Projection down 3 dimensions for aperiodic tiling/quasicrystal

In summary, the conversation revolved around the process of constructing a Penrose tiling or a quasicrystal by projecting down from 5 or 6 dimensions respectively. The steps involved choosing an acute angle, setting up a grid in a higher dimensional space, defining a transformation, finding a surface, and projecting points onto a lower dimensional subspace. The speaker was seeking corrections and a clear explanation of the steps involved.
  • #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.
 
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  • #2
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.
 

1. What is projection down 3 dimensions in the context of aperiodic tiling/quasicrystals?

Projection down 3 dimensions refers to the process of taking a higher-dimensional aperiodic tiling or quasicrystal and projecting it onto a three-dimensional space. This allows us to visualize and study the structure of these complex geometries in a more manageable and understandable way.

2. How is projection down 3 dimensions useful in studying aperiodic tiling/quasicrystals?

Projection down 3 dimensions allows us to see the patterns and symmetries present in aperiodic tiling/quasicrystals, which can provide insights into their underlying mathematical and physical properties. It also helps us to compare and analyze different aperiodic tiling/quasicrystal structures.

3. What are the challenges in projecting down 3 dimensions for aperiodic tiling/quasicrystals?

One of the main challenges is accurately representing the higher-dimensional structure in a three-dimensional space. This can result in distortions and inaccuracies, making it difficult to fully understand and analyze the original geometry. Another challenge is determining the most appropriate projection method for a given aperiodic tiling/quasicrystal.

4. How do scientists choose the projection method for aperiodic tiling/quasicrystals?

The choice of projection method depends on the specific goals of the study and the characteristics of the aperiodic tiling/quasicrystal being analyzed. Some common projection methods include stereographic projection, parallel projection, and perspective projection. Scientists may also use multiple projections to gain a more comprehensive understanding of the structure.

5. Can projection down 3 dimensions help us understand real-world materials?

Yes, projection down 3 dimensions is a valuable tool for studying and designing new materials with aperiodic structures, such as quasicrystals. By visualizing and analyzing these structures, scientists can gain insights into their unique properties and potentially use this knowledge to develop new and improved materials for various applications in fields such as electronics, optics, and biomaterials.

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