I once took notes from a website, and now I cannot find the reference anymore, so I am attempting to reconstruct it, and am sure that I have it all wrong (or that the website did, or both). Hence, I would be grateful for corrections. My apologies for not being able to provide the link.(adsbygoogle = window.adsbygoogle || []).push({});

The website concerned the instructions for constructing a Penrose tiling or a quasicrystal by projecting down from 5 and 6 dimensions respectively. The steps in my sloppy notes appear as follows

[1] choose an acute angle A such that tan(A) is irrational. For example, A= pi/5.

[2] Let n be the number of dimensions of the final covering: 2 for a Penrose tiling, 3 for a quasicrystal.

[3] Set up a grid (lattice) in a n+3 dimensional space D.

[4] Define the transformation T on points in D such that (x_{1},x_{2},…x_{n+2},x_{n+3}) → (x_{n+3},x_{1}, x_{2}…x_{n+2}).

[5] Find the n-dimensional surface S in D (a plane for n=2, a 3D hyperplane for n=3) such that the angle formed between any line in S and T"S will be A. (This point is probably the most confused, as I do not see what T has to do with anything. It would be more natural to set the angle A be between S and the n-dimensional space upon which the result of the following will be projected in step 7.)

[6] Find the centers of the hypercubes in the grid which are closest to the hyperplane.

[7] Project these points onto one of the n-dimensional subspaces, P, of the n+3-space, spanned by vectors lying on the axes. (Another point of confusion. Is the projection necessary? If it is, perhaps the correct condition to find S is that that the angle between S and P is A?)

[8] These centers will form the vertices of the tiling or of the quasicrystal, respectively.

Any and all corrections would be appreciated. If this is completely off the wall, I would appreciate a link which lays out clearly and explicitly the correct steps. (So far I have only found vague analogies or hand-waving, but maybe I am not searching correctly.) Thanks.

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# I Projection down 3 dimensions for aperiodic tiling/quasicrystal

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