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Zeno Rogue

The dual of the famous new hat tiling.

Twenty-Seven on the HyperRogue discord used dual to make it easier to tell which tiles are adjacent in the game, but in HR 12.1k it looked "cursed" because the hats were not centered well (it is better in HR 12.1l). Dui Mauris Football had an idea to make an animation in which the center changes. We get a "wavy" look because the hats have different rotations and mirror symmetries.

In the second animation, the center movement is simpler, but we change the hats (like in mathstodon.xyz/@csk/1100588109). Also the hats are shown.

And the third animation is a calmer version.

25 Mar 2023 at 13:12 | Open on mathstodon.xyz
4 comments
Zeno Rogue

One more interesting thing: the "hats" are 14-gons (counting the long edge as two), what happens when we try to draw a "straight line" by exit each tile via the edge opposite (+7) to the edge we have entered it? (Bulls in Bull Dash move like this...)

The result gets very random, especially compared to the Penrose kite-and-dart (despite both being based on a hierarchical construction).
25 Mar 2023 at 14:38 | Open on mathstodon.xyz
Zeno Rogue

Can we zoom in forever, or do we eventually reach "atoms" with no internal structure?

This is an aperiodic tiling, so what we get depends on where we zoom in. In the first animation we pick a very specific location; in the second animation, we pick a random one.

BTW to keep all our hat threads in one place, here is the other one: mathstodon.xyz/@zenorogue/1100

27 Mar 2023 at 13:50 | Open on mathstodon.xyz
Zeno Rogue

In mathstodon.xyz/@zenorogue/1100 there was a bug: the centers of mirrored hats was supposed to be the mirror images of the centers of non-mirrored hats (in particular, rotate clockwise vs anti-clockwise), but they did not -- here is a fixed version (with some other minor changes). Thanks to Magma for catching this!

27 Mar 2023 at 14:40 | Open on mathstodon.xyz
Zeno Rogue

A version of the "zooming in" animation up in the thread where the color of hats and clusters is based on their rotation (and the saturation is based on hat/cluster type), and the lowest level oscillates between the possible shapes.

28 Mar 2023 at 12:45 | Open on mathstodon.xyz
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