting an octahedron into 48 parts along
every plane of symmetry
After making the 48 sections of an octahedron and experimenting, I found two possible useful sets of modules. The first breaks the octahedron into 16 (non-regular) tetrahedra with this net: 
The faces of each tetrahedron are right-angled and distinct from the others in the same tetrahedron. To form an octahedron, we need eight copies of this tetrahedron, and a further eight of its mirror image.
An octahedron with one of the 16 tetra-
hedra removed
The modules are joined together along their edges, in what I shall call a connection. To get the fluid loop that I want for my flexahedron, I need to ensure that no module face has more than one connection. The two module connections must therefore be on opposite edges, since the modules are tetrahedra. In order for adjacent modules to connect nicely, ie in order to ensure that equivalent faces on each module are adjacent after joining, we must have each shape connected to its mirror image. This means that after we connect the first two modules, we are forced to make the rest of the ring in a certain way, alternating mirror images and always connecting on opposite edges. In a tetrahedron, there are three sets of opposite edges, meaning there are three different ways to attempt to make a loop. Happily, all three work. Surprisingly, changing just the connection edge creates quite a different octa-flexahedron!
Three octa-flexahedra
This isn’t quite what’s on the cover though: the cover shows a net of a flexahedron made of 12 hexahedral modules, each of which includes an edge of the basis octahedron. Each triangular face of the octahedron is formed by three triangles, meeting at the centre of the face of the octahedron: each module has two of these triangles as faces, and four other faces that are on the interior of the octahedron.
Splitting the octahedron into 12 hexahedra
A net of a hexahedron, and a hexahedron
These modules are symmetric, but the connected edges are not symmetrically arranged—there is some chirality here! There are two ways that this arrangement can form an octahedron, and the curved pattern drawn on the surface on the cover distinguishes them. In one orientation, a circle is formed on a face, in the other the lines connect differently and no circle is formed.
While making these shapes module-by-module is practical, one can also create a giant net from which the whole thing can be folded. Doing this for the stretchy flexahedron produced the design on the cover. The thick cyan lines on the image show where the modules connect, and thus the position of the fabric hinges.
So have a go! I’ve achieved my aim of designing my own flexahedron; now I pass the baton over to you. You can cut out the cover and fold along the lines to recreate my flexahedron. Or if you want a real challenge, make your own!
The octa-flexahedra as seen on the cover
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