Editing Golden ratio (section)

===Geometry===
The golden ratio features prominently in geometry. For example, it is intrinsically involved in the internal symmetry of the [[pentagon]], and extends to form part of the coordinates of the vertices of a [[regular dodecahedron#Cartesian coordinates|regular dodecahedron]], as well as those of a [[regular icosahedron]].<ref name=BurgerStarbird /> It features in the [[Kepler triangle]] and [[Penrose tilings]] too, as well as in various other [[polytopes]].

====Construction====
{{multiple image|align=right|direction=vertical|image1=Goldener Schnitt Konstr beliebt.svg|image2=Goldener Schnitt (Äußere Teilung).svg|width=175|footer=Dividing a line segment by interior division (top) and exterior division (bottom) according to the golden ratio.}}

'''Dividing by interior division'''
# Having a line segment {{tmath|AB}}, construct a perpendicular {{tmath|BC}} at point {{tmath|B}}, with {{tmath|BC}} half the length of {{tmath|AB}}. Draw the [[hypotenuse]] {{tmath|AC}}.
# Draw an arc with center {{tmath|C}} and radius {{tmath|BC}}. This arc intersects the hypotenuse {{tmath|AC}} at point {{tmath|D}}.
# Draw an arc with center {{tmath|A}} and radius {{tmath|AD}}. This arc intersects the original line segment {{tmath|AB}} at point {{tmath|S}}. Point {{tmath|S}} divides the original line segment {{tmath|AB}} into line segments {{tmath|AS}} and {{tmath|SB}} with lengths in the golden ratio.

'''Dividing by exterior division'''
# Draw a line segment {{tmath|AS}} and construct off the point {{tmath|S}} a segment {{tmath|SC}} perpendicular to {{tmath|AS}} and with the same length as {{tmath|AS}}.
# Do bisect the line segment {{tmath|AS}} with {{tmath|M}}.
# A circular arc around {{tmath|M}} with radius {{tmath|MC}} intersects in point {{tmath|B}} the straight line through points {{tmath|A}} and {{tmath|S}} (also known as the extension of {{tmath|AS}}). The ratio of {{tmath|AS}} to the constructed segment {{tmath|SB}} is the golden ratio.
Application examples you can see in the articles [[Pentagon#Side length is given|Pentagon with a given side length]], [[Decagon#Construction|Decagon with given circumcircle and Decagon with a given side length]].

Both of the above displayed different [[algorithm]]s produce [[geometric construction]]s that determine two aligned [[line segment]]s where the ratio of the longer one to the shorter one is the golden ratio.

====Golden angle====
{{main|Golden angle}}
[[File:Golden Angle.svg|175px|thumb|{{center|{{math|''g'' ≈ 137.508°}}}}]]
When two angles that make a full circle have measures in the golden ratio, the smaller is called the ''golden angle'', with measure {{tmath|g}}:

<math display=block>\begin{align}
\frac{2\pi - g}{g} &= \frac{2\pi}{2\pi - g} = \varphi,  \\[8mu]
2\pi - g &= \frac{2\pi}{\varphi} \approx 222.5^\circ\!, \\[8mu]
g &= \frac{2\pi}{\varphi^2} \approx 137.5^\circ\!.
\end{align}</math>

This angle occurs in [[phyllotaxis|patterns of plant growth]] as the optimal spacing of leaf shoots around plant stems so that successive leaves do not block sunlight from the leaves below them.<ref name=phyllotaxis />

====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}}

{{clear|left}}

====In triangles and quadrilaterals====

=====Odom's construction=====
[[File:Odom.svg|thumb|upright|Odom's construction: {{math|1=AB&thinsp;:&thinsp;BC = AC&thinsp;:&thinsp;AB = ''φ''&thinsp;:&thinsp;1}}]]

[[George Phillips Odom Jr.|George Odom]] found a construction for {{tmath|\varphi}} involving an [[equilateral triangle]]: if the line segment joining the midpoints of two sides is extended to intersect the [[circumcircle]], then the two midpoints and the point of intersection with the circle are in golden proportion.<ref name=triangleconstruction />

=====Kepler triangle=====
{{main|Kepler triangle}}
{{multiple image
|image1=Kepler triangle.svg|caption1=Geometric progression of areas of squares on the sides of a Kepler triangle
|image2=Kepler and the Deathly Hallows.svg|caption2=An isosceles triangle formed from two Kepler triangles maximizes the ratio of its inradius to side length
|total_width=480}}
The ''Kepler triangle'', named after [[Johannes Kepler]], is the unique [[right triangle]] with sides in [[geometric progression]]:
<math display=block>
1\mathbin:\sqrt{\varphi\vphantom+}\mathbin:\varphi.</math>
These side lengths are the three [[Pythagorean mean]]s of the two numbers {{tmath|\varphi \pm 1}}. The three squares on its sides have areas in the golden geometric progression {{tmath|\textstyle 1\mathbin:\varphi\mathbin:\varphi^2}}.

Among isosceles triangles, the ratio of [[inradius]] to side length is maximized for the triangle formed by two [[Reflection (mathematics)|reflected copies]] of the Kepler triangle, sharing the longer of their two legs.<ref name="Liber mensurationum" /> The same isosceles triangle maximizes the ratio of the radius of a [[semicircle]] on its base to its [[perimeter]].<ref name=bruce />

For a Kepler triangle with smallest side length {{tmath|s}}, the [[area]] and [[acute angle|acute]] [[internal angle]]s are:
<math display=block>\begin{align}
A &= \tfrac12 s^2\sqrt{\varphi\vphantom+}, \\[5mu]
\theta &= \sin^{-1}\frac{1}{\varphi}\approx 38.1727^\circ\!, \\[5mu]
\theta &= \cos^{-1}\frac{1}{\varphi}\approx 51.8273^\circ\!.
\end{align}</math>

=====Golden rectangle=====
{{main|Golden rectangle}}
[[File:Golden Rectangle Construction.svg|175px|thumb|To construct a golden rectangle [[Straightedge and compass construction|with only a straightedge and compass]] in four simple steps:
{|
|-
|Draw a square.
|-
|Draw a line from the midpoint of one side of the square to an opposite corner.
|-
|Use that line as the radius to draw an arc that defines the height of the rectangle.
|-
|Complete the golden rectangle.
|-
|}
]]

The golden ratio proportions the adjacent side lengths of a ''golden rectangle'' in {{tmath|1\mathbin:\varphi}} ratio.{{sfn|Posamentier|Lehmann|2011|p=11}} Stacking golden rectangles produces golden rectangles anew, and removing or adding squares from golden rectangles leaves rectangles still proportioned in {{tmath|\varphi}} ratio. They can be generated by ''golden spirals'', through successive Fibonacci and Lucas number-sized squares and quarter circles. They feature prominently in the [[Regular icosahedron|icosahedron]] as well as in the [[Regular dodecahedron|dodecahedron]] (see section below for more detail).<ref name=BurgerStarbird />

=====Golden rhombus=====
{{main|Golden rhombus}}
A ''golden rhombus'' is a [[rhombus]] whose diagonals are in proportion to the golden ratio, most commonly {{tmath|1\mathbin:\varphi}}.<ref name=hexecontahedron /> For a rhombus of such proportions, its acute angle and obtuse angles are:

<math display=block>\begin{align}
\alpha &= 2\arctan{1\over\varphi}\approx63.43495^\circ\!, \\[5mu]
\beta  &= 2\arctan\varphi=\pi-\arctan2 = \arctan1+\arctan3 \approx 116.56505^\circ\!.
\end{align}</math>

The lengths of its short and long diagonals {{tmath|d}} and {{tmath|D}}, in terms of side length {{tmath|a}} are:

<math display=block>\begin{align}
d &= \frac{2a}{\sqrt{2+\varphi}}
   = 2\sqrt{\frac{3-\varphi}{5}}a \approx 1.05146a, \\[5mu]
D &= 2\sqrt{\frac{2+\varphi}{5}}a \approx 1.70130a.
\end{align}</math>

Its area, in terms of {{tmath|a}} and {{tmath|d}}:

<math display=block>\begin{align}
A &= \sin(\arctan2) \cdot a^2 = {2\over\sqrt5}~a^2 \approx 0.89443a^2, \\[5mu]
A &= {{\varphi}\over2}d^2\approx 0.80902d^2.
\end{align}</math>

Its [[inradius]], in terms of side {{tmath|a}}:

<math display=block>
r = \frac{a}{\sqrt{5}}.
</math>

Golden rhombi form the faces of the [[rhombic triacontahedron]], the two [[golden rhombohedra]], the [[Bilinski dodecahedron]],<ref name="golden rhombohedra" /> and the [[rhombic hexecontahedron]].<ref name=hexecontahedron />

====Golden spiral====
{{main|Golden spiral}}
[[File:FakeRealLogSpiral.svg|thumb|The [[golden spiral]] (red) and its approximation by quarter-circles (green), with overlaps shown in yellow]]
[[File:Golden triangle and Fibonacci spiral.svg|175px|thumb|A [[logarithmic spiral]] whose radius grows by the golden ratio per {{math|108°}} of turn, surrounding nested golden isosceles triangles. This is a different spiral from the [[golden spiral]], which grows by the golden ratio per {{math|90°}} of turn.<ref name=loeb-varney />]]
[[Logarithmic spirals]] are [[self-similar]] spirals where distances covered per turn are in [[geometric progression]]. A logarithmic spiral whose radius increases by a factor of the golden ratio for each quarter-turn is called the [[golden spiral]]. These spirals can be approximated by quarter-circles that grow by the golden ratio,<ref name=quarter-circles /> or their approximations generated from Fibonacci numbers,<ref name=diedrichs /> often depicted inscribed within a spiraling pattern of squares growing in the same ratio. The exact logarithmic spiral form of the golden spiral can be described by the [[Polar coordinate system|polar equation]] with {{tmath|(r,\theta)}}:
<math display=block>r = \varphi^{2\theta/\pi}.</math>

Not all logarithmic spirals are connected to the golden ratio, and not all spirals that are connected to the golden ratio are the same shape as the golden spiral. For instance, a different logarithmic spiral, encasing a nested sequence of golden isosceles triangles, grows by the golden ratio for each {{tmath|108^\circ}} that it turns, instead of the {{tmath|90^\circ}} turning angle of the golden spiral.<ref name=loeb-varney /> Another variation, called the "better golden spiral", grows by the golden ratio for each half-turn, rather than each quarter-turn.<ref name=quarter-circles />

====Dodecahedron and icosahedron====
[[File:Dodecahedron vertices.svg|240px|right|thumb|
{|
|- valign=top
| colspan=2 |
|- valign=top
|[[Cartesian coordinates]] of the [[dodecahedron]] :
<br />
|- valign=top
|<span style="color:#dd4400">{{math|(±1, ±1, ±1)}}</span>
|- valign=top
|<span style="color:#007722">{{math|(0, ±'''φ''', ±{{sfrac|1|'''φ'''}})}}</span>
|- valign=top
|<span style="color:#0011bb">{{math|(±{{sfrac|1|'''φ'''}}, 0, ±'''φ''')}}</span>
|- valign=top
| <span style="color:#cc0055">{{math|(±'''φ''', ±{{sfrac|1|'''φ'''}}, 0)}}</span>
|- valign=top
|A nested cube inside the dodecahedron is represented with <span style="color:#dd4400">dotted</span> lines.
| colspan=2 |
|}
]]

The [[regular dodecahedron]] and its [[dual polyhedron]] the [[regular icosahedron|icosahedron]] are [[Platonic solid]]s whose dimensions are related to the golden ratio. A dodecahedron has {{tmath|12}} regular pentagonal faces, whereas an icosahedron has {{tmath|20}} [[equilateral triangle]]s; both have {{tmath|30}} [[Edge (geometry)|edges]].<ref name ="Regular dodecahedron">{{harvtxt|Livio|2002|pp=70–72}}</ref>

For a dodecahedron of side {{tmath|a}}, the [[radius]] of a circumscribed and inscribed sphere, and [[Midsphere|midradius]] are ({{tmath|r_u}}, {{tmath|r_i}}, and {{tmath|r_m}}, respectively):

{{bi |left=1.6 |1=<math>r_u = a\, \frac{\sqrt{3}\varphi}{2},</math> <math>r_i = a\, \frac{\varphi^2}{2 \sqrt{3-\varphi}},</math> and <math>r_m = a\, \frac{\varphi^2}{2}.</math>}}

While for an icosahedron of side {{tmath|a}}, the radius of a circumscribed and inscribed sphere, and [[Midsphere|midradius]] are:

{{bi |left=1.6 |1=<math>r_u = a\frac{\sqrt{\varphi \sqrt{5}}}{2},</math> <math>r_i = a\frac{\varphi^2}{2 \sqrt{3}},</math> and <math>r_m = a\frac{\varphi}{2}.</math>}}

The volume and surface area of the dodecahedron can be expressed in terms of {{tmath|\varphi}}:

{{bi |left=1.6 |1=<math>A_d = \frac{15\varphi}{\sqrt{3-\varphi}}</math> and <math>V_d = \frac{5\varphi^3}{6-2\varphi}.</math>}}

As well as for the icosahedron:

{{bi|left=1.6|1=<math>A_i = 20\frac{\varphi^{2}}{2}</math> and <math>V_i = \frac{5}{6}(1 + \varphi).</math>}}

[[File:Icosahedron-golden-rectangles.svg|175px|right|thumb|Three golden rectangles touch all of the {{math|12}} vertices of a [[regular icosahedron]].]]

These geometric values can be calculated from their [[Cartesian coordinates]], which also can be given using formulas involving {{tmath|\varphi}}. The coordinates of the dodecahedron are displayed on the figure to the right, while those of the icosahedron are:

<math display=block>
(0,\pm1,\pm\varphi),\ 
(\pm1,\pm\varphi,0),\ 
(\pm\varphi,0,\pm1).
</math>

Sets of three golden rectangles intersect [[perpendicular]]ly inside dodecahedra and icosahedra, forming [[Borromean rings]].<ref name=borromean /><ref name=BurgerStarbird /> In dodecahedra, pairs of opposing vertices in golden rectangles meet the centers of pentagonal faces, and in icosahedra, they meet at its vertices. The three golden rectangles together contain all {{tmath|12}} vertices of the icosahedron, or equivalently, intersect the centers of all {{tmath|12}} of the dodecahedron's faces.<ref name="Regular dodecahedron" />

A [[cube]] can be [[Inscribed figure|inscribed]] in a regular dodecahedron, with some of the diagonals of the pentagonal faces of the dodecahedron serving as the cube's edges; therefore, the edge lengths are in the golden ratio. The cube's volume is {{tmath|2/(2+\varphi)}} times that of the dodecahedron's.<ref name=hume /> In fact, golden rectangles inside a dodecahedron are in golden proportions to an inscribed cube, such that edges of a cube and the long edges of a golden rectangle are themselves in {{tmath|\textstyle \varphi \mathbin: \varphi^{2} }} ratio. On the other hand, the [[octahedron]], which is the dual polyhedron of the cube, can inscribe an icosahedron, such that an icosahedron's {{tmath|12}} vertices touch the {{tmath|12}} edges of an octahedron at points that divide its edges in golden ratio.<ref name="59 Icosahedra" />