Editing Golden ratio (section)

=====Penrose tilings=====
{{main|Penrose tiling}}
[[File:Kite Dart.svg|thumb|The kite and dart tiles of the Penrose tiling. The colored arcs divide each edge in the golden ratio; when two tiles share an edge, their arcs must match.]]
The golden ratio appears prominently in the ''Penrose tiling'', a family of [[aperiodic tiling]]s of the plane developed by [[Roger Penrose]], inspired by [[Johannes Kepler]]'s remark that pentagrams, decagons, and other shapes could fill gaps that pentagonal shapes alone leave when tiled together.<ref name="Tilings and Patterns" /> Several variations of this tiling have been studied, all of whose prototiles exhibit the golden ratio:
*Penrose's original version of this tiling used four shapes: regular pentagons and pentagrams, "boat" figures with three points of a pentagram, and "diamond" shaped rhombi.<ref name=pentaplexity />
*The kite and dart Penrose tiling uses [[kite (geometry)|kites]] with three interior angles of {{tmath|72^\circ}} and one interior angle of {{tmath|144^\circ\!}}, and darts, concave quadrilaterals with two interior angles of {{tmath|36^\circ\!}}, one of {{tmath|72^\circ\!}}, and one non-convex angle of {{tmath|216^\circ\!}}. Special matching rules restrict how the tiles can meet at any edge, resulting in seven combinations of tiles at any vertex. Both the kites and darts have sides of two lengths, in the golden ratio to each other. The areas of these two tile shapes are also in the golden ratio to each other.<ref name="Tilings and Patterns" />
*The kite and dart can each be cut on their symmetry axes into a pair of golden triangles and golden gnomons, respectively. With suitable matching rules, these triangles, called in this context ''Robinson triangles'', can be used as the prototiles for a form of the Penrose tiling.<ref name="Tilings and Patterns" /><ref name=robinson />
*The rhombic Penrose tiling contains two types of rhombus, a thin rhombus with angles of {{tmath|36^\circ}} and {{tmath|144^\circ\!}}, and a thick rhombus with angles of {{tmath|72^\circ}} and {{tmath|108^\circ\!}}. All side lengths are equal, but the ratio of the length of sides to the short diagonal in the thin rhombus equals {{tmath|1\mathbin:\varphi}}, as does the ratio of the sides of to the long diagonal of the thick rhombus. As with the kite and dart tiling, the areas of the two rhombi are in the golden ratio to each other. Again, these rhombi can be decomposed into pairs of Robinson triangles.<ref name="Tilings and Patterns" />

{{multiple image
|align=left
|image1=Penrose Tiling (P1).svg|caption1=Original four-tile Penrose tiling
|image2=PenroseTilingFilled4.svg|caption2=Rhombic Penrose tiling
|total_width=540}}

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