bprentice wrote:The woodcuts by M.C. Escher

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These seem to be a bit too general in nature, and evade my question, which is about the underlying mathematics, not how to represent the pictures. (Obviously I'm familiar with Escher, but my pictures are not paradoxical.)

E.g., can someone recommend a paper that specifically concerns the mapping between isometric projections of surfaces in a cubic lattice and tilings with rhombuses or a class of such isomorphisms. Some of this is obvious in retrospect, but it took me weeks of work to get there. (I have Grünbaum and Shephard's Tilings and Patterns, and if this has a relevant result, I may just not have the wherewithal to understand it.)

So in short, what kind of specialist would find this obvious or at least find the analysis very familiar? (Not a computer scientist, or not me anyway.)

Or another way to put it: I've rediscovered something a little bit interesting. What's it called? It is not called Escher woodcuts or Lisp, and probably not a picture language either. It

is something about projections of monotonically nondecreasing surfaces in a cubic lattice and the connection to tilings and bipartite matchings.

Parts of this are readily explained in terms of geometry, but it took a while to get there. First off, the "Q*bert" isometric projection is onto the plane x+y+z=0, so the axes can be placed as follows.

- Screen Shot 2019-01-27 at 9.22.30 PM.png (32.85 KiB) Viewed 1433 times

The 2 colors come from this observation (italics because this is really the key to explaining most of what's going on):

When you follow an adjacent rhombus in the same direction, you increase or decrease x, y, or z, thus increasing or decreasing x+y+z accordingly.

When you follow a rhombus around a bend, you increase (decrease) one axis and (decrease) increase the other axis, leaving x+y+z unchanged.

The colors (red and blue in the Q*bert picture) are based on whether x+y+z is odd or even, assuming cubes are placed at integer coordinates. Hence, they alternate on straight paths, but stay the same around bends.

Here is the part that is at least fairly interesting. We have 3 distinct discrete systems that are equivalent:

(1) A bipartite graph in a triangular grid in which an edge between two adjacent triangles may be included in a matching.

(2) A rhombus tiling in which each rhombus covers two adjacent grid triangles and rhombuses are colored the same as adjacent ones with a different orientation, or the opposite color to an adjacent one with the same orientation.

(3) A surface of exposed square facets in a cube lattice, projected as rhombuses onto the plane x+y+z=0, and colored according to whether their x+y+z is odd or even (parity).

The connection between (1) and (2) is a classic result used, among other things, for counting packings of rectangular grids by dominoes. The connection between 2 and 3 is a bit more subtle, though I am sure it is an old, standard result (where?).

So if you remove or add a cube to the surface (3), the operation on the isometric projection is effectively a local retiling of 3 rhombuses (2) and this is also explicable as switching matching edges with unmatched edges in an alternating path in a bipartite graph (1).

I find this surprising. Anyone else? And someone who does not find it surprising may be able to help.