Some of the greatest mathematical minds of all ages, from
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
pentagons. According to one story, 5th-century BC mathematician
Hippasus discovered that the golden ratio was neither a whole number nor a fraction (it is
Pythagoreans.Euclid's Elements (c. 300 BC) provides several
propositions and their proofs employing the golden ratio,[c] and contains its first known definition which proceeds as follows:
A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.[d]
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
Luca Pacioli named his book Divina proportione (
1509) after the ratio; the book, largely plagiarized from
Piero della Francesca, explored its properties including its appearance in some of the
Platonic solids.Leonardo da Vinci, who illustrated Pacioli's book, called the ratio the sectio aurea ('golden section'). 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. 16th-century mathematicians such as
Rafael Bombelli solved geometric problems using the ratio.
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.
zome construction system, developed by
Steve Baer in the late 1960s, is based on the
symmetry system of the
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 gained in interest after
Dan Shechtman's Nobel-winning 1982 discovery of
quasicrystals with icosahedral symmetry, which were soon afterward explained through analogies to the Penrose tiling.
The golden ratio is an
irrational number. Below are two short proofs of irrationality:
Contradiction from an expression in lowest terms
the whole is the longer part plus the shorter part;
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
is to as is to
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 √5
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 all non-
squarenatural numbers are irrational.
The absolute value of this quantity () corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, ).
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 :
It is in fact the simplest form of a continued fraction, alongside its reciprocal form:
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
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 :
Successive powers of the golden ratio obey the Fibonacci
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
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.
When two angles that make a full circle have measures in the golden ratio, the smaller is called the golden angle, with measure
This angle occurs in
patterns of plant growth as the optimal spacing of leaf shoots around plant stems so that successive leaves do not block sunlight from the leaves below them.
Pentagonal symmetry system
Pentagon and pentagram
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 Dividing both sides by yields (see
§ Calculation above),
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.
Pentagonal and pentagrammic geometry permits us to calculate the following values for :
The triangle formed by two diagonals and a side of a regular pentagon is called a golden triangle or sublime triangle. It is an acute
isosceles triangle with apex angle 36° and base angles 72°. Its two equal sides are in the golden ratio to its base. The triangle formed by two sides and a diagonal of a regular pentagon is called a golden gnomon. It is an obtuse isosceles triangle with apex angle 108° and base angle 36°. Its base is in the golden ratio to its two equal sides. The pentagon can thus be subdivided into two golden gnomons and a central golden triangle. The five points of a
regular pentagram are golden triangles, as are the ten triangles formed by connecting the vertices of a
regular decagon to its center point.
Bisecting one of the base angles of the golden triangle subdivides it into a smaller golden triangle and a golden gnomon. Analogously, any acute isosceles triangle can be subdivided into a similar triangle and an obtuse isosceles triangle, but the golden triangle is the only one for which this subdivision is made by the angle bisector, because it is the only isosceles triangle whose base angle is twice its apex angle. The angle bisector of the golden triangle subdivides the side that it meets in the golden ratio, and the areas of the two subdivided pieces are also in the golden ratio.
If the apex angle of the golden gnomon is
trisected, the trisector again subdivides it into a smaller golden gnomon and a golden triangle. The trisector subdivides the base in the golden ratio, and the two pieces have areas in the golden ratio. Analogously, any obtuse triangle can be subdivided into a similar triangle and an acute isosceles triangle, but the golden gnomon is the only one for which this subdivision is made by the angle trisector, because it is the only isosceles triangle whose apex angle is three times its base angle.
The golden ratio appears prominently in the Penrose tiling, a family of
aperiodic tilings of the plane developed by
Roger Penrose, inspired by
Johannes Kepler's remark that pentagrams, decagons, and other shapes could fill gaps that pentagonal shapes alone leave when tiled together. Several variations of this tiling have been studied, all of whose prototiles exhibit the golden ratio:
Penrose's original version of this tiling used four shapes: regular pentagons and pentagrams, "boat" figures with three points of a pentagram, and "diamond" shaped rhombi.
The kite and dart Penrose tiling uses
kites with three interior angles of 72° and one interior angle of 144°, and darts, concave quadrilaterals with two interior angles of 36°, one of 72°, and one non-convex angle of 216°. Special matching rules restrict how the tiles can meet at any edge, resulting in seven combinations of tiles at any vertex. Both the kites and darts have sides of two lengths, in the golden ratio to each other. The areas of these two tile shapes are also in the golden ratio to each other.
The kite and dart can each be cut on their symmetry axes into a pair of golden triangles and golden gnomons, respectively. With suitable matching rules, these triangles, called in this context Robinson triangles, can be used as the prototiles for a form of the Penrose tiling.
The rhombic Penrose tiling contains two types of rhombus, a thin rhombus with angles of 36° and 144°, and a thick rhombus with angles of 72° and 108°. All side lengths are equal, but the ratio of the length of sides to the short diagonal in the thin rhombus equals , as does the ratio of the sides of to the long diagonal of the thick rhombus. As with the kite and dart tiling, the areas of the two rhombi are in the golden ratio to each other. Again, these rhombi can be decomposed into pairs of Robinson triangles.
Original four-tile Penrose tiling
Rhombic Penrose tiling
In triangles and quadrilaterals
George Odom found a construction for involving an
equilateral triangle: if the line segment joining the midpoints of two sides is extended to intersect the
circumcircle, then the two midpoints and the point of intersection with the circle are in golden proportion.
These side lengths are the three
Pythagorean means of the two numbers . The three squares on its sides have areas in the golden geometric progression .
Among isosceles triangles, the ratio of
inradius to side length is maximized for the triangle formed by two
reflected copies of the Kepler triangle, sharing the longer of their two legs. The same isosceles triangle maximizes the ratio of the radius of a
semicircle on its base to its
Draw a line from the midpoint of one side of the square to an opposite corner.
Use that line as the radius to draw an arc that defines the height of the rectangle.
Complete the golden rectangle.
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).
Logarithmic spirals are
self-similar spirals where distances covered per turn are in
geometric progression. A logarithmic spiral whose radius increases by a factor of the golden ratio for each quarter-turn is called the
golden spiral. These spirals can be approximated by quarter-circles that grow by the golden ratio, or their approximations generated from Fibonacci numbers, often depicted inscribed within a spiraling pattern of squares growing in the same ratio. The exact logarithmic spiral form of the golden spiral can be described by the
polar equation with :
Not all logarithmic spirals are connected to the golden ratio, and not all spirals that are connected to the golden ratio are the same shape as the golden spiral. For instance, a different logarithmic spiral, encasing a nested sequence of golden isosceles triangles, grows by the golden ratio for each 108° that it turns, instead of the 90° turning angle of the golden spiral. Another variation, called the "better golden spiral", grows by the golden ratio for each half-turn, rather than each quarter-turn.
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
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 also can be given using formulas involving . 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.
cube can be
inscribed in a regular dodecahedron, with some of the diagonals of the pentagonal faces of the dodecahedron serving as the cube's edges; therefore, the edge lengths are in the golden ratio. The cube's volume is times that of the dodecahedron's. 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.
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
. 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 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 
architectLe Corbusier, famous for his contributions to the
moderninternational 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."
In addition to the golden ratio, Le Corbusier based the system on
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.
Leonardo da Vinci's illustrations of
polyhedra in Pacioli's 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 Leonardo's 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.
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 
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".
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.
However, some have argued that many apparent manifestations of the golden ratio in nature, especially in regard to animal dimensions, are fictitious.
Isingferromagnet (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
Examples of disputed observations of the golden ratio include the following:
Specific proportions in the bodies of vertebrates (including humans) are often claimed to be in the golden ratio; for example the ratio of successive
metacarpal bones (finger bones) has been said to approximate 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 shells of mollusks such as the
nautilus are often claimed to be in the golden ratio. The growth of nautilus shells follows a
logarithmic spiral, and it is sometimes erroneously claimed that any logarithmic spiral is related to the golden ratio, or sometimes claimed that each new chamber is golden-proportioned relative to the previous one. However, measurements of nautilus shells do not support this claim.
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
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.
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.
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.
Section d'Or ('Golden Section') was a collective of
painters, sculptors, poets and critics associated with
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. (Several authors have claimed that Seurat employed the golden ratio in his paintings, but Seurat’s writings and paintings suggest that he employed simple whole-number ratios and any approximation of the golden ratio was coincidental.) 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.
^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 .
^Other names include the golden mean, golden section,golden cut,golden proportion, golden number,medial section, and divine section.
^Euclid, Elements, Book II, Proposition 11; Book IV, Propositions 10–11; Book VI, Proposition 30; Book XIII, Propositions 1–6, 8–11, 16–18.
Summerson, John (1963).
Heavenly Mansions and Other Essays on Architecture. New York: W.W. Norton. 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.
Schielack, Vincent P. (1987). "The Fibonacci Sequence and the Golden Ratio". The Mathematics Teacher. 80 (5): 357–358.
JSTOR27965402. This source contains an elementary derivation of the golden ratio's value.
Odom, George; van de Craats, Jan (1986). "E3007: The golden ratio from an equilateral triangle and its circumcircle". Problems and solutions. The American Mathematical Monthly. 93 (7): 572.
du Val, Patrick; Flather, H.T.;
Petrie, J.F. (1938).
The Fifty-Nine Icosahedra. Vol. 6. University of Toronto Studies. 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.
Muller, J. M. (2006). Elementary functions : algorithms and implementation (2nd ed.). Boston: Birkhäuser. p. 93.
^Le Corbusier, The Modulor, p. 35, as cited in
Padovan, Richard (1999). Proportion: Science, Philosophy, Architecture. Taylor & Francis. p. 320.
ISBN9781135811112. Both the paintings and the architectural designs make use of the golden section
Devlin, Keith (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.
Tosto, Pablo (1969). La composición áurea en las artes plásticas [The golden composition in the plastic arts] (in Spanish). Hachette. pp. 134–144.
^Tschichold, Jan (1991).
The Form of the Book. Hartley & Marks. p. 43 Fig 4.
ISBN0-88179-116-4. 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.
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?
Cox, Simon (2004).
Cracking the Da Vinci Code. Barnes & Noble. p. 62. The Golden Ratio also crops up in some very unlikely places: widescreen televisions, postcards, credit cards and photographs all commonly conform to its proportions.
^Posamentier & Lehmann 2011, chapter 4, footnote 12: "The Togo flag was designed by the artist Paul Ahyi (1930–2010), who claims to have attempted to have the flag constructed in the shape of a golden rectangle".
Lendvai, Ernő (1971). Béla Bartók: An Analysis of His Music. London: Kahn and Averill.
Man, John (2002).
Gutenberg: How One Man Remade the World with Word. Wiley. pp. 166–167.
ISBN9780471218234. 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 [Preschool of Aesthetics] (in German). Leipzig: Breitkopf & Härtel. pp. 190–202.
Herz-Fischler, Roger (2000). The Shape of the Great Pyramid. Wilfrid Laurier University Press.
ISBN0-88920-324-5. 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. 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
Van Mersbergen, Audrey M. (1998). "Rhetorical Prototypes in Architecture: Measuring the Acropolis with a Philosophical Polemic". Communication Quarterly. 46 (2): 194–213.