the golden ratio

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.


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

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).

These two triangles are at the origin of the Penrose tilings.

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 center of 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!

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.

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

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".

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.

animation

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

During the closing ceremony of the Sydney 2000 Olympic Games a regular dodecahedron sat enthroned in the center 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!

animation

references:	• Fibonacci Numbers and the Golden Section (site) • Alexander Bogomolny's logo • Aux origines du nombre d'or (video by El Jj, in French)	• 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...

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)

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August 1999
updated 07-09-2012