Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
Classical Physics
Quantum Physics
Quantum Interpretations
Special and General Relativity
Atomic and Condensed Matter
Nuclear and Particle Physics
Beyond the Standard Model
Cosmology
Astronomy and Astrophysics
Other Physics Topics
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
Classical Physics
Quantum Physics
Quantum Interpretations
Special and General Relativity
Atomic and Condensed Matter
Nuclear and Particle Physics
Beyond the Standard Model
Cosmology
Astronomy and Astrophysics
Other Physics Topics
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Physics
Other Physics Topics
Projection down 3 dimensions for aperiodic tiling/quasicrystal
Reply to thread
Message
[QUOTE="nomadreid, post: 6003355, member: 112452"] 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 (x[SUB]1[/SUB],x[SUB]2[/SUB],…x[SUB]n+2[/SUB],x[SUB]n+3[/SUB]) → (x[SUB]n+3[/SUB],x[SUB]1[/SUB], x[SUB]2[/SUB]…x[SUB]n+2[/SUB]). [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. [/QUOTE]
Insert quotes…
Post reply
Forums
Physics
Other Physics Topics
Projection down 3 dimensions for aperiodic tiling/quasicrystal
Back
Top