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A type of notation that is useful for describing regular tilings is the Schläfli symbol. It is stated as {# edges per polygon, # polygons meeting at a vertex}. In this notation, the three regular tilings are:

  • {3,6} for triangular tiling—each tile has 3 sides and 6 tiles meet at each vertex.
  • {4,4} for square tiling—each tile has 4 sides and 4 tiles meet at each vertex.
  • {6,3} for hexagonal tiling—each tile has 6 sides and 3 tiles meet at each vertex.

These are all defined for regular polygons.

I am trying to understand if we can use the symbol to also describe tilings of regular polygons but using irregular tiles such as a rhomb. And if we can't, is there an alternative?

The below figure illustrates the question where we are not considering the whole plane as above, but the bounded tessellation of the hexagon using irregular tiles:

enter image description here

  • In figure (a) a valid Schläfli symbol is used alongside all the tiles that complete the tessellation.
  • Figure (b) is where the question really starts: the tiling uses $T_i$ which is a rhomb as a tile so that the symbol 'becomes' (if you can in fact use the symbol in this way) {4,3} - the tile has four sides and three of them meet at each vertex.
  • Figure (c) similarly: {4,2} - the tile has four sides and 2 of them meet at each vertex.

The question is thus: is this a valid usage of the Schläfli symbol?

EDIT

It seems that Coxeter-Dynkin diagrams may be a good direction to explore for the general case: https://en.wikipedia.org/wiki/Coxeter–Dynkin_diagram#:~:text=Coxeter–Dynkin%20diagrams%20can%20explicitly,diagram%20with%20permutations%20of%20markups.

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No. At least that's not how Schläfli symbols are normally used.

As you rightly say, they are for tessellations of the plane by regular polygons. One of the standard generalisations involves tessellations of other uniform spaces, namely the hyperbolic plane and the sphere. In particular, {4,3} is a valid symbol for the tessellation of the sphere which you get if you "inflate" the surface of a cube into a sphere.

If you consider partitions of the regular polygons ($n$-gons with some fixed $n$) into identical parts consisting of a few sectors (triangles formed by a side and the centre), then maybe a more natural way to describe such partition is just to say how many sectors does such a part include (necessarily a divisor of $n$).

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  • $\begingroup$ Great, thanks. Very helpful. It is interesting that there does not seem to be (at least I cannot find one) a standard way of describing a general tessellation of an n-gon. $\endgroup$
    – Astrid
    Commented Sep 17, 2023 at 18:04

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