View source for Golden rectangle

{{Short description|Rectangle with side lengths in the golden ratio}}
[[File:SimilarGoldenRectangles.svg |thumb |{{sfrac|a|b}} = {{sfrac|a+b|a}} = φ.]]
In [[geometry]], a '''golden rectangle''' is a [[rectangle]] with side lengths in [[golden ratio]] <math>\tfrac{1 + \sqrt{5}}{2} :1,</math> or {{tmath|\varphi :1,}} with {{tmath|\varphi}} approximately equal to {{math|1.618}} or {{math|89/55.}}<!-- If you want to add more decimals, please consider that the longer side of a rectangle with a shorter side of 1 meter should be measured more accurately than to the nearest millimeter to make the next decimal meaningful! Note that 1.618... is an irrational number, and φ is an exact number. Also the is number NOT inside math tags so people can copy/paste it. -->

Golden rectangles exhibit a special form of [[self-similarity]]: if a square is added to the long side, or removed from the short side, the result is a golden rectangle as well.

==Construction==
{{multiple image
|total_width=230
|direction=vertical
|image1=Golden Rectangle Construction.svg
|caption1=Construction of a golden rectangle.{{efn|<math>(\tfrac{1}{2})^2 + 1^2 = \tfrac{5}{2^2}</math>}}
|image2=Golden vs Fibonacci Spiral.svg
|caption2=Golden spiral and intersecting diagonals.
}}
Owing to the [[Pythagorean theorem]], the diagonal dividing one half of a square equals the radius of a circle whose outermost point is the corner of a golden rectangle added to the square.<ref>{{Cite book |last1=Posamentier |first1=Alfred S. |author-link1=Alfred S. Posamentier |last2=Lehmann |first2=Ingmar |title=The Glorious Golden Ratio |year=2011 |publisher=[[Prometheus Books]] |location=New York |page=11 |url=https://books.google.com/books?id=Gw-lqvE6fNgC&pg=PT11 |isbn=9-781-61614-424-1}}</ref> Thus, a golden rectangle can be [[Straightedge and compass construction|constructed with only a straightedge and compass]] in four 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

A distinctive feature of this shape is that when a [[square (geometry)|square]] section is added—or removed—the product is another golden rectangle, having the same [[aspect ratio]] as the first. Square addition or removal can be repeated infinitely, in which case corresponding corners of the squares form an infinite sequence of points on the [[golden spiral]], the unique [[logarithmic spiral]] with this property. Diagonal lines drawn between the first two orders of embedded golden rectangles will define the intersection point of the diagonals of all the embedded golden rectangles; [[Clifford A. Pickover]] referred to this point as "the Eye of God".<ref>{{cite book |last=Pickover |first=Clifford A. |author-link=Clifford A. Pickover |title=The Loom of God: Mathematical Tapestries at the Edge of Time |year=1997 |publisher=Plenum Press |location=New York |pages=167–175 |isbn=0-3064-5411-4}}</ref>

===Golden whirl===
[[File:Golden_rectangle_whirl.svg |thumb |upright=1.5 |A whirl of golden rectangles.]]
Divide a square into four [[Congruence (geometry)|congruent]] [[right triangle]]s with legs in ratio {{math|1 : 2}} and arrange these in the shape of a golden rectangle, enclosing a [[Similarity (geometry)|similar]] rectangle that is scaled by factor {{tmath|\tfrac{1}{\varphi} }} and rotated about the centre by {{tmath|\arctan(\tfrac{1}{2}).}} Repeating the construction at successively smaller scales results in four infinite sequences of adjoining right triangles, tracing a [[mice problem|whirl]] of converging golden rectangles.<ref>{{cite book |last=Walser |first=Hans |title=Spiralen, Schraubenlinien und spiralartige Figuren |language=de |date=2022 |publisher=[[Springer Spektrum]] |location=Berlin, Heidelberg |pages=75–76 |doi=10.1007/978-3-662-65132-2 |isbn=978-3-662-65131-5}}</ref>

The logarithmic spiral through the vertices of adjacent triangles has [[Pitch angle of a spiral|polar slope]] <math>k =\frac{\ln( \varphi)}{\arctan( \tfrac{1}{2})} .</math> The [[parallelogram]] between the pair of upright grey triangles has perpendicular diagonals in ratio {{tmath|\varphi}}, hence is a [[golden rhombus]].

If the triangle has legs of lengths {{math|1}} and {{math|2}} then each discrete spiral has length <math>\varphi^2 =\sum_{n=0}^{\infty} \varphi^{-n} .</math> The areas of the triangles in each spiral region sum to <math>\varphi =\sum_{n=0}^{\infty} \varphi^{-2n} ;</math> the perimeters are equal to {{tmath|5 +\sqrt{5} }} (grey) and {{tmath|4\varphi}} (yellow regions).

==History==
The proportions of the golden rectangle have been observed as early as the [[Babylonia]]n ''[[Tablet of Shamash]]'' {{nowrap|(c. 888–855 BC)}},<ref>{{cite book |last=Olsen |first=Scott |title=The Golden Section: Nature's Greatest Secret |year=2006 |publisher=Wooden Books |location=Glastonbury |page=3 |url=https://books.google.com/books?id=RxFdDwAAQBAJ&pg=PA3 |isbn=978-1-904263-47-0}}</ref> though [[Mario Livio]] calls any knowledge of the golden ratio before the [[Ancient Greeks]] "doubtful".<ref>{{cite web |last=Livio |first=Mario |author-link=Mario Livio |title=The Golden Ratio in Art: Drawing heavily from The Golden Ratio |year=2014 |page=6 |url=https://www.math.ksu.edu/~cjbalm/Quest/Breakouts/GR.pdf |access-date=2019-09-11}}</ref>

According to Livio, since the publication of [[Luca Pacioli]]'s ''[[Divina proportione]]'' in 1509, "the Golden Ratio started to become available to artists in theoretical treatises that were not overly mathematical, that they could actually use."<ref>{{cite book |last=Livio |first=Mario |author-link=Mario Livio |year=2002 |title=The Golden Ratio: The Story of Phi, The World's Most Astonishing Number |publisher=[[Random House|Broadway Books]] |location=New York |page=136 |url=https://archive.org/details/goldenratio00mari/page/136 |isbn=0-7679-0816-3}}</ref>

The 1927 [[Villa Stein]] designed by [[Le Corbusier]], some of whose [[Golden ratio#Architecture|architecture utilizes the golden ratio]], features dimensions that closely approximate golden rectangles.<ref>Le Corbusier, ''The Modulor'', p. 35, as cited in: {{cite book |last=Padovan |first=Richard |author-link=Richard Padovan |title=Proportion: Science, Philosophy, Architecture |year=1999 |publisher=Taylor and Francis |location=London |page=320 |isbn=0-419-22780-6}} "Both the paintings and the architectural designs make use of the golden section".</ref>

==Relation to regular polygons and polyhedra==
{{multiple image
|image1=Euclid XIII.10.svg
|caption1=Construction of half-golden rectangle (central right triangle) from polygons.
|image2=Icosahedron-golden-rectangles.svg
|caption2=Three golden rectangles in an [[icosahedron]].
|total_width=480}}

[[Euclid]] gives an alternative construction of the golden rectangle using three polygons [[Circumscribed circle|circumscribed]] by congruent circles: a regular [[decagon]], [[hexagon]], and [[pentagon]]. The respective lengths ''a'', ''b'', and ''c'' of the sides of these three polygons satisfy the equation ''a''<sup>2</sup>&nbsp;+&nbsp;''b''<sup>2</sup>&nbsp;=&nbsp;''c''<sup>2</sup>, so line segments with these lengths form a [[Special right triangles#Sides of regular polygons|right triangle]] (by the converse of the [[Pythagorean theorem]]). The ratio of the side length of the hexagon to the decagon is the golden ratio, so this triangle forms half of a golden rectangle.<ref>{{cite web |url=http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/propXIII10.html |title=Euclid's Elements, Book XIII, Proposition 10 |last=Joyce |first=David E. |author-link=David E. Joyce (mathematician)|year=2014 |publisher=Department of Mathematics, Clark University |access-date=2024-09-13}}</ref>

The [[convex hull]] of two opposite edges of a regular [[icosahedron]] forms a golden rectangle. The twelve vertices of the icosahedron can be decomposed in this way into three mutually-perpendicular golden rectangles, whose boundaries are linked in the pattern of the [[Borromean rings]].<ref>{{Cite book |title=The Heart of Mathematics: An Invitation to Effective Thinking |first1=Edward B. |last1=Burger |author-link1=Edward Burger |first2=Michael P. |last2=Starbird |author-link2=Michael Starbird |publisher=Springer |location=New York |year=2005 |page=382 |url=https://books.google.com/books?id=M-qK8anbZmwC&pg=PA382 |isbn=978-1931914413}}</ref>

==Relation to angles of the golden triangle==
[[File:Golden_rectangle_segmented.svg |thumb |upright=1.5 |Powers of {{math|''φ''}} within a golden rectangle.]]

Assume a golden rectangle has been constructed as indicated above, with height {{math|1}}, length {{tmath|\varphi}} and [[diagonal]] length <math> \sqrt{\varphi^2 +1}</math>. The triangles on the diagonal have [[Altitude (triangle)|altitudes]] <math>1 /\sqrt{1 +\varphi^{-2}}\,;</math> each perpendicular foot divides the diagonal in ratio {{tmath|\varphi^2.}}

If a horizontal line is drawn through the intersection point of the diagonal and the internal [[Edge (geometry)|edge]] of the square, the original golden rectangle and the two scaled copies along the diagonal have linear sizes in the ratios <math>\varphi^2 :\varphi :1\,,</math> the square and rectangle opposite the diagonal both have areas equal to {{tmath|\varphi^{-2}.}}<ref>Analogue to the construction in: {{cite journal |last=Crilly |first=Tony |date=1994 |title=A supergolden rectangle |journal=[[The Mathematical Gazette]] |volume=78 |issue=483 |pages=320–325 |doi=10.2307/3620208|jstor=3620208 }}</ref>

Relative to [[Vertex (geometry)|vertex]] {{math|A}}, the coordinates of feet of altitudes {{math|U}} and {{math|V}} are <math>\left( \tfrac{1}{\sqrt{5}}, \tfrac{1}{\varphi \sqrt{5}} \right)</math> and <math>\left( \tfrac{\varphi^2}{\sqrt{5}}, \tfrac{\varphi}{\sqrt{5}} \right)</math>; the length of line segment {{tmath|\overline{U V} }} is equal to altitude {{tmath|h.}}

If the diagram is further subdivided by perpendicular lines through {{math|U}} and {{math|V}}, the lengths of the diagonal and its subsections can be expressed as [[trigonometric functions]] of arguments 72 and 36 degrees, the angles of the [[Golden triangle (mathematics)|golden triangle]]:

[[File:Pentagon_segments.svg |thumb |upright=1.5 |Diagonal segments of the golden rectangle measure nested pentagons. The ratio {{math|size=94%|AU:SV}} is {{math|''φ''{{sup|2}}.}}]]
:<math>\begin{align}
\overline{A B} + \overline{A S} &=\tan(72)\\
\overline{A B} =\sqrt{\varphi^2 +1} &=2\sin(72)\\
\overline{A V} =\varphi /\overline{A S} &=\cot(36)\\
\overline{A S} =\sqrt{1 +\varphi^{-2}} &=2\sin(36)\\
\overline{U V} =1 /\overline{A S} &=\cot(36) /\varphi\\
\overline{S B} =\overline{A S} /\varphi &=\tan(36)\\
\overline{U S} =2 /(\varphi\overline{A B}) &=2\cot(72)\\
\overline{A U} =1 /\overline{A B} &=\varphi\cot(72)\\
\overline{U V} - \overline{A U} &=\cot(72)\\
\overline{S V} =(2 -\varphi) /\overline{A B} &=\cot(72) /\varphi,\end{align}</math>

:with {{tmath|1=\varphi =2\cos(36).}}

Both the lengths of the diagonal sections and the trigonometric values are elements of quartic [[Algebraic number field|number field]] <math>K =\mathbb{Q}\left( \sqrt{(5 +\sqrt{5}) /2} \right).</math>

The [[golden rhombus]] with edge {{tmath|\tfrac{1}{2} }} has diagonal lengths equal to {{tmath|\overline{U V} }} and {{tmath|\overline{A U}.}} The [[Pentagon#Regular pentagons|regular pentagon]] with side length <math>\tfrac{2}{\varphi} =\sec(36)</math> has area {{tmath|5\overline{A U}.}} Its [[Pentagram#Geometry|five diagonals]] divide the pentagon into golden triangles and gnomons, and an upturned, scaled copy at the centre. Since the regular pentagon is defined by its side length and the angles of the golden triangle, it follows that all measures can be expressed in powers of {{tmath|\varphi }} and the diagonal segments of the golden rectangle, as illustrated above.<ref>{{MathWorld |id=Pentagram |title=Pentagram}}</ref>

[[File:Golden_pentatonic_scales.svg |thumb |upright=2.2 |Intervals on the diagonal of the golden rectangle.]]
Interpreting the diagonal sections as [[monochord|musical string lengths]] results in a set of ten corres&shy;ponding [[Pitch (music)|pitches]], one of which doubles at the [[octave]]. Mapping the [[Interval (music)|intervals]] in [[logarithmic scale]] — with the 'golden octave' equal to {{tmath|\varphi^4}} — shows [[equal temperament|equally tempered]] [[semitone]]s, [[minor third]]s and one [[major second]] in the span of an [[eleventh]]. An analysis in musical terms is substantiated by the single exceptional pitch proportional to {{tmath|\overline{U S} }}, that approximates the [[harmonic seventh]] within remarkable one [[Cent (music)|cent]] accuracy.{{efn |1=Absent in modern [[12 equal temperament|12 equal]], this interval is accurately rendered in [[Quarter-comma meantone#Size of intervals|quarter-comma meantone]] temperament. Relative to base note D, the two [[augmented sixth]]s E♭ - c♯ and B♭ - g♯ with frequency ratio 5<sup>5/2</sup>/2<sup>5</sup> are only 3 cents short of just ratio 7/4.}}

This set of ten tones can be partitioned into two [[Mode (music)|modes]] of the [[pentatonic scale]]: the palindromic 'Egyptian' mode (red dots, Chinese {{Audio|Ruibin_diao.ogg|''rui bin diao''}} [[Guqin tunings|guqin tuning]]) and the stately 'blues minor' mode (blue dots, Chinese {{Audio|Mangong_diao.ogg|''man gong diao''}} tuning).

==See also==
*[[Golden triangle (mathematics)|Golden triangle]] – Triangle with sides in the golden ratio
*{{annotated link|Kepler triangle}}
*{{annotated link|Golden rhombus}}
*[[Square root of 5#Relation to the golden ratio and Fibonacci numbers|Square root of 5]] – Algebraic relationship between √5 and φ
*{{annotated link|Rabatment of the rectangle}}

==Notes==
{{notelist}}

==References==
{{reflist}}

==External links==
{{Commons category|Golden rectangle}}
* {{MathWorld |id=GoldenRectangle |title=Golden rectangle}}
* [https://arxiv.org/abs/physics/0411195 The golden mean and the physics of aesthetics]
* [https://www.researchgate.net/publication/323869376_An_example_of_constructive_defining_From_a_golden_rectangle_to_golden_quadrilaterals_and_Beyond From golden rectangle to golden quadrilaterals]
* [https://www.huygens-fokker.org/bpsite/833cent.html Heinz Bohlen's 833 cents φ scale]
* [https://anaphoria.com/wilsonintroMERU.html Ervin Wilson's recurrent sequence scales]

{{Metallic ratios}}

[[Category:Golden ratio]]
[[Category:Elementary geometry]]
[[Category:Types of quadrilaterals]]