Golden rectangle

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|image1=Golden Rectangle Construction.svg

|caption1=Construction of a golden rectangle.{{efn|$(\backslash tfrac\{1\}\{2\})^2\; +\; 1^2\; =\; \backslash tfrac\{5\}\{2^2\}$}}

|image2=Golden vs Fibonacci Spiral.svg

|caption2=Golden spiral and intersecting diagonals.

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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.{{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}} Thus, a golden rectangle can be constructed with only a straightedge and compass in four steps:

1. Draw a square
2. Draw a line from the midpoint of one side of the square to an opposite corner
3. Use that line as the radius to draw an arc that defines the height of the rectangle
4. Complete the golden rectangle

A distinctive feature of this shape is that when a 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".{{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}}

File:Golden_rectangle_whirl.svg

Divide a square into four congruent right triangles with legs in ratio {{math|1 : 2}} and arrange these in the shape of a golden rectangle, enclosing a 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 whirl of converging golden rectangles.{{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}}

The logarithmic spiral through the vertices of adjacent triangles has polar slope $k\; =\backslash frac\{\backslash ln(\; \backslash varphi)\}\{\backslash arctan(\; \backslash tfrac\{1\}\{2\})\}\; .$ 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 $\backslash varphi^2\; =\backslash sum\_\{n=0\}^\{\backslash infty\}\; \backslash varphi^\{-n\}\; .$ The areas of the triangles in each spiral region sum to $\backslash varphi\; =\backslash sum\_\{n=0\}^\{\backslash infty\}\; \backslash varphi^\{-2n\}\; ;$ the perimeters are equal to {{tmath|5 +\sqrt{5} }} (grey) and {{tmath|4\varphi}} (yellow regions).

The proportions of the golden rectangle have been observed as early as the Babylonian Tablet of Shamash {{nowrap|(c. 888–855 BC)}},{{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}} though Mario Livio calls any knowledge of the golden ratio before the Ancient Greeks "doubtful".{{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}}

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."{{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=Broadway Books |location=New York |page=136 |url=https://archive.org/details/goldenratio00mari/page/136 |isbn=0-7679-0816-3}}

The 1927 Villa Stein designed by Le Corbusier, some of whose architecture utilizes the golden ratio, features dimensions that closely approximate golden rectangles.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".

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|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.

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Euclid gives an alternative construction of the golden rectangle using three polygons 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 a² + b² = c², so line segments with these lengths form a 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.{{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}}

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.{{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}}

File:Golden_rectangle_segmented.svg

Assume a golden rectangle has been constructed as indicated above, with height {{math|1}}, length {{tmath|\varphi}} and diagonal length $\backslash sqrt\{\backslash varphi^2\; +1\}$. The triangles on the diagonal have altitudes $1\; /\backslash sqrt\{1\; +\backslash varphi^\{-2\}\}\backslash ,;$ 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 of the square, the original golden rectangle and the two scaled copies along the diagonal have linear sizes in the ratios $\backslash varphi^2\; :\backslash varphi\; :1\backslash ,,$ the square and rectangle opposite the diagonal both have areas equal to {{tmath|\varphi^{-2}.}}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 }}

Relative to vertex {{math|A}}, the coordinates of feet of altitudes {{math|U}} and {{math|V}} are $\backslash left(\; \backslash tfrac\{1\}\{\backslash sqrt\{5\}\},\; \backslash tfrac\{1\}\{\backslash varphi\; \backslash sqrt\{5\}\}\; \backslash right)$ and $\backslash left(\; \backslash tfrac\{\backslash varphi^2\}\{\backslash sqrt\{5\}\},\; \backslash tfrac\{\backslash varphi\}\{\backslash sqrt\{5\}\}\; \backslash right)$; 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:

File:Pentagon_segments.svg

:$\backslash begin\{align\}$

\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}

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

Both the lengths of the diagonal sections and the trigonometric values are elements of quartic number field $K\; =\backslash mathbb\{Q\}\backslash left(\; \backslash sqrt\{(5\; +\backslash sqrt\{5\})\; /2\}\; \backslash right).$

The golden rhombus with edge {{tmath|\tfrac{1}{2} }} has diagonal lengths equal to {{tmath|\overline{U V} }} and {{tmath|\overline{A U}.}} The regular pentagon with side length $\backslash tfrac\{2\}\{\backslash varphi\}\; =\backslash sec(36)$ has area {{tmath|5\overline{A U}.}} Its 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.{{MathWorld |id=Pentagram |title=Pentagram}}

File:Golden_pentatonic_scales.svg

Interpreting the diagonal sections as musical string lengths results in a set of ten corres­ponding pitches, one of which doubles at the octave. Mapping the intervals in logarithmic scale — with the 'golden octave' equal to {{tmath|\varphi^4}} — shows equally tempered semitones, minor thirds 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 accuracy.{{efn |1=Absent in modern 12 equal, this interval is accurately rendered in quarter-comma meantone temperament. Relative to base note D, the two augmented sixths E♭ - c♯ and B♭ - g♯ with frequency ratio 5^5/2/2⁵ are only 3 cents short of just ratio 7/4.}}

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

• Golden triangle – Triangle with sides in the golden ratio
• {{annotated link|Kepler triangle}}
• {{annotated link|Golden rhombus}}
• Square root of 5 – Algebraic relationship between √5 and φ
• {{annotated link|Rabatment of the rectangle}}

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{{reflist}}

{{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