[math-fun] Don't cry for me, Pentagonia
AG/RWG: Adam Goucher has made a startling observation (quoted without permission): ---- I was reading the third edition of 'Minskys and Trinskys' by Gosper, Holloway and (Ziegler Hunts)^2: http://www.blurb.com/books/2172660-minskys-trinskys-3rd-edition The pattern of disjoint D10-symmetric snowflakes (or 'Rastermen', as they're referred to in the book) shown in Figure 31 appears to be affinely isomorphic to the pattern obtained by marking Penrose tilings: http://condellpark.com/kd/penrmark24bg.gif This suggests that it should be possible to create a very integer-esque method of drawing an affined Penrose tiling on a square grid... -- 1. I point out there are a number of 5-symmetric-looking pictures in this book, but page 31 is not one of them. 2. Is it even possible for anything drawn on an integer grid to be affine to a Penrose tiling? Related question: for which N, can the vertices of a regular N-gon be affined to distinct integer coordinates?
2. Is it even possible for anything drawn on an integer grid to be affine to a Penrose tiling? Related question: for which N, can the vertices of a regular N-gon be affined to distinct integer coordinates?
--Answer: Theorem: it is impossible to affine a regular N-gon to get distinct integer coordinates for its vertices, unless N=2,3,4,6. Proof idea: (A) The integer linear combinations of the vertex vectors of a regular N-gon, form a dense set. (Except for the Ns listed.) (B) The integer linear combinations of a finite set of points with distinct integer coordinates, form a set of density <=1, which indeed is necessarily periodic (period arises from LCM). We know (B) does not hold for any N besides those listed because of 2D crystallography. QED. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
APG said Figure 31, not page 31. The book may or may not adequately explain how Holloway "circularized" it. I got http://www.tweedledum.com/rwg/rhombicp5.svg by skewing a trinsky. --rwg On 2015-06-13 06:54, Warren D Smith wrote:
AG/RWG: Adam Goucher has made a startling observation (quoted without permission): ---- I was reading the third edition of 'Minskys and Trinskys' by Gosper, Holloway and (Ziegler Hunts)^2: http://www.blurb.com/books/2172660-minskys-trinskys-3rd-edition The pattern of disjoint D10-symmetric snowflakes (or 'Rastermen', as they're referred to in the book) shown in Figure 31 appears to be affinely isomorphic to the pattern obtained by marking Penrose tilings: http://condellpark.com/kd/penrmark24bg.gif This suggests that it should be possible to create a very integer-esque method of drawing an affined Penrose tiling on a square grid...
-- 1. I point out there are a number of 5-symmetric-looking pictures in this book, but page 31 is not one of them. 2. Is it even possible for anything drawn on an integer grid to be affine to a Penrose tiling? Related question: for which N, can the vertices of a regular N-gon be affined to distinct integer coordinates?
participants (2)
-
rwg -
Warren D Smith