## Friday, November 9, 2012

### Hexaflexagation Visualization

If you've never heard of a hexaflexagon, watch this.   In brief, hexaflexagons are cool because they have really weird geometry.  They have too many sides.  If you've never seen a hexahexaflexagon, watch this.  Hexahexaflexagons have even more too many sides.  If you find yourself hexiflexagonally inclined, make one yourself and play with it for a while.  Then, once you've accidentally sunk way too much time into trying to understand your new toy and you're really frustrated that the sides keep disappearing and new ones keep appearing out of nowhere, watch this:

For the more set-theoretically inclined:

Node set: {1o, 1*, 1#, 2o, 2*, 2#, 3o, 3*, 3#, 4o, 4*, 5o, 5*, 6o, 6*}

("o" stands for "circle" (a circle drawn around the center of the hexagon), "*" is "star" (which looks more like a snow flake), and "#" is "box", which is actually a small hexagon drawn around the center.)

"Open" for top side: {(1*, 5*), (1*, 3o), (1o, 3o), (2*, 1*), (2*, 4*), (2o, 1*), (3*, 2*), (3o, 6o), (3o, 2*), (4*, 3*), (5*, 20), (6*, 1o)}

"Open" for bottom side: {(1#, 6o), (1#, 2#), (1o, 2#), (2#, 3#), (2#, 5o), (2o, 3#), (3*, 1#), (3#, 4o), (4o, 2o), (5o, 1o), (6o, 3*)}

"Flip" for top side: {(1*, 2#), (1o, 6o), (2*, 3#), (2o, 5o), (3*, 4o), (3o, 1#), (4*, 2o), (5*, 1o), (6*, 3*)}

"Flip" for bottom side: {(1#, 3o), (1o, 5*), (2#, 1*), (2o, 4*), (3*, 6*), (3#, 2*), (4o, 3*), (5o, 2o), (6o, 1o)}

But it's way more fun on a balloon, right?