A chiral aperiodic monotile

There was a rumour that if they found a chiral aperiodic monotile then they might call it a Vampire Tile, because it doesn&#x27;t have a reflection.<p>Seems like they didn&#x27;t go with that.<p>I also see that the old discussion has come up: &quot;But what can it be used for?&quot;<p>These sorts of things are pursued because they are fun, and there&#x27;s a community of people who find it interesting. Is Rachmaninoff&#x27;s second piano concerto useful? Is Bach&#x27;s Toccata and Fugue in D minor (BWV 565)[0] useful? Is Rodin&#x27;s &quot;The Thinker&quot; useful?<p>No. And for each of them there are people who Simply. Don&#x27;t. Care.<p>So it is with Pure Maths.<p>The difference is that sometimes things people pursued simply out of interest or curiosity turn out to be useful. It might be decades down the line, but it happens, and you never know in advance which bit of maths they will be.<p>So maybe Chiral Aperiodic Tilings will turn out to be useful, maybe not. Maybe the work done to create them is what will turn out to be useful. Maybe not.<p>It&#x27;s not the point.<p>[0] Interestingly, this might not be by Bach, and some claim it&#x27;s not in D minor.

This excites me much more than the original result, which I considered to use two tiles[1]. The fact that it’s a such tiny modification of the original result is crazy. Even if you don’t intend to read the paper, look at the illustrations of the hierarchical substitution algorithm at the top of pages 6 and 7, those are just beautiful.<p>[1] The authors discuss various historic definitions of tilings and whether reflections should be allowed or not (they argue that most definitions allow them). For me, the answer is simple: nature is chiral, you can’t reflect things willy-nilly. Puzzle pieces, bathroom tiles, even polygons in 3D rendering all have distinguishable sides.

The recently discovered 2D &quot;hat&quot; monotile tiles aperiodically but allows reflections (and must allow reflections to tile?).<p>The paper discusses a (2D) monotile whose shape will allow an aperiodic tiling without reflections.

Hmmm... I wonder if these spectres are potentially a basis for a new form of cryptographic algorithms. Unique non-repeating sequences exclusively derived from a set of rules and an initial state in multiple dimensions sounds like a promising candidate.

Is that finally the single tile you can use to cover whole space aperiodically without flipping it upside down?

I wish I could buy ceramic Penrose P3 tiles to put on my floor or wall. They 2 different tiles instead of 1, but they&#x27;re simple diamond shapes, and they tile aperiodically.

How could you address each tile, to create an aperiodic tile map? Would be a neat tech demo. Like HyperRogue, for example. <a href="https:&#x2F;&#x2F;www.roguetemple.com&#x2F;z&#x2F;hyper&#x2F;" rel="nofollow">https:&#x2F;&#x2F;www.roguetemple.com&#x2F;z&#x2F;hyper&#x2F;</a>

Can anyone explain why people are interested in tilings?

Only slightly related: anyone know how to make porcelain or other normal tiles? If I wanted to redo the bathroom in these or whatever, can they be made?

If I was a billionaire I&#x27;d have such custom-made tiling installed in my home&#x27;s hallways and terrace.

I wonder if the authors anticipated this result when publishing their first paper, or if they were primarily motivated by &quot;complaints&quot; that their hat tile (and other tiles in the associated spectrum) required reflection. Certainly they <i>mention</i> this question, but my question is whether they completely anticipated it.

Random thought&#x2F;nerd snipe hazard: would a Wolfram-style classification of cellular automata running on this tiling kick up anything interesting?

Seems useful for creating unique and modern designs.

Can someone please let me buy actual tiles shaped like this? I would like to renovate my bathroom

I wonder if there is an aperiodic monotile that may tile a sphere.