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I Projection down 3 dimensions for aperiodic tiling/quasicrystal

  1. May 25, 2018 #1

    nomadreid

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    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.

    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 (x1,x2,…xn+2,xn+3) → (xn+3,x1, x2…xn+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.
     
  2. jcsd
  3. May 30, 2018 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
  4. May 30, 2018 #3

    nomadreid

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    So, let me make the question easier. Can anyone send a good link to explain more in detail how one reflects down from a 5-dimensional (resp 6-dimensional) lattice down to a Penrose tiling (resp quasicrystal approximation)? Thanks.
     
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