Editing Golden ratio (section)

====Pentagonal symmetry system====

=====Pentagon and pentagram=====
[[File:Pentagram-phi.svg|right|thumb|A pentagram colored to distinguish its line segments of different lengths. The four
lengths are in golden ratio to one another.]]

In a [[pentagon#Regular pentagons|regular pentagon]] the ratio of a diagonal to a side is the golden ratio, while intersecting diagonals section each other in the golden ratio. The golden ratio properties of a regular pentagon can be confirmed by applying [[Ptolemy's theorem]] to the quadrilateral formed by removing one of its vertices. If the quadrilateral's long edge and diagonals are {{tmath|a}}, and short edges are {{tmath|b}}, then Ptolemy's theorem gives {{tmath|1=\textstyle a^2 = b^2 + ab}}. Dividing both sides by {{tmath|ab}} yields (see {{slink|#Calculation}} above),
<math display=block>
\frac ab = \frac{a + b}{a} = \varphi.
</math>

The diagonal segments of a pentagon form a [[pentagram]], or five-pointed [[star polygon]], whose geometry is quintessentially described by {{tmath|\varphi}}. Primarily, each intersection of edges sections other edges in the golden ratio. The ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (that is, a side of the inverted pentagon in the pentagram's center) is {{tmath|\varphi}}, as the four-color illustration shows.

Pentagonal and pentagrammic geometry permits us to calculate the following values for {{tmath|\varphi}}:
<math display=block>\begin{align}
\varphi &= 1+2\sin(\pi/10) = 1 + 2\sin 18^\circ\!, \\[5mu]
\varphi &= \tfrac12\csc(\pi/10) = \tfrac12\csc 18^\circ\!, \\[5mu]
\varphi &= 2\cos(\pi/5)=2\cos 36^\circ\!, \\[5mu]
\varphi &= 2\sin(3\pi/10)=2\sin 54^\circ\!.
\end{align}</math>

=====Golden triangle and golden gnomon=====
{{main|Golden triangle (mathematics)}}
[[File:Golden triangle (math).svg|235px|right|thumb|A [[Golden triangle (mathematics)|golden triangle]] {{mvar|ABC}} can be subdivided by an angle bisector into a smaller golden triangle {{mvar|CXB}} and a golden gnomon {{mvar|XAC}}.]]
The triangle formed by two diagonals and a side of a regular pentagon is called a ''golden triangle'' or ''sublime triangle''. It is an acute [[isosceles triangle]] with apex angle {{tmath|36^\circ}} and base angles {{tmath|72^\circ\!}}.<ref name=fletcher /> Its two equal sides are in the golden ratio to its base.<ref name=loeb /> The triangle formed by two sides and a diagonal of a regular pentagon is called a ''golden gnomon''. It is an obtuse isosceles triangle with apex angle {{tmath|108^\circ}} and base angle {{tmath|36^\circ\!}}. Its base is in the golden ratio to its two equal sides.<ref name=loeb /> The pentagon can thus be subdivided into two golden gnomons and a central golden triangle. The five points of a [[Pentagram|regular pentagram]] are golden triangles,<ref name=loeb /> as are the ten triangles formed by connecting the vertices of a [[regular decagon]] to its center point.<ref name=miller />

Bisecting one of the base angles of the golden triangle subdivides it into a smaller golden triangle and a golden gnomon. Analogously, any acute isosceles triangle can be subdivided into a similar triangle and an obtuse isosceles triangle, but the golden triangle is the only one for which this subdivision is made by the angle bisector, because it is the only isosceles triangle whose base angle is twice its apex angle. The angle bisector of the golden triangle subdivides the side that it meets in the golden ratio, and the areas of the two subdivided pieces are also in the golden ratio.<ref name=loeb />

If the apex angle of the golden gnomon is [[Angle trisection|trisected]], the trisector again subdivides it into a smaller golden gnomon and a golden triangle. The trisector subdivides the base in the golden ratio, and the two pieces have areas in the golden ratio. Analogously, any obtuse triangle can be subdivided into a similar triangle and an acute isosceles triangle, but the golden gnomon is the only one for which this subdivision is made by the angle trisector, because it is the only isosceles triangle whose apex angle is three times its base angle.<ref name=loeb />

=====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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