Cloverleaf Flexagons

Flexagons are a pretty famous area of recreational mathematics. They are strange objects comprising loads of folded paper that look quite rigid. They can, however, be flexed in some way to reveal faces previously hidden inside their numerous folds.

The first type of a flexagon created was a hexaflexagon, which, as its name suggests, is hexagonal. Interestingly, Martin Gardner’s first article was exactly about them*. There exists a general method for creating hexaflexagons with 3*2N for all N≥0. 6-faced ones seem to have the highest regard/effort ratio. I have also heard about hexaflexagons with 4, 5, and 7 sides, but I don’t know how to make them.

* It was the most popular article in Scientific American until an article about the Game of Life came out.

There also exist rectangular flexagons. I can recall making a 12-faced one a few years ago.

Anyway, the purpose of this article is to present a brand new type of flexagons I have invented. All my flexagons have four faces, but their shapes and even symmetry groups can vary greatly. Instead of tediously describing the general method of creating “cloverleaf flexagons”, I will show you two concrete examples.

Print this out and cut along thin solid lines. All other lines are folds; it’s recommended to flex each one back and forth before assembling the flexagon. Then flatten all dotted lines and turn all thick lines into valley folds. Finally, add some glue (for model 1) or sellotape (for model 2). You should now be able to enjoy cloverflexing.

It turns out that the first flexagon is cloverleaf-shaped (hence the name of the entire family), while the second one looks like a square. It’s, however, impossible to flex the latter one just by folding it in half; it should actually be folded into Square Base.

If you feel that you understand cloverleaf flexagons, here are a few open problems just for you:

  1. (Easy and rewarding) Generalise my flexagons to allow more faces.
  2. (Easy and less rewarding) Find out which of my flexagons have planar embeddings.
  3. (Challenging and very rewarding) More shapes!!!

P. S. I accidentally folded the squarish flexagon incorrectly and spent the following 10 minutes trying to get it back to normal shape. It may constitute a pretty complex puzzle.

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