the golden ratio

phi formulas
It is interesting to point out that every sequence defined as the Fibonacci sequence by f(n+1)=f(n)+f(n-1) leads to the golden ratio, no matter what the two initial values f(0) and f(1) are: f(n+1)/f(n) -> φ.  But the golden ratio is also the irrational number the worst approximated by rational numbers because all the integer parts in its continuous fraction are all equal to 1!
Remark: Any non-zero natural integer is written uniquely as sum of non-consecutive numbers of the Fibonacci sequence (Zeckendorf's theorem).
example: 33=21+8+3+1   by successive subtractions of numbers from the Fibonacci sequence (the largest possible):  33-21=12   12-8=4   4-3=1

three constructions of the golden ratio

We construct 0, U and G such as OG/OU=φ, thus OG=φ if we choose OU as unit.
construction 1 construction 2 construction 3
starting from a rectangular triangle OIV with OV=2×OI
- classical construction -
starting from an equilateral triangle and a square
- Michel Bataille -
starting from three segments with same length
MO=NU=PG (N midpoint of [OM] and P of [UN])
- Jo Niemeyer -

classical golden figures

figures d'or construction of
the regular pentagon

A simple knot made with a strip of paper, and then carefully flatted is a "golden knot"; just fold over one of the strip's ends and you get a complete pentagram (convex regular pentagon with its five diagonals which are the sides of a regular star pentagon). The ratio of the sides of this two regular pentagons is the golden ratio φ.
A rectangle whose length/width ratio is the golden ratio is a golden rectangle. Its construction is simple (with AB=2AU, ABCD and ABC'D' are two golden rectangles, and we get the first by adding a square to the second).
The vertices of three golden rectangles two by two orthogonal are the vertices of a regular icosahedron; more generally two opposite edges of a regular icosahedron define a golden rectangle (thus there are 15).
A rhombus whose diagonal's ratio is the golden ratio is a golden rhombus (its vertices are the midpoints of the sides of a golden rectangle).
The pentagram shows several golden sections and several examples of the two types of golden triangles: isosceles triangles with ratio of the sides equal to φ, their angles measure  72°-36°-72°  and  36°-108°-36°  (remark: cos π/5 = φ/2).

An ellipse inscribed in a golden rectangle is a golden ellipse (ratio of the axis equal to φ).
Its foci F' and F are easy to construct (M being a point on the ellipse, if AB=1 then  MF'+MF = F'F² = φ). Its area is πφ (for AB=1).
Its eccentricity e is equal to the distance focus-directrix d (if AB=2 then  e = d = 1/√φ).

With sequences of embedded golden rectangles or triangles, we get easily nice "golden spirals". These drawings are close to logarithmic spirals, also known as equiangle spirals (constant tangent angle) or Bernoulli's spirals; they are invariant by similitude. On Bernoulli's tombstone is engraved "eadem mutata resurgo" which can be translated by "moved, I rise again the same".
On the left drawing the centre if similitude is the intersection of the rectangles' diagonals, the ratio 1/φ, and the angle -π/2; the radius is thus multiplied by φ4≅6,9 at each turn. On the spiral on the right this coefficient is around 5, while it is around 3 for the nautilus' shell!

spirales d'or

Miss Nature uses them to provide harmonious growths (flowers, fruits, shells, horns... galaxies).

Plant species implant scales (pine cone), seeds (sunflower)... on a spiral by creating an object every 137,5...°. These objects are then laid-out in arcs of spirals oriented in the two directions; the numbers of arcs (in the two directions) are two consecutive numbers of the Fibonacci's sequence.
The golden angle above corresponds to the division of the circle into two arcs with lengths proportional to 1 and φ:  a/1 = b/φ = (a+b)/(1+φ) = 2π/φ² rad  or  360°/φ².
pine cone & sunflower flower
Golden rectangle, golden ellipse, pentagram, golden spiral... are generally considered as especially harmonious. It's not surprising that the golden ratio is widely used in painting and in architecture.
It appears for example in the glass pyramid of the Louvre, in Paris: for a square base with side 2, the altitude is √φ (thus the main altitude of each of its lateral face is φ).
Remark: the rectangular triangle with sides 1, √φ and φ (converse of Pythagoras: 1+φ = φ²) leads to an other outstanding angle  α = 51.827...°  with  cos(α) = 1/φ  and  sin(α) = 1/√φ
the pyramid of Louvre

the Penrose tilings

By assembling two identical golden triangles we get two pairs of quadrilaterals (dart and kite - skinny and fat rhombi) which allow to carry out nice tilings with 5-fold symmetry (stars and suns below).
An other example of a tilling of this type designed by Nicolas Hannachi (one my notice an assembling condition).
star-sun 1 star-sun 2
Penrose pavers 1 Penrose pavers 2

To get non-periodic tilings of the plane one must use a simple matching rule: tiles have to meet in a way such that each arc of each tile is connected with the arcs on the neighboured tiles. Since both sets use golden triangles, it makes sense that a kite and dart tiling can be translated into a rhombi tiling, and vice versa.

Penrose kites-darts Penrose rhombs

Discovered by Roger Penrose in 1974 these tilings have curious properties; among them:
  • any finite region of a Penrose tiling appears an infinity of times in any Penrose tiling,
  • there are small regions with a 5-fold symmetry,
  • if the tiling grows infinitely the ratio of the numbers of tiles in each type tends toward φ!
  • a rhombi tilling reveals a pattern of two types of overlapping decagons; the ratio of the two populations is φ!

a golden ring (Alexander Bogomolny)

With a strip long enough we may make five golden knots at regular intervals. By gluing the two ends together we get this nice pentagonal ring which is a Möbius strip: it has only one side and one "edge". gold knots

a golden pyramid

The pentagram is the template of an outstanding pyramid whose lateral faces are golden triangles. If the radius of its base is 1 then its altitude is φ.

If we assemble twelve of such pyramids on the faces of the regular dodecahedron then we get the small stellated dodecahedron.


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the gold of Sydney...   (medals and mathematics)

During the closing ceremony of the Sydney 2000 Olympic Games a regular dodecahedron sat enthroned in the centre of the olympic stadium; the idea to flatten it in order to make a podium consisting of two half templates was quite original and constituted a nice symbol: each face of this polyhedron is a regular pentagon whose diagonal and side are in the golden ratio!

Sydney

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references: •  Phi - The golden number  (site)
•  Fibonacci Numbers and the Golden Section  (site)      
•  Alexander Bogomolny's logo
•  Le nombre d'or  by Thérèse Eveilleau (in French)
•  Le nombre d'or  by Robert Chalavoux (in French)
•  Le tournesol  (student's work in French)
•  ->  all that glitters is not gold!: the end of a myth with Geogebra...
•  shell of molluscs  (shellfish shell) by J.B. Roux (web page in French)
•  Penrose tilings  by Eric Hwang   -   Penrose tilings of the plane  by Ianiv Schweber
•  Penrose tiling applet ( download penrose.jar then open it with Java)
•  a Penrose tiling generator by J.-L. Sigrist
As we noticed, φ appears not only in the mathematical reality, sometimes in an unexpected way, but also in nature. The aesthetics of this ratio has been shown without ambiguousness; it is thus a nice example of "mathematical beauty".
A good reading: The Divine Proportion. A Study in Mathematical Beauty  by Herbert E. Huntley (Dover Publications Inc. New-York - 1970)


August 1999
updated 07-09-2012