- #1
nomadreid
Gold Member
- 1,670
- 204
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.
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.