The golden ratio is also called the golden mean or golden section (Latin: sectio aurea). Other names include extreme and mean ratio, medial section, divine proportion (Latin: proportio divina), divine section (Latin: sectio divina), golden proportion, golden cut, and golden number.
Mathematicians since Euclid have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and in a golden rectangle, which may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as financial markets, in some cases based on dubious fits to data. The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other parts of vegetation.
Some 20th-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio, believing this to be aesthetically pleasing. These often appear in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio.
Two quantities and are said to be in the golden ratio if
One method for finding the value of is to start with the left fraction. Through simplifying the fraction and substituting in
Multiplying by gives
which can be rearranged to
Using the quadratic formula, two solutions are obtained:
Because is the ratio between positive quantities, is necessarily the positive root. However, the negative root resolves to the negative inverse, , which shares many properties with the golden ratio.
According to Mario Livio,
Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. ... Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.— The Golden Ratio: The Story of Phi, the World's Most Astonishing Number
Ancient Greek mathematicians first studied the golden ratio because of its frequent appearance in geometry; the division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons. According to one story, 5th-century BC mathematician Hippasus discovered that the golden ratio was neither a whole number nor a fraction (an irrational number), surprising Pythagoreans. Euclid's Elements (c. 300 BC) provides several propositions and their proofs employing the golden ratio,[b] and contains its first known definition which proceeds as follows:
The golden ratio was studied peripherally over the next millennium. Abu Kamil (c. 850–930) employed it in his geometric calculations of pentagons and decagons; his writings influenced that of Fibonacci (Leonardo of Pisa) (c. 1170–1250), who used the ratio in related geometry problems but did not observe that it was connected to the Fibonacci numbers.
Luca Pacioli named his book Divina proportione (1509) after the ratio, and explored its properties including its appearance in some of the Platonic solids. Leonardo da Vinci, who illustrated the aforementioned book, called the ratio the sectio aurea ('golden section'). 16th-century mathematicians such as Rafael Bombelli solved geometric problems using the ratio.
German mathematician Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to the golden ratio; this was rediscovered by Johannes Kepler in 1608. The first known decimal approximation of the (inverse) golden ratio was stated as "about " in 1597 by Michael Maestlin of the University of Tübingen in a letter to Kepler, his former student. The same year, Kepler wrote to Maestlin of the Kepler triangle, which combines the golden ratio with the Pythagorean theorem. Kepler said of these:
Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into extreme and mean ratio. The first we may compare to a mass of gold, the second we may call a precious jewel.
18th-century mathematicians Abraham de Moivre, Daniel Bernoulli, and Leonhard Euler used a golden ratio-based formula which finds the value of a Fibonacci number based on its placement in the sequence; in 1843, this was rediscovered by Jacques Philippe Marie Binet, for whom it was named "Binet's formula". Martin Ohm first used the German term goldener Schnitt ('golden section') to describe the ratio in 1835. James Sully used the equivalent English term in 1875.
By 1910, mathematician Mark Barr began using the Greek letter Phi () as a symbol for the golden ratio.[d] It has also been represented by tau (), the first letter of the ancient Greek τομή ('cut' or 'section').
The zome construction system, developed by Steve Baer in the late 1960s, is based on the symmetry system of the icosahedron/dodecahedron, and uses the golden ratio ubiquitously. Between 1973 and 1974, Roger Penrose developed Penrose tiling, a pattern related to the golden ratio both in the ratio of areas of its two rhombic tiles and in their relative frequency within the pattern. This led to Dan Shechtman's early 1980s discovery of quasicrystals, some of which exhibit icosahedral symmetry.
The golden ratio is an irrational number. Below are two short proofs of irrationality:
Contradiction from an expression in lowest terms
the whole is to the longer part as the longer part is to the shorter part.
If we call the whole and the longer part then the second statement above becomes
To say that the golden ratio is rational means that is a fraction where and are integers. We may take to be in lowest terms and and to be positive. But if is in lowest terms, then the equally valued is in still lower terms. That is a contradiction that follows from the assumption that is rational.
By irrationality of √
Another short proof – perhaps more commonly known – of the irrationality of the golden ratio makes use of the closure of rational numbers under addition and multiplication. If is rational, then is also rational, which is a contradiction if it is already known that the square root of a non-square natural number is irrational.
The golden ratio is also closely related to the polynomial
which has roots and
Golden ratio conjugate and powers
The conjugate root to the minimal polynomial is
The absolute value of this quantity () corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, ), and is sometimes referred to as the golden ratio conjugate or silver ratio.[e] It is denoted here by the capital Phi ():
This illustrates the unique property of the golden ratio among positive numbers, that
or its inverse:
The conjugate and the defining quadratic polynomial relationship lead to decimal values that have their fractional part in common with :
The sequence of powers of contains these values more generally, any power of is equal to the sum of the two immediately preceding powers:
As a result, one can easily decompose any power of into a multiple of and a constant. The multiple and the constant are always adjacent Fibonacci numbers. This leads to another property of the positive powers of :
Continued fraction and square root
It is in fact the simplest form of a continued fraction, alongside its reciprocal form:
The convergents of these continued fractions ( ... or ...) are ratios of successive Fibonacci numbers. The consistently small terms in its continued fraction explain why the approximants converge so slowly. This makes the golden ratio an extreme case of the Hurwitz inequality for Diophantine approximations, which states that for every irrational , there are infinitely many distinct fractions such that,
This means that the constant cannot be improved without excluding the golden ratio. It is in fact the smallest number that must be excluded to generate closer approximations of such Lagrange numbers.
A continued square root form for can be obtained from , yielding:
Relationship to Fibonacci and Lucas numbers
Fibonacci numbers and Lucas numbers have an intricate relationship with the golden ratio. In the Fibonacci sequence, each number is equal to the sum of the preceding two, starting with the base sequence :
The sequence of Lucas numbers (not to be confused with the generalized Lucas sequences, of which this is part) is like the Fibonacci sequence, in-which each term is the sum of the previous two, however instead starts with :
In other words, if a Fibonacci and Lucas number is divided by its immediate predecessor in the sequence, the quotient approximates .
For example, and .
These approximations are alternately lower and higher than and converge to as the Fibonacci and Lucas numbers increase.
Closed-form expressions for the Fibonacci and Lucas sequences that involve the golden ratio are:
Combining both formulas above, one obtains a formula for that involves both Fibonacci and Lucas numbers:
Between Fibonacci and Lucas numbers one can deduce which simplifies to express the limit of the quotient of Lucas numbers by Fibonacci numbers as equal to the square root of five:
Successive powers of the golden ratio obey the Fibonacci recurrence, i.e.
The reduction to a linear expression can be accomplished in one step by using:
This identity allows any polynomial in to be reduced to a linear expression, as in:
Consecutive Fibonacci numbers can also be used to obtain a similar formula for the golden ratio, here by infinite summation:
In particular, the powers of themselves round to Lucas numbers (in order, except for the first two powers, and , are in reverse order):
and so forth. The Lucas numbers also directly generate powers of the golden ratio; for :
Rooted in their interconnecting relationship with the golden ratio is the notion that the sum of third consecutive Fibonacci numbers equals a Lucas number, that is ; and, importantly, that .
Both the Fibonacci sequence and the sequence of Lucas numbers can be used to generate approximate forms of the golden spiral (which is a special form of a logarithmic spiral) using quarter-circles with radii from these sequences, differing only slightly from the true golden logarithmic spiral. Fibonacci spiral is generally the term used for spirals that approximate golden spirals using Fibonacci number-sequenced squares and quarter-circles.
The golden ratio features prominently in geometry. For example, it is intrinsically involved in the internal symmetry of the pentagon, and extends to form part of the coordinates of the vertices of a regular dodecahedron, as well as those of a 5-cell. It features in the Kepler triangle and Penrose tilings too, as well as in various other polytopes.
Dividing by interior division
- Having a line segment construct a perpendicular at point with half the length of Draw the hypotenuse
- Draw an arc with center and radius This arc intersects the hypotenuse at point
- Draw an arc with center and radius This arc intersects the original line segment at point Point divides the original line segment into line segments and with lengths in the golden ratio.
Dividing by exterior division
- Draw a line segment and construct off the point a segment perpendicular to and with the same length as
- Do bisect the line segment with
- A circular arc around with radius intersects in point the straight line through points and (also known as the extension of ). The ratio of to the constructed segment is the golden ratio.
Application examples you can see in the articles Pentagon with a given side length, Decagon with given circumcircle and Decagon with a given side length.
Two arcs that make a circle can be proportioned in golden ratio, therein generating a golden angle, :
Logarithmic spirals are self-similar spirals where distances covered per turn are in geometric progression. Importantly, isosceles golden triangles can be encased by a golden logarithmic spiral, such that successive turns of a spiral generate new golden triangles inside. This special case of logarithmic spirals is called the golden spiral, and it exhibits continuous growth in golden ratio. That is, for every turn, there is a growth factor of . As mentioned above, these golden spirals can be approximated by quarter-circles generated from Fibonacci and Lucas number-sized squares that are tiled together. In their exact form, they can be described by the polar equation with :
As with any logarithmic spiral, for with at right angles:
Its polar slope can be calculated using alongside from above,
It has a complementary angle, :
Golden spirals can be symmetrically placed inside pentagons and pentagrams as well, such that fractal copies of the underlying geometry are reproduced at all scales.
In triangles, quadrilaterals, and pentagons
George Odom has given a remarkably simple construction for involving an equilateral triangle: if an equilateral triangle is inscribed in a circle and the line segment joining the midpoints of two sides is produced to intersect the circle in either of two points, then these three points are in golden proportion. This result is a straightforward consequence of the intersecting chords theorem and can be used to construct a regular pentagon, a construction that attracted the attention of the noted Canadian geometer H. S. M. Coxeter who published it in Odom's name as a diagram in the American Mathematical Monthly accompanied by the single word "Behold!" 
The Kepler triangle can also be understood as the right triangle formed by three squares whose areas are also in golden geometric progression .
Fittingly, the Pythagorean means for are precisely , , and . It is from these ratios that we are able to geometrically express the fundamental defining quadratic polynomial for with the Pythagorean theorem; that is, .
The inradius of an isosceles triangle is greatest when the triangle is composed of two mirror Kepler triangles, such that their bases lie on the same line. Also, the isosceles triangle of given perimeter with the largest possible semicircle is one from two mirrored Kepler triangles.
A square pyramid defined by medial right triangles in golden ratio is sometimes termed a golden pyramid.
A golden triangle is characterized as an isosceles with the property that bisecting the angle produces new acute and obtuse isosceles triangles and that are similar to the original, as well as in leg to base length ratios of and , respectively.
The acute isosceles triangle is sometimes called a sublime triangle, and the ratio of its base to its equal-length sides is . Its apex angle is equal to:
Both base angles of the isosceles golden triangle equal degrees each, since the sum of the angles of a triangle must equal degrees. It is the only triangle to have its three angles in ratio. A regular pentagram contains five acute sublime triangles, and a regular decagon contains ten, as each two vertices connected to the center form acute golden triangles.
The obtuse isosceles triangle is sometimes called a golden gnomon, and the ratio of its base to its other sides is the reciprocal of the golden ratio, . The measure of its apex angle is:
Its two base angles equal each. It is the only triangle whose internal angles are in ratio. It's base angles, being equal to , are the same measure as that of the acute golden triangle's apex angle. Five golden gnomons can be created from adjacent sides of a pentagon whose non-coincident vertices are joined by a diagonal of the pentagon.
Appropriately, the ratio of the area of the obtuse golden gnomon to that of the acute sublime triangle is in golden ratio. Bisecting a base angle inside a sublime triangle produces a golden gnomon, and another a sublime triangle. Bisecting the apex angle of a golden gnomon in ratio produces two new golden triangles, too. Golden triangles that are decomposed further like this into pairs of isosceles and obtuse golden triangles are known as Robinson triangles.
The golden ratio proportions the adjacent side lengths of a golden rectangle in ratio. Stacking golden rectangles produces golden rectangles anew, and removing or adding squares from golden rectangles leaves rectangles still proportioned in ratio. They can be generated by golden spirals, through successive Fibonacci and Lucas number-sized squares and quarter circles. They feature prominently in the icosahedron as well as in the dodecahedron (see section below for more detail).
The lengths of its short and long diagonals and , in terms of side length are:
Its area, in terms of ,and :
Its inradius, in terms of side :
Pentagon and pentagram
In a regular pentagon the ratio of a diagonal to a side is the golden ratio, while intersecting diagonals section each other in the golden ratio. The golden ratio properties of a regular pentagon can be confirmed by applying Ptolemy's theorem to the quadrilateral formed by removing one of its vertices. If the quadrilateral's long edge and diagonals are and short edges are then Ptolemy's theorem gives which yields,
The diagonal segments of a pentagon form a pentagram, or five-pointed star polygon, whose geometry is quintessentially described by . Primarily, each intersection of edges sections other edges in the golden ratio. The ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (that is, a side of the inverted pentagon in the pentagram's center) is as the four-color illustration shows.
A pentagram has ten isosceles triangles: five are acute sublime triangles, and five are obtuse golden gnomons. In all of them, the ratio of the longer side to the shorter side is These can be decomposed further into pairs of golden Robinson triangles, which become relevant in Penrose tilings.
Otherwise, pentagonal and pentagrammic geometry permits us to calculate the following values for :
The golden ratio appears prominently in Penrose tilings, which are aperiodic tilings of the plane that contain pentagonal symmetry. They were developed by Roger Penrose in his attempt to find a solution to tiling the plane with pentagonal symmetries, inspired by Johannes Kepler's remark that pentagrams, decagons, and other shapes could fill gaps that pentagonal shapes alone leave when tiled together. Three types of Penrose tilings exist with different prototiles that exhibit golden symmetry: the original P1 tiling, the kite and dart P2 tiling, and the rhombus P3 tiling.
- The original P1 tiling contains different matching rules for how pentagons, pentagrams, "boat" figures that are roughly 3/5ths of a star, and "diamond" shaped rhombi can come together to tile the plane.
- The kite and dart P2 tiling contains kites with three interior angles of degrees and one interior angle of degrees, and darts with two interior angles of degrees, one of degrees, and one of degrees. Both the kites and darts are composed themselves of golden Robinson triangles, and as such the ratio of the short side to the long side in both the kite and dart is . A consequence of this is that the ratio of the area of the kites and darts is also . In total, there are seven possible combinations of kites and darts that generate all possible P2 Penrose tilings, which are determined from special matching rules.
- The rhombus P3 tiling contains two types of rhomuses, a thin t rhomb with two and two degree angles, and a thick T romb with two and two degree angles. Like the P2 tiling, this tiling's rhombic prototiles can be decomposed into golden Robinson triangles, making the ratio of the length of sides to that of the short diagonal in the thin t rhomb equal to , as well for the sides of the thick T rhomb to its long diagonal. As with the P2 tiling, the ratio of the areas of these two prototiles is in golden ratio.
In the dodecahedron and icosahedron
The regular dodecahedron and its dual polyhedron the icosahedron are Platonic solids that have symmetries essentially interrelated with the golden ratio. An icosahedron is made of regular pentagonal faces, whereas the icosahedron is made of equilateral triangles; both with edges.
Circumscribing and inscribed circles of both the dodecahedron and icosahedron produce radii that are in proportion to , as are their surface area and volume. For a dodecahedron of side , the radius of a circumscribed and inscribed sphere, and midradius are (, and , respectively):
While for an icosahedron of side , the radius of a circumscribed and inscribed sphere, and midradius are:
The volume and surface area of the dodecahedron can be expressed in terms of :
As well as for the icosahedron:
These geometric values can be calculated from their Cartesian coordinates, which are in proportion to both and its conjugate, . The coordinates of the dodecahedron are displayed on the figure above, while those of the icosahedron are the cyclic permutations of:
Sets of three golden rectangles intersect perpendicularly inside dodecahedra and icosahedra, forming Borromean rings. In dodecahedra, pairs of opposing vertices in golden rectangles meet the centers of pentagonal faces, and in icosahedra, they meet at its vertices. In all, the three golden rectangles contain vertices of the icosahedron, or equivalently, intersect the centers of of the dodecahedron's faces.
Inscribed cubes inside dodecahedra have a volume that is times that of the dodecahedron's, with each other's sides in golden ratio to one another. In fact, golden rectangles inside a dodecahedron are in golden proportions to an inscribed cube, such that edges of a cube and the long edges of a golden rectangle are themselves in ratio. On the other hand, the octahedron, which is the dual polyhedron of the cube, can inscribe an icosahedron, such that an icosahedron's vertices touch the edges of an octahedron at points that divide its edges in golden ratio.
Inside full icosahedral symmetry Ih
The icosahedron and the dodecahedron share the same set of rotational symmetries (I) and symmetry order (Ih, or full icosahedral symmetry), that includes transformations by a combination of rotations and reflections. Their full symmetry group is H3 (), which contains the subgroup I that is isomorphic to the alternating group on five letters, A5. I acts on the compound of five cubes (UC9), compound of five octahedra (W23), and both enantimorphic forms of the compound of five tetrahedra (W24), which form the compound of ten tetrahedra (W25). All of these uniform polyhedron compounds share the faceting or convex hull of the dodecahedron and icosahedron, and therefore contain golden symmetry. For example,
- the compound of five tetrahedra can be constructed via sections that assemble into three-piece pyramids, whose sides have lengths , , and .
- The compound of five cubes can be generated by rotating cubes about through angles of for .
- The compound of four octahedra has a convex hull that is an icosidodecahedron, with a special ratio of its side to circumradius equal to , and a hemiface that is a decagon; which is noteworthy since this makes the icosidodecahedron and the decagon two of three regular polytopes to have an edge-to-circumradius ratio of .
The icosidodecahedron, which shares symmetry order Ih, is also one of only three quasiregular convex polyhedra, the other being the rhombic dodecahedron and the tetrahedron. It's dual polyhedron, the rhombic triacontahedron, which is the stellation core of the compound of five cubes, also shares full icosahedral symmetry Ih. It is made of faces that are golden rhombi whose diagonals are in ratio, and collectively are the convex hull of the icosahedron and dodecahedron. Furthermore, a rhombic triacontahedron can be constructed with golden rhombohedra ( acute and obtuse), which are trigonal trapezohedra with six congruent golden rhombi that contain diagonals in ratio. Under its Cartesian coordinates, the edge length of a rhombic triacontahedron is , with golden rhombi faces that have diagonal lengths of and .
More deeply, icosahedral symmetry contains distinct Coxeter group generators represented by rotation matrices, reflection matrices, and rotoreflection matrices, most of which contain . For example, I can be generated by a rotation of S1,2 () with the following rotation matrix, through the axis :
Four groups in total, out of seven, contain matrices that are in proportion to (R1, S0,1, S1,2, and V0,1,2), with the remaining symmetry matrices containing values of and .
Inside other polyhedra
Dodecahedra and icosahedra, through various geometric operations including truncations and alternations, produce other semi-regular Archimedean solids that also share full icosahedral and golden symmetry. These include the truncated icosahedron, truncated dodecahedron, and rhombicosidodecahedron, with basis vectors , , and , respectively. Their dual Catalan solids also all contain golden symmetry, reflected primarily in their Cartesian coordinates that are in proportion with .
The rhombic enneacontahedron, a zonohedron with Ih symmetry that resembles the rhombic triacontahedron, contains rhombi whose short and long diagonals are in ratio of . These rhombi have angles approximating and degrees, and make up of rhombic faces in the polyhedron. It is the dual polyhedron of the rectified truncated icosahedron, a near-miss Johnson solid.
Jessen's icosahedron is a non-convex vertex-transitive shaky polyhedron with the same number of faces, edges, and vertices as a regular icosahedron yet constructed from equilateral and obtuse isosceles triangles that meet at degrees. Jessen's icosahedron can be constructed by segmenting an octahedron's sides in golden ratio, as with the icosahedron, and proceeding instead to reverse the respective short and long segments.
Stellations of the dodecahedron and icosahedron generate another family of polyhedra that contain Ih symmetry and therefore symmetries. Of particular interest are the Kepler-Poinsot polyhedra, which are the only (four) regular star polyhedra. The small stellated dodecahedron can be constructed by adding pentagonal pyramids with faces that are golden isosceles sublime triangles on top of the pentagonal faces of a dodecahedron. Similarly, the great stellated dodecahedron can be constructed by adding triangular pyramids, whose faces are also golden isosceles sublime triangles, on top of the triangular faces of an icosahedron. In the great dodecahedron, intersecting pentagonal faces produce facelets that are golden gnomons.
The rhombic hexecontahedron, which is one of stellations of the rhombic triacontahedron, contains golden rhombi as faces whose diagonals are in ratio. It can also be dissected into golden rhombohedra, as with the rhombic triacontahedron. A closely related solid is the great rhombic triacontahedron, which is a non-convex isohedral and isotoxal polyhedron with golden rhombi as faces. It can be constructed from the rhombic triacontahedron by expanding the size of its faces by a factor of . The obtuse degree angle inside golden rhombi of the rhombic enneacontahedron is also the dihedral angle of the medial triambic icosahedron (or great triambic icosahedron), which is one of five abstract polyhedra, and the stellation of the icosahedron.
In higher dimensions
Of interest in four dimensions, the golden ratio primarily appears in the 5-cell, 120-cell, and 600-cell, which is the dual polychoron to the 120-cell; all of which are regular 4-dimensional polytopes.
The 5-cell, which is the four-dimensional analogue of the tetrahedron, has an orthogonal projection isomorphic to the complete graph K5 that can be represented as the union of the graphs of a regular pentagon and pentagram where every pair of distinct vertices in the pentagon is joined by unique edges of the pentagram. The simplest set of Cartesian coordinates of the 5-cell, which is composed of five tetrahedra, are:
The 120-cell, itself the four-dimensional analogue of the dodecahedron, is made of dodecahedra whose faceting also fits 5-cells. It has an edge-length of and circumradius of , with vertices that are the permutations, and even permutations, respectively, of:
The dihedral angle of a 120-cell is , which is the same as the dihedral angle of adjacent golden rhombi in a rhombic triacontahedron, as well as equal to twice the acute angle of a sublime triangle.
On the other hand, the 600-cell, which is composed of tetrahedra, has an edge to circumradius ratio of like the icosidodecahedron and the decagon; making the icosidodecahedron the equatorial cross-section of the 600-cell. By duality of the 120-cell, it contains vertices. These vertices are the same as the vertices of the 24-cell (which is another regular 4-polytope), plus the vertices contained in the snub 24-cell, which is one of three unique semi-regular 4-polytopes. The vertices belonging to the snub 24-cell are constructed from partitioning the edges of the 24-cell in golden ratio along the direction of a common vector, yielding a snub truncation of the 24-cell. Indeed, the vertices of the snub 24-cell, which contain a total of tetrahedral and dodecahedral cells, can be obtained by taking even permutations of
An emergent property of the 600-cell is its construction of roots of the exceptional Lie group E8. vertices of the 600-cell can originate from permutiations of , vertices from permutations of and vertices from even permutations of The first vertices correspond to the vertices of a tesseract, while the second vertices correspond to those of the 24-cell. Together, these vertices form half the roots of the E8 group, which is equivalent to the order Weyl group and Coxeter group H4 of the 120-cell and the 600-cell, as well as the binary icosahedral group.
These vertices define specific Hamiltonian hypercomplex quaternions which are more specifically called the icosians. The semi-regular 8-dimensional polytope 421, which is the final finite semi-regular polytope in all dimensions, projects from the H4/E8 Coxeter plane two copies of the vertices of the 600-cell, one smaller than the other, that are scaled in ratio; therein generating the complete root vectors of the E8 root system. Essentially, this means that , by construction of 421, creates an isomorphism of the eight-dimensional octonions , which are preserved by G2(F2) ∈ E8; with the isotopy group of the integral octonions as the perfect double cover of the rotation groups of the E8 lattice.
The vertices of the 24-cell on S3 projects the exceptional Lie algebra F4 through its Weyl/Coxeter group action, which can be identified with the quaternions and more specifically the full ring of Hurwitz integral quaternions H ∈ , as well as both the binary tetrahedral group 2T and binary octahedral group 2O. In similar fashion, then, the octonions can be maped from E8 by 421 and therefore by extension the 600-cell, which is fundamentally structured by , and contains to the binary icosahedral group 2I (itself a subset of ).
There are no analogues of the 120-cell and 600-cell, or equivantly, the dodecahedron and icosahedron, in dimensions , which makes the fourth dimension the final dimension in-which regular -polytopes have fundamental root pentagonal symmetry.
The golden ratio's decimal expansion can be calculated via root-finding methods, such as Newton's method or Halley's method, on the equation or on (to compute first). The time needed to compute digits of the golden ratio using Newton's method is essentially , where is the time complexity of multiplying two -digit numbers. This is considerably faster than known algorithms for and . An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers and each over digits, yields over significant digits of the golden ratio. The decimal expansion of the golden ratio  has been calculated to an accuracy of ten trillion () digits.
The golden ratio and inverse golden ratio have a set of symmetries that preserve and interrelate them. They are both preserved by the fractional linear transformations – this fact corresponds to the identity and the definition quadratic equation. Further, they are interchanged by the three maps – they are reciprocals, symmetric about and (projectively) symmetric about More deeply, these maps form a subgroup of the modular group isomorphic to the symmetric group on letters, corresponding to the stabilizer of the set of standard points on the projective line, and the symmetries correspond to the quotient map – the subgroup consisting of the identity and the -cycles, in cycle notation fixes the two numbers, while the -cycles interchange these, thus realizing the map.
In the complex plane, the 5th roots of unity for have a pentagonal representation. For , the primitive roots of unity do not form quadratic integers, however the sum of its roots of unity and its complex conjugate, z + z = 2 Re z, that is an element of the ring Z[1 + √/2], does form a quadratic integer (as with ). Such sums for two pairs of non-real fifth roots of unity are and .
When the golden ratio is used as the base of a numeral system (see golden ratio base, sometimes dubbed phinary or -nary), quadratic integers in the ring – that is, numbers of the form for – have terminating representations, but rational fractions have non-terminating representations.
The golden ratio also appears in hyperbolic geometry, as the maximum distance from a point on one side of an ideal triangle to the closer of the other two sides: this distance, the side length of the equilateral triangle formed by the points of tangency of a circle inscribed within the ideal triangle, is 
An infinite series can be derived to express :
The golden ratio appears in the theory of modular functions as well. For , let
Applications and observations
The Swiss architect Le Corbusier, famous for his contributions to the modern international style, centered his design philosophy on systems of harmony and proportion. Le Corbusier's faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described as "rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned."
Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci's "Vitruvian Man", the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture.
In addition to the golden ratio, Le Corbusier based the system on human measurements, Fibonacci numbers, and the double unit. He took suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor system. Le Corbusier's 1927 Villa Stein in Garches exemplified the Modulor system's application. The villa's rectangular ground plan, elevation, and inner structure closely approximate golden rectangles.
Another Swiss architect, Mario Botta, bases many of his designs on geometric figures. Several private houses he designed in Switzerland are composed of squares and circles, cubes and cylinders. In a house he designed in Origlio, the golden ratio is the proportion between the central section and the side sections of the house.
Divina proportione (Divine proportion), a three-volume work by Luca Pacioli, was published in 1509. Pacioli, a Franciscan friar, was known mostly as a mathematician, but he was also trained and keenly interested in art. Divina proportione explored the mathematics of the golden ratio. Though it is often said that Pacioli advocated the golden ratio's application to yield pleasing, harmonious proportions, Livio points out that the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions. Pacioli also saw Catholic religious significance in the ratio, which led to his work's title.
Leonardo da Vinci's illustrations of polyhedra in Divina proportione have led some to speculate that he incorporated the golden ratio in his paintings. But the suggestion that his Mona Lisa, for example, employs golden ratio proportions, is not supported by Leonardo's own writings. Similarly, although the Vitruvian Man is often shown in connection with the golden ratio, the proportions of the figure do not actually match it, and the text only mentions whole number ratios.
Salvador Dalí, influenced by the works of Matila Ghyka, explicitly used the golden ratio in his masterpiece, The Sacrament of the Last Supper. The dimensions of the canvas are a golden rectangle. A huge dodecahedron, in perspective so that edges appear in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.
A statistical study on 565 works of art of different great painters, performed in 1999, found that these artists had not used the golden ratio in the size of their canvases. The study concluded that the average ratio of the two sides of the paintings studied is with averages for individual artists ranging from (Goya) to (Bellini). On the other hand, Pablo Tosto listed over 350 works by well-known artists, including more than 100 which have canvasses with golden rectangle and proportions, and others with proportions like and 
Books and design
According to Jan Tschichold,
There was a time when deviations from the truly beautiful page proportions and the Golden Section were rare. Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimeter.
According to some sources, the golden ratio is used in everyday design, for example in the proportions of playing cards, postcards, posters, light switch plates, and widescreen televisions.
Ernő Lendvai analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale, though other music scholars reject that analysis. French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. The golden ratio is also apparent in the organization of the sections in the music of Debussy's Reflets dans l'eau (Reflections in Water), from Images (1st series, 1905), in which "the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the phi position".
The musicologist Roy Howat has observed that the formal boundaries of Debussy's La Mer correspond exactly to the golden section. Trezise finds the intrinsic evidence "remarkable", but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.
Though Heinz Bohlen proposed the non-octave-repeating 833 cents scale based on combination tones, the tuning features relations based on the golden ratio. As a musical interval the ratio 1.618... is 833.090... cents (Play (help·info)).
Johannes Kepler wrote that "the image of man and woman stems from the divine proportion. In my opinion, the propagation of plants and the progenitive acts of animals are in the same ratio".
The psychologist Adolf Zeising noted that the golden ratio appeared in phyllotaxis and argued from these patterns in nature that the golden ratio was a universal law. Zeising wrote in 1854 of a universal orthogenetic law of "striving for beauty and completeness in the realms of both nature and art".
However, some have argued that many apparent manifestations of the golden ratio in nature, especially in regard to animal dimensions, are fictitious.
The quasi-one-dimensional Ising ferromagnet CoNb2O6 (cobalt niobate) has 8 predicted excitation states (with E8 symmetry), that when probed with neutron scattering, showed its lowest two were in golden ratio. Specifically, these quantum phase transitions during spin excitation, which occur at near absolute zero temperature, showed pairs of kinks in its ordered-phase to spin-flips in its paramagnetic phase; revealing, just below its critical field, a spin dynamics with sharp modes at low energies approaching the golden mean.
There is no known general algorithm to arrange a given number of nodes evenly on a sphere, for any of several definitions of even distribution (see, for example, Thomson problem or Tammes problem). However, a useful approximation results from dividing the sphere into parallel bands of equal surface area and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. This method was used to arrange the 1500 mirrors of the student-participatory satellite Starshine-3.
The golden ratio is a critical element to golden-section search as well.
Examples of disputed observations of the golden ratio include the following:
- Some specific proportions in the bodies of many animals (including humans) and parts of the shells of mollusks are often claimed to be in the golden ratio. There is a large variation in the real measures of these elements in specific individuals, however, and the proportion in question is often significantly different from the golden ratio. The ratio of successive phalangeal bones of the digits and the metacarpal bone has been said to approximate the golden ratio. The nautilus shell, the construction of which proceeds in a logarithmic spiral, is often cited, usually with the erroneous idea that any logarithmic spiral is related to the golden ratio, but sometimes with the claim that each new chamber is golden-proportioned relative to the previous one. However, measurements of nautilus shells do not support this claim.
- Historian John Man states that both the pages and text area of the Gutenberg Bible were "based on the golden section shape". However, according to his own measurements, the ratio of height to width of the pages is 
- Studies by psychologists, starting with Gustav Fechner c. 1876, have been devised to test the idea that the golden ratio plays a role in human perception of beauty. While Fechner found a preference for rectangle ratios centered on the golden ratio, later attempts to carefully test such a hypothesis have been, at best, inconclusive.
- In investing, some practitioners of technical analysis use the golden ratio to indicate support of a price level, or resistance to price increases, of a stock or commodity; after significant price changes up or down, new support and resistance levels are supposedly found at or near prices related to the starting price via the golden ratio. The use of the golden ratio in investing is also related to more complicated patterns described by Fibonacci numbers (e.g. Elliott wave principle and Fibonacci retracement). However, other market analysts have published analyses suggesting that these percentages and patterns are not supported by the data.
The Great Pyramid of Giza (also known as the Pyramid of Cheops or Khufu) has been analyzed by pyramidologists as having a doubled Kepler triangle as its cross-section. If this theory were true, the golden ratio would describe the ratio of distances from the midpoint of one of the sides of the pyramid to its apex, and from the same midpoint to the center of the pyramid's base. However, imprecision in measurement caused in part by the removal of the outer surface of the pyramid makes it impossible to distinguish this theory from other numerical theories of the proportions of the pyramid, based on pi or on whole-number ratios. The consensus of modern scholars is that this pyramid's proportions are not based on the golden ratio, because such a basis would be inconsistent both with what is known about Egyptian mathematics from the time of construction of the pyramid, and with Egyptian theories of architecture and proportion used in their other works.
The Parthenon's façade (c. 432 BC) as well as elements of its façade and elsewhere are said by some to be circumscribed by golden rectangles. Other scholars deny that the Greeks had any aesthetic association with golden ratio. For example, Keith Devlin says, "Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation." Midhat J. Gazalé affirms that "It was not until Euclid ... that the golden ratio's mathematical properties were studied."
From measurements of 15 temples, 18 monumental tombs, 8 sarcophagi, and 58 grave stelae from the fifth century BC to the second century AD, one researcher concluded that the golden ratio was totally absent from Greek architecture of the classical fifth century BC, and almost absent during the following six centuries. Later sources like Vitruvius (first century BC) exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions.
The Section d'Or ('Golden Section') was a collective of painters, sculptors, poets and critics associated with Cubism and Orphism. Active from 1911 to around 1914, they adopted the name both to highlight that Cubism represented the continuation of a grand tradition, rather than being an isolated movement, and in homage to the mathematical harmony associated with Georges Seurat. The Cubists observed in its harmonies, geometric structuring of motion and form, the primacy of idea over nature, an absolute scientific clarity of conception. However, despite this general interest in mathematical harmony, whether the paintings featured in the celebrated 1912 Salon de la Section d'Or exhibition used the golden ratio in any compositions is more difficult to determine. Livio, for example, claims that they did not, and Marcel Duchamp said as much in an interview. On the other hand, an analysis suggests that Juan Gris made use of the golden ratio in composing works that were likely, but not definitively, shown at the exhibition. Art historian Daniel Robbins has argued that in addition to referencing the mathematical term, the exhibition's name also refers to the earlier Bandeaux d'Or group, with which Albert Gleizes and other former members of the Abbaye de Créteil had been involved.
Piet Mondrian has been said to have used the golden section extensively in his geometrical paintings, though other experts (including critic Yve-Alain Bois) have discredited these claims.
- If the constraint on and each being greater than zero is lifted, then there are actually two solutions, one positive and one negative, to this equation. is defined as the positive solution. The negative solution is The sum of the two solutions is and the product of the two solutions is
- Euclid, Elements, Book II, Proposition 11; Book IV, Propositions 10–11; Book VI, Proposition 30; Book XIII, Propositions 1–6, 8–11, 16–18.
- "῎Ακρον καὶ μέσον λόγον εὐθεῖα τετμῆσθαι λέγεται, ὅταν ᾖ ὡς ἡ ὅλη πρὸς τὸ μεῖζον τμῆμα, οὕτως τὸ μεῖζον πρὸς τὸ ἔλαττὸν."
- After Classical Greek sculptor Phidias (c. 490–430 BC); Barr later wrote that he thought it unlikely that Phidias actually used the golden ratio.
- Not to be confused with the silver mean, also known as the silver ratio.
- Not to be confused with the congruence subgroup
- OEIS: A001622
- Weisstein, Eric W. "Golden Ratio". mathworld.wolfram.com. Retrieved 2020-08-10.
- Livio 2003, pp. 3, 81.
- Dunlap, Richard A., The Golden Ratio and Fibonacci Numbers, World Scientific Publishing, 1997
- Euclid, Elements, Book 6, Definition 3.
- Fink, Karl; Beman, Wooster Woodruff; Smith, David Eugene (1903). A Brief History of Mathematics: An Authorized Translation of Dr. Karl Fink's Geschichte der Elementar-Mathematik (2nd ed.). Chicago: Open Court Publishing Co. p. 223.
- Summerson John, Heavenly Mansions: And Other Essays on Architecture (New York: W.W. Norton, 1963) p. 37. "And the same applies in architecture, to the rectangles representing these and other ratios (e.g. the 'golden cut'). The sole value of these ratios is that they are intellectually fruitful and suggest the rhythms of modular design."
- Jay Hambidge, Dynamic Symmetry: The Greek Vase, New Haven CT: Yale University Press, 1920
- William Lidwell, Kritina Holden, Jill Butler, Universal Principles of Design: A Cross-Disciplinary Reference, Gloucester MA: Rockport Publishers, 2003
- Pacioli, Luca. De divina proportione, Luca Paganinem de Paganinus de Brescia (Antonio Capella) 1509, Venice.
- Strogatz, Steven (September 24, 2012). "Me, Myself, and Math: Proportion Control". The New York Times.
- Weisstein, Eric W. "Golden Ratio Conjugate". MathWorld.
- Livio 2003, p. 6.
- Livio 2003, p. 4: "... line division, which Euclid defined for ... purely geometrical purposes ..."
- Livio 2003, pp. 7–8.
- Livio 2003, pp. 4–5.
- Livio 2003, p. 78.
- Hemenway, Priya (2005). Divine Proportion: Phi In Art, Nature, and Science. New York: Sterling. pp. 20–21. ISBN 978-1-4027-3522-6.
- Livio 2003, p. 3.
- Fitzpatrick, Richard (2007). Euclid's Elements of Geometry. Translated by Richard Fitzpatrick. p. 156. ISBN 978-0615179841.
- Livio 2003, pp. 88–96.
- Livio 2003, pp. 131–132.
- Baravalle, H. V. (1948). "The geometry of the pentagon and the golden section". Mathematics Teacher. 41: 22–31. doi:10.5951/MT.41.1.0022.
- Livio 2003, p. 141.
- Schreiber, Peter (1995). "A Supplement to J. Shallit's Paper "Origins of the Analysis of the Euclidean Algorithm"". Historia Mathematica. 22 (4): 422–424. doi:10.1006/hmat.1995.1033.
- Livio 2003, pp. 151–152.
- "The Golden Ratio". The MacTutor History of Mathematics archive. Retrieved 2007-09-18.
- Weisstein, Eric W. "Binet's Fibonacci Number Formula". MathWorld.
- Herz-Fischler, Roger (1987). A Mathematical History of Division in Extreme and Mean Ratio. Wilfrid Laurier University Press. ISBN 978-0889201521.
- Posamentier, Alfred S.; Lehmann, Ingmar (2011). The Glorious Golden Ratio. Prometheus Books. p. 8. ISBN 9-781-61614-424-1.
- Posamentier, Alfred S.; Lehmann, Ingmar (2011). The Glorious Golden Ratio. Prometheus Books. p. 285. ISBN 9-781-61614-424-1.
- Cook, Theodore Andrea (1914). The Curves of Life. London: Constable and Company Ltd. p. 420.
- Barr, Mark (1929). "Parameters of beauty". Architecture (NY). Vol. 60. p. 325. Reprinted: "Parameters of beauty". Think. Vol. 10–11. International Business Machines Corporation. 1944.
- Livio 2003, p. 5.
- Weisstein, Eric W. "Golden Ratio". MathWorld.
- Gardner, Martin (2001). The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems. W.W. Norton & Company. pp. 77, 88. ISBN 978-0393020236.
- Gerlin, Andrea (October 5, 2011). "Tecnion's Shechtman Wins Nobel in Chemistry for Quasicrystals Discovery". Bloomberg. Archived from the original on December 5, 2014. Retrieved January 4, 2019.
- Jaric, Marko V. (2012), Introduction to the Mathematics of Quasicrystals, Elsevier, p. x, ISBN 978-0323159470,
Although at the time of the discovery of quasicrystals the theory of quasiperiodic functions had been known for nearly sixty years, it was the mathematics of aperiodic Penrose tilings, mostly developed by Nicolaas de Bruijn, that provided the major influence on the new field.
- Livio 2003, pp. 203–209.
- Goldman, Alan I.; et al. (1996). "Quasicrystalline Materials". American Scientist. 84 (3): 230–241.
- Weisstein, Eric W. (2002). "Golden Ratio Conjugate". CRC Concise Encyclopedia of Mathematics, Second Edition, pp. 1207–1208. CRC Press. ISBN 978-1420035223.
- Max. Hailperin; Barbara K. Kaiser; Karl W. Knight (1998). Concrete Abstractions: An Introduction to Computer Science Using Scheme. Brooks/Cole Pub. Co. ISBN 978-0-534-95211-2.
- Hardy, G. H. (2008). An introduction to the theory of numbers. E. M. Wright, D. R. Heath-Brown, Joseph H. Silverman (6th ed.). Oxford: Oxford University Press. p. 209. ISBN 978-0-19-921985-8. OCLC 214305907.
- Tattersall, James Joseph (2005). Elementary number theory in nine chapters (2nd ed.). Cambridge University Press. p. 28. ISBN 978-0-521-85014-8.
- Parker, Matt (2014). "13". Things to Make and Do in the Fourth Dimension. Farrar, Straus and Giroux. p. 284. ISBN 978-0-374-53563-6.
- Freitas, Pedro J. (2021-01-25). "The Golden Angle is not Constructible". arXiv:2101.10818 [math.HO].
- Chris and Penny. "Quandaries and Queries". Math Central. Retrieved 23 October 2011.
- Busard, Hubert L. L. (April–June 1968). "L'algèbre au Moyen Âge : le "Liber mensurationum" d'Abû Bekr". Journal des Savants (in French and Latin). 1968 (2): 65–124. doi:10.3406/jds.1968.1175. Archived from the original on 2022-01-12. Retrieved 2022-01-12. See problem 51, reproduced on p. 98
- Bruce, Ian (1994). "Another instance of the golden right triangle" (PDF). Fibonacci Quarterly. 32 (3): 232–233. MR 1285752. Archived (PDF) from the original on 2022-01-29. Retrieved 2022-01-29.
- Loeb, Arthur (1992). Concepts and Images: Visual Mathematics. Boston: Birkhäuser Boston. p. 180. ISBN 0-8176-3620-X.
- Weisstein, Eric W. "Golden Triangle". mathworld.wolfram.com. Retrieved 2019-12-26.
- Tilings Encyclopedia. 1970. Archived from the original on 2009-05-24.
- Weisstein, Eric W. "Golden Gnomon". mathworld.wolfram.com. Retrieved 2022-06-08.
- Posamentier, Alfred S.; Lehmann, Ingmar (2011). The Glorious Golden Ratio. Prometheus Books. p. 11. ISBN 9-781-61614-424-1.
- Burger, Edward B.; Starbird, Michael P. (2005). The Heart of Mathematics: An Invitation to Effective Thinking. Springer. p. 382. ISBN 9781931914413.
- Weisstein, Eric W. "Golden Rhombus". MathWorld. Retrieved 2022-06-13.
- Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 978-0-7167-1193-3.
- Austin, David (2005a). "Penrose Tiles Talk Across Miles". Providence: American Mathematical Society.
- Penrose, Roger (1978). "Pentaplexity" (PDF). Eureka. Vol. 39. p. 32.
- Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. pp. 537–547. ISBN 978-0-7167-1193-3.
- Ramachandrarao, P (2000). "On the fractal nature of Penrose tiling" (PDF). Current Science. 79: 364.
- Weisstein, Eric W. "Regular Dodecahedron". MathWorld. Retrieved 2022-06-13.
- Weisstein, Eric W. "Regular Icosahedron". MathWorld. Retrieved 2022-06-13.
- Gunn, Charles; Sullivan, John M. (2008), "The Borromean Rings: A video about the New IMU logo", in Sarhangi, Reza; Séquin, Carlo H. (eds.), Bridges Leeuwarden: Mathematics, Music, Art, Architecture, Culture, London: Tarquin Publications, pp. 63–70, ISBN 978-0-9665201-9-4; Video at "The Borromean Rings: A new logo for the IMU", International Mathematical Union
- Knott, Ron (26 September 2016). "The Golden Geometry of Solids or Phi in 3 dimensions". Ron Knott's Mathematics Pages. Retrieved 2022-03-19.
- Coxeter, H.S.M.; du Val, Patrick; Flather, H.T.; Petrie, J.F. (1938). The Fifty-Nine Icosahedra. Vol. 6. University of Toronto Studies (Mathematical Series). p. 4.
Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the "golden section.
- Weisstein, Eric W. "Icosahedral Group". MathWorld. Retrieved 2022-06-13.
- Toth, Gabor (2002). Finite Möbius Groups, Minimal Immersions of Spheres, and Moduli. Universitext. New York: Springer. pp. 13–15. doi:10.1007/978-1-4613-0061-8. ISBN 978-1-4612-6546-7.
- Weisstein, Eric W. "Tetrahedron 5-Compound". MathWorld. Retrieved 2022-06-13.
- Weisstein, Eric W. "Cube 5-Compound". MathWorld. Retrieved 2022-06-13.
- Weisstein, Eric W. "Octahedron 5-Compound". MathWorld. Retrieved 2022-06-13.
- Denney, Tomme; Hooker, Da'Shay; Johnson, De'Janeke; Robinson, Tianna; Butler, Majid; Claiborne, Sandernishe (2020). "The geometry of H4 polytopes". Advances in Geometry. 20 (3): 433–444. arXiv:1912.06156. doi:10.1515/advgeom-2020-0005. S2CID 220367622.
- Weisstein, Eric W. "Rhombic Triacontahedron". MathWorld. Retrieved 2022-06-13.
- Litvin, D. B. (1991). "The icosahedral point groups". Acta Crystallographica. pp. 70–73. Retrieved 2022-06-13.
- Kabai, Sándor (2007). "Rhombic Enneacontahedron with 30 Icosahedra". Wolfram Demonstrations Project. MathWorld. Retrieved 2022-06-13.
- Weisstein, Eric W. "Jessen's Orthogonal Icosahedron". MathWorld. Retrieved 2022-06-13.
- "Polygons Related to the Golden Ratio, and Associated Figures in Geometry, Part 1: Triangles". RobertLovesPi. 2014-12-06. Retrieved 2022-06-13.
- Weisstein, Eric W. "Rhombic Hexecontahedron". MathWorld. Retrieved 2022-06-13.
- Weisstein, Eric W. "Great Rhombic Triacontahedron". MathWorld. Retrieved 2022-06-13.
- Weisstein, Eric. "Pentatope". MathWorld. Retrieved 6 June 2022.
- Weisstein, Eric W. "120-Cell". MathWorld. Retrieved 2022-06-12.
- Coxeter, H.S.M. (1973) . Regular Polytopes (3rd ed.). New York: Dover. pp. 151–153.
- Tomme, Denney; Da'Shay, Hooker; De'Janeke, Johnson; Tiana, Robinson; Majid, Butler; Sandernisha, Claiborne (2019-12-12). "The Geometry of H4 Polytopes". arXiv:1912.06156 [math.MG].
- Weisstein, Eric W. "600-Cell". MathWorld. Retrieved 2022-06-12.
- Choi, Jihyun; Lee, Jae-Hyouk (2018-09-07). "Binary Icosahedral Group and 600-Cell". Symmetry. 10 (8): 12–14. Bibcode:2018Symm...10..326C. doi:10.3390/sym10080326. Retrieved 2022-06-13.
- Baez, John C. (2018). "From the Icosahedron to E8". London Math. Soc. Newsletter. 476: 18–23. arXiv:1712.06436.
- Richter, David A. (2005-08-16). "The Great 600-Cell, A Zome Model". Western Michigan University. Retrieved 2022-06-13.
...one obtains a polytope containing not only the vertices of the 'usual' 600-cell, but also those of the 600-cell which has been scaled up by the Golden Ratio. These vertices are represented by black balls in the Zome model. As it turns out, the edges which are present in this model are images of the edges of the quasicrystalline projection of Gosset's 8-dimensional semiregular polytope into 3-space.
- Crowell, Lawrence B. (2005). Quantum fluctuations of spacetime. New Jersey: World Scientific. p. 329. ISBN 9812565159.
- Baez, John C. (2002). "The Octonions". Bulletin of the American Mathematical Society. 39 (2): 145–205. arXiv:math/0105155. doi:10.1090/S0273-0979-01-00934-X. ISSN 0273-0979. MR 1886087. S2CID 586512.
- Adams, J. Frank (1996). Lectures on exceptional Lie groups. Chicago Lectures in Mathematics. University of Chicago Press. ISBN 978-0-226-00526-3. MR 1428422.
- Muller, J. M. (2006). Elementary functions : algorithms and implementation (2nd ed.). Boston: Birkhäuser. p. 93. ISBN 978-0817643720.
- Yee, Alexander J. (13 March 2021). "Records Set by y-cruncher". numberword.org. Two independent computations done by Clifford Spielman.
- Horocycles exinscrits : une propriété hyperbolique remarquable, cabri.net, retrieved 2009-07-21.
- Brian Roselle, "Golden Mean Series"
- Brendt, B. et al. "The Rogers–Ramanujan Continued Fraction"
- Weisstein, Eric W. "Pisot Number". MathWorld.
- Le Corbusier, The Modulor p. 25, as cited in Padovan, Richard, Proportion: Science, Philosophy, Architecture (1999), p. 316, Taylor and Francis, ISBN 0-419-22780-6
- Frings, Marcus, The Golden Section in Architectural Theory, Nexus Network Journal vol. 4 no. 1 (Winter 2002).
- Le Corbusier, The Modulor, p. 35, as cited in Padovan, Richard, Proportion: Science, Philosophy, Architecture (1999), p. 320. Taylor & Francis. ISBN 0-419-22780-6: "Both the paintings and the architectural designs make use of the golden section".
- Urwin, Simon. Analysing Architecture (2003) pp. 154–155, ISBN 0-415-30685-X
- Livio 2003, pp. 134–135.
- Hart, George W. (1999). "Leonardo da Vinci's Polyhedra". George W. Hart. Retrieved March 10, 2019.
- Livio, Mario (November 1, 2002). "The golden ratio and aesthetics". Plus Magazine. Retrieved November 26, 2018.
- Keith Devlin (May 2007). "The Myth That Will Not Go Away". Retrieved September 26, 2013.
Part of the process of becoming a mathematics writer is, it appears, learning that you cannot refer to the golden ratio without following the first mention by a phrase that goes something like 'which the ancient Greeks and others believed to have divine and mystical properties.' Almost as compulsive is the urge to add a second factoid along the lines of 'Leonardo Da Vinci believed that the human form displays the golden ratio.' There is not a shred of evidence to back up either claim, and every reason to assume they are both false. Yet both claims, along with various others in a similar vein, live on.
- Donald E. Simanek. "Fibonacci Flim-Flam". Archived from the original on January 9, 2010. Retrieved April 9, 2013.
- Salvador Dalí (2008). The Dali Dimension: Decoding the Mind of a Genius (DVD). Media 3.14-TVC-FGSD-IRL-AVRO.
- Hunt, Carla Herndon and Gilkey, Susan Nicodemus. Teaching Mathematics in the Block pp. 44, 47, ISBN 1-883001-51-X
- Olariu, Agata, Golden Section and the Art of Painting Available online
- Tosto, Pablo, La composición áurea en las artes plásticas – El número de oro, Librería Hachette, 1969, pp. 134–144
- Jan Tschichold. The Form of the Book, p. 43 Fig 4. "Framework of ideal proportions in a medieval manuscript without multiple columns. Determined by Jan Tschichold 1953. Page proportion 2:3. margin proportions 1:1:2:3, Text area proportioned in the Golden Section. The lower outer corner of the text area is fixed by a diagonal as well."
- Tschichold, Jan (1991). The Form of the Book. Hartley & Marks. pp. 27–28. ISBN 0-88179-116-4.
Jones, Ronald (1971). "The golden section: A most remarkable measure". The Structurist. 11: 44–52.
Who would suspect, for example, that the switch plate for single light switches are standardized in terms of a Golden Rectangle?
Johnson, Art (1999). Famous problems and their mathematicians. Libraries Unlimited. p. 45. ISBN 978-1-56308-446-1.
The Golden Ratio is a standard feature of many modern designs, from postcards and credit cards to posters and light-switch plates.
- Stakhov & Olsen 2009, p. 21. "A credit card has a form of the golden rectangle."
Cox, Simon (2004). Cracking the Da Vinci code: the unauthorized guide to the facts behind Dan Brown's bestselling novel. Barnes & Noble Books. p. 62. ISBN 978-0-7607-5931-8.
The Golden Ratio also crops up in some very unlikely places: widescreen televisions, postcards, credit cards and photographs all commonly conform to its proportions.
- Lendvai, Ernő (1971). Béla Bartók: An Analysis of His Music. London: Kahn and Averill.
- Livio 2003, p. 190.
- Smith, Peter F. The Dynamics of Delight: Architecture and Aesthetics (New York: Routledge, 2003) p. 83, ISBN 0-415-30010-X
- Roy Howat (1983). Debussy in Proportion: A Musical Analysis. Cambridge University Press. ISBN 978-0-521-31145-8.
- Simon Trezise (1994). Debussy: La Mer. Cambridge University Press. p. 53. ISBN 978-0-521-44656-3.
- "An 833 Cents Scale: An experiment on harmony", Huygens-Fokker.org. Accessed December 1, 2012.
- Livio 2003, p. 154.
- Richard Padovan (1999). Proportion. Taylor & Francis. pp. 305–306. ISBN 978-0-419-22780-9.
- Padovan, Richard (2002). "Proportion: Science, Philosophy, Architecture". Nexus Network Journal. 4 (1): 113–122. doi:10.1007/s00004-001-0008-7.
- Zeising, Adolf (1854). Neue Lehre van den Proportionen des meschlischen Körpers. preface.
- Pommersheim, James E., Tim K. Marks, and Erica L. Flapan, eds. 2010. "Number Theory: A Lively Introduction with Proofs, Applications, and Stories". John Wiley and Sons: 82.
- Coldea, R.; Tennant, D.A.; Wheeler, E.M.; Wawrzynksa, E.; Prabhakaran, D.; Telling, M.; Habicht, K.; Smeibidl, P.; Keifer, K. (2010). "Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry". Science. 327 (5962): 177–180. arXiv:1103.3694. Bibcode:2010Sci...327..177C. doi:10.1126/science.1180085. PMID 20056884. S2CID 206522808.
- "A Disco Ball in Space". NASA. 2001-10-09. Retrieved 2007-04-16.
- Pheasant, Stephen (1998). Bodyspace. London: Taylor & Francis. ISBN 978-0-7484-0067-6.
- van Laack, Walter (2001). A Better History Of Our World: Volume 1 The Universe. Aachen: van Laach GmbH.
- Falbo, Clement (March 2005). "The golden ratio—a contrary viewpoint". The College Mathematics Journal. 36 (2): 123–134. doi:10.1080/07468342.2005.11922119. JSTOR 30044835.
- Moscovich, Ivan, Ivan Moscovich Mastermind Collection: The Hinged Square & Other Puzzles, New York: Sterling, 2004[page needed]
- Peterson, Ivars (1 April 2005). "Sea shell spirals". Science News. Archived from the original on 3 October 2012. Retrieved 10 November 2008.
- Man, John, Gutenberg: How One Man Remade the World with Word (2002) pp. 166–167, Wiley, ISBN 0-471-21823-5. "The half-folio page (30.7 × 44.5 cm) was made up of two rectangles—the whole page and its text area—based on the so called 'golden section', which specifies a crucial relationship between short and long sides, and produces an irrational number, as pi is, but is a ratio of about 5:8."
- Fechner, Gustav (1876). Vorschule der Ästhetik. Leipzig: Breitkopf & Härtel. pp. 190–202.
- Livio 2003, p. 7.
- For instance, Osler writes that "38.2 percent and 61.8 percent retracements of recent rises or declines are common," in Osler, Carol (2000). "Support for Resistance: Technical Analysis and Intraday Exchange Rates" (PDF). Federal Reserve Bank of New York Economic Policy Review. 6 (2): 53–68.
- Roy Batchelor and Richard Ramyar, "Magic numbers in the Dow," 25th International Symposium on Forecasting, 2005, p. 13, 31. "Not since the 'big is beautiful' days have giants looked better", Tom Stevenson, The Daily Telegraph, Apr. 10, 2006, and "Technical failure", The Economist, Sep. 23, 2006, are both popular-press accounts of Batchelor and Ramyar's research.
- Herz-Fischler, Roger (2000). The Shape of the Great Pyramid. Waterloo, Ontario: Wilfrid Laurier University Press. ISBN 0-88920-324-5. MR 1788996. The entire book surveys many alternative theories for this pyramid's shape. See Chapter 11, "Kepler triangle theory", pp. 80–91, for material specific to the Kepler triangle, and p. 166 for the conclusion that the Kepler triangle theory can be eliminated by the principle that "A theory must correspond to a level of mathematics consistent with what was known to the ancient Egyptians." See note 3, p. 229, for the history of Kepler's work with this triangle.
- Rossi, Corinna (2004). Architecture and Mathematics in Ancient Egypt. Cambridge University Press. pp. 67–68. ISBN 978-0-521-82954-0.
there is no direct evidence in any ancient Egyptian written mathematical source of any arithmetic calculation or geometrical construction which could be classified as the Golden Section ... convergence to , and itself as a number, do not fit with the extant Middle Kingdom mathematical sources; see also extensive discussion of multiple alternative theories for the shape of the pyramid and other Egyptian architecture, pp. 7–56
- Rossi, Corinna; Tout, Christopher A. (2002). "Were the Fibonacci series and the Golden Section known in ancient Egypt?". Historia Mathematica. 29 (2): 101–113. doi:10.1006/hmat.2001.2334. MR 1896969.
- Markowsky, George (January 1992). "Misconceptions about the Golden Ratio" (PDF). The College Mathematics Journal. Mathematical Association of America. 23 (1): 2–19. doi:10.2307/2686193. JSTOR 2686193. Archived (PDF) from the original on 2020-12-11. Retrieved 2012-06-29.
It does not appear that the Egyptians even knew of the existence of much less incorporated it in their buildings
- Livio 2003, pp. 74–75.
- Van Mersbergen, Audrey M., "Rhetorical Prototypes in Architecture: Measuring the Acropolis with a Philosophical Polemic", Communication Quarterly, Vol. 46 No. 2, 1998, pp. 194–213.
- Devlin, Keith J. (2009) . The Math Instinct: Why You're a Mathematical Genius (Along with Lobsters, Birds, Cats, and Dogs). New York: Basic Books. p. 54. ISBN 978-1-56025-672-4.
- Gazalé, Midhat J., Gnomon: From Pharaohs to Fractals, Princeton University Press, 1999, p. 125. ISBN 0-691-00514-1
- Patrice Foutakis, "Did the Greeks Build According to the Golden Ratio?", Cambridge Archaeological Journal, vol. 24, n° 1, February 2014, pp. 71–86.
- Le Salon de la Section d'Or, October 1912, Mediation Centre Pompidou
- Jeunes Peintres ne vous frappez pas !, La Section d'Or: Numéro spécial consacré à l'Exposition de la "Section d'Or", première année, n° 1, 9 octobre 1912, pp. 1–7 Archived 2020-10-30 at the Wayback Machine, Bibliothèque Kandinsky
- Herbert, Robert, Neo-Impressionism, New York: The Solomon R. Guggenheim Foundation, 1968[page needed]
- Livio 2003, p. 169.
- Camfield, William A., Juan Gris and the Golden Section, Art Bulletin, 47, no. 1, March 1965, 128–134. 68
- Green, Christopher, Juan Gris, Whitechapel Art Gallery, London, 18 September–29 November 1992; Staatsgalerie Stuttgart 18 December 1992–14 February 1993; Rijksmuseum Kröller-Müller, Otterlo, 6 March–2 May 1993, Yale University Press, 1992, pp. 37–38, ISBN 0300053746
- Cottington, David, Cubism and Its Histories, Barber Institute's critical perspectives in art history series, Critical Perspectives in Art History, Manchester University Press, 2004, pp. 112, 142, ISBN 0719050049
- Roger Allard, Sur quelques peintre, Les Marches du Sud-Ouest, June 1911, pp. 57–64. In Mark Antliff and Patricia Leighten, A Cubism Reader, Documents and Criticism, 1906-1914, The University of Chicago Press, 2008, pp. 178–191, 330.
- Bouleau, Charles, The Painter's Secret Geometry: A Study of Composition in Art (1963) pp. 247–248, Harcourt, Brace & World, ISBN 0-87817-259-9
- Livio 2003, pp. 177–178.
- Livio, Mario (2003) . The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (First trade paperback ed.). New York City: Broadway Books. ISBN 978-0-7679-0816-0.
- Stakhov, Alexey P.; Olsen, Scott (2009). The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science. Singapore: World Scientific Publishing. ISBN 978-981-277-582-5.
- Doczi, György (2005) . The Power of Limits: Proportional Harmonies in Nature, Art, and Architecture. Boston: Shambhala Publications. ISBN 978-1-59030-259-0.
- Hemenway, Priya (2005). Divine Proportion: Phi In Art, Nature, and Science. New York: Sterling. ISBN 978-1-4027-3522-6.
- Huntley, H. E. (1970). The Divine Proportion: A Study in Mathematical Beauty. New York: Dover Publications. ISBN 978-0-486-22254-7.
- Joseph, George G. (2000) . The Crest of the Peacock: The Non-European Roots of Mathematics (New ed.). Princeton, NJ: Princeton University Press. ISBN 978-0-691-00659-8.
- Sahlqvist, Leif (2008). Cardinal Alignments and the Golden Section: Principles of Ancient Cosmography and Design (3rd Rev. ed.). Charleston, SC: BookSurge. ISBN 978-1-4196-2157-4.
- Schneider, Michael S. (1994). A Beginner's Guide to Constructing the Universe: The Mathematical Archetypes of Nature, Art, and Science. New York: HarperCollins. ISBN 978-0-06-016939-8.
- Scimone, Aldo (1997). La Sezione Aurea. Storia culturale di un leitmotiv della Matematica. Palermo: Sigma Edizioni. ISBN 978-88-7231-025-0.
- Walser, Hans (2001) [Der Goldene Schnitt 1993]. The Golden Section. Peter Hilton trans. Washington, DC: The Mathematical Association of America. ISBN 978-0-88385-534-8.
- "Golden ratio", Encyclopedia of Mathematics, EMS Press, 2001 
- "Golden Section" by Michael Schreiber, Wolfram Demonstrations Project, 2007.
- Weisstein, Eric W. "Golden Ratio". MathWorld.
- Knott, Ron. "The Golden section ratio: Phi". Information and activities by a mathematics professor.
- The Pentagram & The Golden Ratio. Green, Thomas M. Updated June 2005. Archived November 2007. Geometry instruction with problems to solve.
- The Myth That Will Not Go Away, by Keith Devlin, addressing multiple allegations about the use of the golden ratio in culture.
- Spurious golden spirals collected by Randall Munroe
- YouTube lecture on Zeno's mice problem and logarithmic spirals