Golden ratio in the context of "Straightedge and compass construction"

In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself.

Among irrational numbers are the ratio π of a circle's circumference to its diameter, Euler's number e, the golden ratio φ, and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational.

In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio ${\displaystyle (1+{\sqrt {5}})/2}$ is an algebraic number, because it is a root of the polynomial ${\displaystyle x^{2}-x-1}$, i.e., a solution to the equation ${\displaystyle x^{2}-x-1=0}$, and the complex number ${\displaystyle 1+i}$ is algebraic because it is a root of the polynomial ${\displaystyle x^{4}+4}$. Algebraic numbers include all integers, rational numbers, and n-th roots of integers.

Algebraic complex numbers are closed under addition, subtraction, multiplication and division, and hence form a field, denoted ${\displaystyle {\overline {\mathbb {Q} }}}$. The set of algebraic real numbers ${\displaystyle {\overline {\mathbb {Q} }}\cap \mathbb {R} }$ is also a field.

In the geometry of spirals, the pitch angle or pitch of a spiral is the angle made by the spiral with a circle through one of its points, centered at the center of the spiral. Equivalently, it is the complementary angle to the angle made by the vector from the origin to a point on the spiral, with the tangent vector of the spiral at the same point. Pitch angles are used to characterize the steepness of spirals, such as in astronomy to describe spiral galaxies.

Logarithmic spirals, for example, are characterized by the property that the pitch angle remains invariant for all points of the spiral. Two logarithmic spirals are congruent when they have the same pitch angle, but otherwise are not congruent. For instance, only the golden spiral has pitch angle$${\displaystyle \arctan \left({\frac {\ln \varphi }{\pi /2}}\right)\approx 17^{\circ },}$$where ${\displaystyle \varphi }$ denotes the golden ratio; logarithmic spirals with other angles are not golden spirals.

The rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron and can be seen as an elongated rhombic icosahedron.

The ratio of the long diagonal to the short diagonal of each face is exactly equal to the golden ratio, φ, so that the acute angles on each face measure 2 arctan(⁠1/φ⁠) = arctan(2), or approximately 63.43°. A rhombus so obtained is called a golden rhombus.

Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty. Mathematics can be discerned in arts such as music, dance, painting, architecture, sculpture, and textiles. This article focuses, however, on mathematics in the visual arts.

Mathematics and art have a long historical relationship. Artists have used mathematics since the 4th century BC when the Greek sculptor Polykleitos wrote his Canon, prescribing proportions conjectured to have been based on the ratio 1:√2 for the ideal male nude. Persistent popular claims have been made for the use of the golden ratio in ancient art and architecture, without reliable evidence. In the Italian Renaissance, Luca Pacioli wrote the influential treatise De divina proportione (1509), illustrated with woodcuts by Leonardo da Vinci, on the use of the golden ratio in art. Another Italian painter, Piero della Francesca, developed Euclid's ideas on perspective in treatises such as De Prospectiva Pingendi, and in his paintings. The engraver Albrecht Dürer made many references to mathematics in his work Melencolia I. In modern times, the graphic artist M. C. Escher made intensive use of tessellation and hyperbolic geometry, with the help of the mathematician H. S. M. Coxeter, while the De Stijl movement led by Theo van Doesburg and Piet Mondrian explicitly embraced geometrical forms. Mathematics has inspired textile arts such as quilting, knitting, cross-stitch, crochet, embroidery, weaving, Turkish and other carpet-making, as well as kilim. In Islamic art, symmetries are evident in forms as varied as Persian girih and Moroccan zellige tilework, Mughal jali pierced stone screens, and widespread muqarnas vaulting.

A golden triangle, also called a sublime triangle, is an isosceles triangle in which the duplicated side is in the golden ratio ${\displaystyle \varphi }$ to the base side: