"How chiral is my tile?" Explanation for beginners how to tell if a tile or tiling is or is not chiral (if its boolean) or how chiral it is.

Question

From the new arXiv preprint Smith et al. (2023) A chiral aperiodic monotile (based on the earlier Smith et al. (2023) An aperiodic monotile):

A tile T is a closed topological disk in the plane, and a monohedral tiling admitted by it is a countable collection $\text{T} = \{ {T_1, T_2, . . . \} }$ of congruent copies of T with disjoint interiors, whose union is the entire plane. Each $T_i$ is of the form $g_iT$ for some planar isometry $g_i$. We say that a monohedral tiling T is a chiral tiling if for every pair $T_i, T_j ∈ \text{T}$ , there is an orientation preserving isometry mapping $T_i$ to $T_j$. We then define a weakly chiral aperiodic monotile to be a tile whose chiral tilings are all non-periodic (and that admits at least one such tiling), and a strictly chiral aperiodic monotile to be a tile that admits only chiral non-periodic tilings. Following Klein, the weak case is aperiodic if we decree reflections to be off limits, even if the tile admits periodic tilings when reflections are allowed. The strict case remains both chiral and aperiodic in the presence of reflections. With these definitions in hand, we ask: Do there exist any weakly or strictly chiral aperiodic monotiles?¹

¹We might call this the “vampire einstein” problem, as we are seeking a shape that is not accompanied by its reflection.

I've tried to do some search engining but I've only gotten either high level papers or articles announcing this new tile.

To help me get started understanding this:

Question: "How chiral is my tile?" Explanation for beginners how to tell if a tile or a tiling is or is not chiral (if its a boolean) or how chiral it is.

If I look for images of tiling, I can apply duck typing (i.e. if it looks like a duck and quacks like a duck) and I see right away that there are many tilings that have some spiraling-like appearance, but there must be some objective test one can apply.

somewhat related and still unanswered What is 12-fold Stampfli-inflation tiling and where/how can I recognize it in this analysis of dodecagonal 30° twisted bilayer graphene quasicrystal? — uhoh, Jun 07 '23 at 03:30

score 1 · Accepted Answer · answered Jun 07 '23 at 08:46

The definition was given in your quotation. A tiling by a single tile is chiral if all tiles are related by an orientation preserving isometry - that is, they are all just translations and rotations of each other with no reflections allowed. That seems like a pretty simple definition to me, but my perspective could be skewed.

They similarly define weakly and strictly chiral aperiodic monotiles in terms of the possible tilings that the monotile admits.

Really, is just comes down to the the following two questions for your given monotile:

Given all possible tilings by your monotile that only use translations and rotations of the monotile (not relfections), are these tilings all non-periodic? -- If yes, then your monotile is a weakly chiral aperiodic monotile.
Given all possible tilings by your monotile that use translations, rotations and reflections do both of the following hold? a) all of these tilings are non-periodic, and b) the tilings only use translations and rotations of the monotile (not reflections). -- If yes, then your monotile is a strictly chiral aperiodic monotile.

To be a weakly chiral aperiodic monotile and not strictly chiral aperiodic, your monotile has to be such that if you disallow reflections, then all tilings by it are non-periodic and if you then allow reflections, then there exist periodic tilings by that tile (like with the tile $Tile(1,1)$).

This is very helpful. While it might feel like you're reiterating, at least for me (asking an explicit "beginner" question) you're both translating and teaching. This will make it easier for me to read further on my own. And speaking of both duck typing (in the question) and aperiodic order, this may (or may not) be something of interest: Proposing a 2D quasicrystal; what are the necessary and sufficient conditions? (If it looks like a duck and quacks like a duck, or...?) — uhoh, Jun 07 '23 at 13:50
With regard to that question, there's a strong mathematical sense in which 'weak model sets' (sets coming from a cut and project method with some restrictions on the window) and 'point sets whose diffraction is pure point' are essentially the same thing. This is the basis of much of Nicolae Strungaru's work so you might like to contact Nicu. — Dan Rust, Jun 07 '23 at 14:19
Oh that's quite helpful, thanks! I'll start with Strungaru's writings first, which guide me to the early papers, e.g. "Indeed, the pivotal article (11) by Bombieri / Taylor written right after the discovery of quasicrystals has in its title the question ‘Which distributions of matter diffract?’. Of course, in order to answer this question one needs to be more specific: How is the distribution of matter modeled? What is meant by ‘diffract’?" — uhoh, Jun 07 '23 at 21:09

"How chiral is my tile?" Explanation for beginners how to tell if a tile or tiling is or is not chiral (if its boolean) or how chiral it is.

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