Flexagons as Polygon Rings

Les Pook has written two classic books on flexagon theory. His second book, Serious Fun with Flexagons, is a rigorous, comprehensive analysis of many kinds of flexagons, the nets from which they are constructed and some of their interesting properties.

In this and the following articles we’ll learn a bit about Pook’s work.

Flexagons as Polygon Rings

Les Pook’s approach is extremely systematic, focusing on the structure of all possible flexagons. Very broadly speaking, his definition of a flexagon (in its main state) is a polygon ring where each polygon is a stack of hinged leaves and the polygons are hinged either along their edges or at their vertices. The angle between two edges is known as the hinge angle.

The use of the word ring may be a bit misleading, since when you fold up a tri-hexa-flexagon for example, it doesn’t have a “hole” in the middle, but mathematically it is still a ring, and in any case, if you “unruffled” it you will get a twisted ring of leaves – as you can see in the top two left hand photos in the image above (1 and 2).

Some of Pook’s flexagons really are rings even in their main state, like the flexagon whose main state is a ring of hexagons in the image above (3). For fans of nomenclature, this is the octadecagonal ring hexagon hexa-hexaflexagon! It’s worthwhile noting that Pook’s flexagon definition as polygon rings allows for skew flexagons – flexagons that don’t lie flat.

Polygon Rings

There are different kinds of polygon rings depending on the number and shape of the polygons that make them:

• Regular even edge polygon rings – these consist of an even number of identical, convex, regular polygons hinged at their edges, all of them at an equal distance from the centre of the ring. A regular polygon is a polygon whose angles are all the same and whose sides are all equal, for example, an equilateral triangle, a square, a regular pentagon etc. All hinge angles are the same. The tri-hexa-flexagon and the octadecagonal ring hexagon hexa-hexaflexagon are both examples of regular even edge polygon rings. Note that the word “regular” in “regular even edge polygon rings” refers to the overall shape of the ring, not the individual polygons that make it (although they too are regular polygons).
• Regular odd edge polygon rings – the same as regular even edge polygon rings, except that there is an odd number of polygons in the ring. In the image above (8) you can see the main state of a hepta-flexagon which is a ring of seven equilateral triangles. This flexagon cannot lie flat – it is a skew flexagon! In fact, the first flexagon that has a flat odd edge polygon ring has a 20-gon as its constituent polygon!
• Compound polygon rings – these consist of an even number of identical, convex, regular polygons hinged at their edges, all of them at an equal distance from the center of the ring. Alternate hinge angles are the same. A flexagon whose main state is a compound ring of eight squares is shown in the image above (5).
• Irregular, polygon rings – rings that are neither regular nor compound. The six square main state of the flexagon in the image above (6) is an irregular ring, because not all the squares are at an equal distance from the center of the ring.
• Rings of irregular polygons – not to be confused with irregular polygon rings, these are rings made of irregular polygons i.e. non equilateral triangles, rectangles etc. The main state of Ann Schwartz’s octa-dodeca-flexagon, made from silver 45-45-90 triangles, shown in the image above (7), is one such a ring.
• Vertex polygon rings – rings of polygons hinged at their vertices instead of their edges. A vertex ring is shown in the image above (4). There are regular, compound, irregular polygon and irregular vertex rings. Since it’s practically impossible to create a model with joined vertices that don’t tear, we’ve added small tabs to each vertex as the joint.

Flexagon Sectors

Les Pook builds flexagons from templates, constructed from a minimum number of what’s known as sectors. A flexagon in its main state can usually be divided into identical units. These units are called sectors. The tri-hexa-flexagon, for example, is made out of 3 identical sectors joined at a common edge. Each sector has two pats, one of them with 1 equilateral triangle leaf, and the other with 2 equilateral triangle leaves.

The difference in the number of leaves in each pat is the reason why there are 3 sectors and not 6. The sectors must be completely identical. The hexa-hexa-flexagon also has 3 sectors. In the next step we’ll learn how to make different families of flexagons from Pook’s sector templates – or as he calls them – fundamental nets.