Editing Golden ratio (section)

====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 />