Animated Penrose Tiling (part 2)

Let us carry out the procedure... We begin with whatever initial configuration we want (the axiom) for instance, a decagon made up by five kites Then we apply the subdivision procedure on each triangle, obtaining a more refined subdivision Look: the new triangles can piece together again kites and darts Finally, we blow up all the tiles to the original size and repeat once more the procedure Now we have to split also some obtuse triangles The procedure obeys the adjacency constraints between the tiles again we blow up the tiles to the original size Third step... Fourth step... A central decagon exactly alike the initial configuration has appeared! Fifth step... Sixth step... Seventh step... Eight step... We can go on, tiling a bigger and bigger part of the plane This is the result after nine steps In conclusion, we have a procedure which tiles all the plane obeying the adjacancy constaints. Due to the initial configuration (a decagon made up by five kites), the resulting tiling will have a rotational symmetry (by 72 degrees) and also axial symmetries (reflections across five axes by the center) as suggested by some of the motions that you can see There are precisely ten symmetries, forming the dihedral group "D5" (symmetries of a pentagon) The group can be generated by a rotation of 72 degrees ...and a reflection across a suitable axis by the center We can also modify a little the silhouette of the DART and the KITE, obtaining new tiles in the spirit of Escher's artworks Le we employ the new forms in the tessellation... The result has the same structure as the KITE and DART tiling Here is the result after five steps of deflation/inflation A remarkable option can be found by a suitable choice of two different rhombi However, there is a close relation with Kite and Dart, as you can see in the animation The rhombi are split in two isosceles triangles which are equal to those we have already seen, apart from the proportions We can perform a procedure of deflation/inflation starting from a suitable subdivision of the golden triangles similarly to the subdivision of Dart and Kite After six steps of the new procedure of deflation/inflation starting from a suitable initial configuration we obtain the tiling with rhombi... Roger Penrose tried to split a large regular pentagon into six smaller pentagons which side lenght is the inverse square of the golden ratio of the large side Repeating the subdivision a few times, some holes remain Some of them have the shape of a regular pentagon, hence they can be included in the subdivision procedure on the contrary, some other holes have different shapes: rhombus, crown, star The three different colors of the pentagons are useful to mark adjacency constraints which induce a non periodic tiling. In such a way we get a set of six tiles which have the aperiodicity property Up to now it is not known if there exists a single tile which has the aperiodicity property!