I have such an interesting observation: if I take a square grid and rotate it over itself by atan(3/4) , it forms a structure which has four axes of reflection symmetry:

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The resulting structure is really mesmerizing, I see a lot of symmetric shapes in it: octagons, stars, rhombuses. And all of them appear inside this structure in different sizes, namely scaled by an integer factor:
enter image description here

Also looking at minimal periods, I notice that exactly the same structure I can create from rhombus grids, put over itself at 90 degrees. Namely rhombuses which have height of double of its width.

And the same structure, only with one additional square grid over it, I can create from a parallelogramm. Once I have made a question about this parallelogram (Very special geometric shape - parallelogram (No name yet?))

The relation between all shapes which spawn these grids:
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To spawn it from parallelogram I copy-reflect the parallelogram grid and then copy-rotate the whole by 90 degrees:

enter image description here


  • How this particular structure is classified, or named?
  • Do you know of any articles about it or any applications?

To be exact, I made up two structures, but since they are almost the same, I hope this does not add much confusion.

  • $\begingroup$ Not this specific structure, but the structures that you're looking at are very closely related to Penrose tilings (and indeed, there could easily be an 'inflatable' tiling lurking here). Some phrases to start with are 'quasicrystal', 'aperiodic tiling',and 'Ammann line' : en.wikipedia.org/wiki/Robert_Ammann for the latter... $\endgroup$ – Steven Stadnicki Dec 6 '16 at 0:54
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    $\begingroup$ @StevenStadnicki Indeed, replication of shapes inside this structure is well seen. This is however periodic and much simpler than Penrose tiling and other aperiodic structures. $\endgroup$ – Mikhail V Dec 6 '16 at 0:59
  • $\begingroup$ I suggest that you ask your question to the Tiling List at groups.google.com/forum/#!forum/tiling There are tiling experts on that list that may have seen something like this (or not). $\endgroup$ – W. Edwin Clark Dec 12 '16 at 19:05
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    $\begingroup$ mathpuzzle.com/kitesbricks.html has some discussion. $\endgroup$ – Ed Pegg Dec 29 '16 at 18:56

Such kind of things are called multi-grids. The kind of multi-grids you drew are called singular. Non-singular or regular multi-grids, where no three different lines intersect, provide a general method for constructing non-periodic tilings of the plane. They were used (maybe itroduced?) by N. de Bruijn in his paper Algebraic theory of Penrose tilings. Later on de Bruijn's student Beenker used tetra-grids to describe Ammann-Beenker tilings -take a look at references therein.

The multi-grids method for constructing non-periodic tilings of the plane is generalized by the projection method (also cut-and-project method) and is related to the mathematical theory of quasi-crystals. Such method consists in projecting a stripe of a higher dimensional lattice onto a totally irrational subspace.

The relation between projection tilings and the inflational structure was studied by Harriss and Lamb.


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