Golden rectangle

Template:Sfrac = Template:Sfrac = φ.

In geometry, a golden rectangle is a rectangle with side lengths in golden ratio $\frac{1+\sqrt{5}}{2}:1,$ or Template:Tmath with Template:Tmath approximately equal to Template:Math or Template:Math

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

Template:Multiple image 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. 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".

Golden whirl

A whirl of golden rectangles.

Divide a square into four congruent right triangles with legs in ratio Template:Math and arrange these in the shape of a golden rectangle, enclosing a similar rectangle that is scaled by factor Template:Tmath and rotated about the centre by Template:Tmath Repeating the construction at successively smaller scales results in four infinite sequences of adjoining right triangles, tracing a whirl of converging golden rectangles.

The logarithmic spiral through the vertices of adjacent triangles has polar slope $k=\frac{\ln (\phi )}{\arctan (\frac{1}{2})}.$ The parallelogram between the pair of upright grey triangles has perpendicular diagonals in ratio Template:Tmath, hence is a golden rhombus.

If the triangle has legs of lengths Template:Math and Template:Math then each discrete spiral has length ${\phi }^2=\sum _{n=0}^{\mathrm{\infty }}{\phi }^{-n}.$ The areas of the triangles in each spiral region sum to $\phi =\sum _{n=0}^{\mathrm{\infty }}{\phi }^{-2n};$ the perimeters are equal to Template:Tmath (grey) and Template:Tmath (yellow regions).

History

The proportions of the golden rectangle have been observed as early as the Babylonian Tablet of Shamash Template:Nowrap, though Mario Livio calls any knowledge of the golden ratio before the Ancient Greeks "doubtful".

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."

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

Relation to regular polygons and polyhedra

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.

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.

Relation to angles of the golden triangle

Powers of Template:Math within a golden rectangle.

Assume a golden rectangle has been constructed as indicated above, with height Template:Math, length Template:Tmath and diagonal length $\sqrt{{\phi }^2+1}$. The triangles on the diagonal have altitudes $1/\sqrt{1+{\phi }^{-2}};$ each perpendicular foot divides the diagonal in ratio Template:Tmath

If an 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 ${\phi }^2:\phi :1,$ the square and rectangle opposite the diagonal both have areas equal to Template:Tmath

Relative to vertex Template:Math, the coordinates of feet of altitudes Template:Math and Template:Math are $(\frac{1}{\sqrt{5}},\frac{1}{\phi \sqrt{5}})$ and $(\frac{{\phi }^2}{\sqrt{5}},\frac{\phi }{\sqrt{5}})$; the length of line segment Template:Tmath is equal to altitude Template:Tmath

If the diagram is further subdivided by perpendicular lines through Template:Math and Template:Math, 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:

Diagonal segments of the golden rectangle measure nested pentagons. The ratio Template:Math is Template:Math

$\begin{array}{cc}\overline{AB}+\overline{AS}& =\tan (72)\\ \overline{AB}=\sqrt{{\phi }^2+1}& =2\sin (72)\\ \overline{AV}=\phi /\overline{AS}& =\cot (36)\\ \overline{AS}=\sqrt{1+{\phi }^{-2}}& =2\sin (36)\\ \overline{UV}=1/\overline{AS}& =\cot (36)/\phi \\ \overline{SB}=\overline{AS}/\phi & =\tan (36)\\ \overline{US}=2/(\phi \overline{AB})& =2\cot (72)\\ \overline{AU}=1/\overline{AB}& =\phi \cot (72)\\ \overline{UV}-\overline{AU}& =\cot (72)\\ \overline{SV}=(2-\phi )/\overline{AB}& =\cot (72)/\phi ,\end{array}$

with Template:Tmath

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

The golden rhombus with edge Template:Tmath has diagonal lengths equal to Template:Tmath and Template:Tmath The regular pentagon with side length $\frac{2}{\phi }=\sec (36)$ has area Template:Tmath 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 Template:Tmath and the diagonal segments of the golden rectangle, as illustrated above.

Intervals on the diagonal of the golden rectangle.

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 Template:Tmath — 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 Template:Tmath, that approximates the harmonic seventh within remarkable one cent accuracy.Template:Efn

This set of ten tones can be partitioned into two modes of the pentatonic scale: the palindromic 'Egyptian' mode (red dots, Chinese Template:Audio guqin tuning) and the stately 'blues minor' mode (blue dots, Chinese Template:Audio tuning).

See also

• Golden triangle – Triangle with sides in the golden ratio
• Template:Annotated link
• Template:Annotated link
• Square root of 5 – Algebraic relationship between √5 and φ
• Template:Annotated link

External links

• Template:MathWorld
• The golden mean and the physics of aesthetics
• From golden rectangle to golden quadrilaterals
• Heinz Bohlen's 833 cents φ scale
• Ervin Wilson's recurrent sequence scales