Mathematics and Art - A Penrose Tiling
"The whole is greater than the sum of its parts"
- Aristotle (384 BC – 322 BC)
A Penrose tiling is a non-periodic tiling generated by an aperiodic set of prototiles. Penrose tilings are named after mathematician and physicist Roger Penrose who investigated these sets in the 1970s.
A Penrose tiling has many remarkable properties, most notably:
- It is non-periodic, which means that it lacks any translational symmetry.
- It is self-similar, so the same patterns occur at larger and larger scales. Thus, the tiling can be obtained through “inflation” (or “deflation”) and any finite patch from the tiling occurs infinitely many times.
- It is a quasicrystal: implemented as a physical structure a Penrose tiling will produce Bragg diffraction and its diffractogram reveals both the fivefold symmetry and the underlying long range order.
The three types of Penrose tiling:The original pentagonal Penrose tiling (P1), Kite and dart tiling (P2) and Rhombus tiling (P3).They have many common features: in each case, the tiles are constructed from shapes related to the pentagon (and hence to the golden ratio), but the basic tile shapes need to be supplemented by matching rules in order to tile aperiodically; these rules may be described using labeled vertices or edges, or patterns on the tile faces – alternatively the edge profile can be modified (e.g. by indentations and protrusions) to obtain an aperiodic set of prototiles.
Image: Penrose Tiling at http://en.wikipedia.org/wiki/Penrose_tiling.
Some substitution rules includes: Overlapping Robinson Triangle I & II - Discovered by P. Gummelt and Danzer’s 7-fold variant.
Rule Image: (I) A substitution rule where the tiles are allowed to overlap. The image left indicates, that the yellow and the green tiles do overlap. It is unknown whether these tilings are mld to the Penrose Rhomb tilings. Rule Image: (II) As Overlapping Robinson Triangles I, this is a variant of the Penrose Rhomb tiling, using only one prototile, and the tiles are allowed to overlap. Here, the overlap happens after applying the substitution rule twice on one tile.
Danzer’s 7-fold variant - Rule Image: Substitution tiling with isosceles triangles as prototiles allow several variations: For each tile in the first order supertiles, one can choose whether it is a left-handed or a right-handed version. By playing around with these possibilities, one obtains this variant from Danzer’s 7-fold.