# golden ratio (φ)

The golden ratio.

The golden ratio appears in Dali's The Last Supper.

The golden ratio (φ) is a remarkable number that, like pi and *e*,
pops up all over the place in mathematics but, in some ways, has a more
'human' connection in that it seems to be linked to aesthetics. Its name,
which is also given as the **golden mean**, the **golden
section**, the **golden number**, and the **divine
proportion**, reflects this sense of a harmonious or pleasing ideal.

The golden ratio is an irrational
number of the type known as an algebraic
number (in contrast with pi and *e*, which are transcendental)
and is represented by the Greek letter φ (phi). It can be defined in
various ways. For example, it is the only number equal to its own reciprocal
plus 1, i.e. φ = (1/φ so that φ^{2} = φ + 1. From
this comes the quadratic equation φ^{2} – φ – 1 = 0 of which
the golden ratio is the positive solution, (1 + √(5))/2 = 1.6180339887...
The golden ratio is also approximated by the ratio of successive terms in
the Fibonacci sequence; in fact, *F*(*n*+1)/*F*(*n*) gets closer and closer to phi as *n* tends to infinity. Because 1/(1 – φ) = φ, the continued
fraction representation of φ is

φ = 1+1/(1+1/(1+1/(1+1/(1+1/... = [1;1,1,1,1, ...]

Two quantities are said to be in the golden ratio, if the ratio of the larger
one, *a*, to the smaller one, *b*, is the same as the ratio of
the smaller one to their difference, i.e., *a*/*b* = *b*/(*a* – *b*).
The so-called **golden rectangle** is one whose sides *a* and *b* stand in the golden ratio. It is famously said to have great
aesthetic appeal and is closely approximated by the dimensions of the front
of the Parthenon in Rome. Leonardo da Vinci's masterpiece the Mona Lisa
is said to have a face that is framed by a golden rectangle; what is certain
is that Leonardo was a
close personal friend of Luca Pacioli, who
published a three-volume treatise on the golden ratio, *Divina Proportione*,
in 1509. The Swiss-French architect and painter Le Corbusier designed an
entire proportional system called the "Modulor," that was based on the golden
ratio. The Modulor was supposed to provide a standardized system that would
automatically confer harmonious proportions to everything, from door handles
to high-rise buildings. Another artist who deliberately used the golden
ratio is the surrealist Salvador Dali. The ratio of the dimensions of Dali's
Sacrament of the Last Supper is equal to the golden ratio. Dali also incorporated
in the painting a huge dodecahedron (a twelve-faced Platonic solid in which
each side is a pentagon) engulfing the supper table.

The dodecahedron, which according to Plato is the solid "which the god used for embroidering the constellations on the whole heaven," is intimately related to the golden ratio – both the surface area and the volume of a dodecahedron of unit edge length are simple functions of the golden ratio. In fact, φ turns up frequently in figures that have pentagonal symmetry. For instance the ratio of a regular pentagon's side and diagonal is equal to φ, and the vertices of a regular icosahedron are located on three orthogonal golden rectangles. The golden ratio is also related to Penrose tiling and to the plastic number.

### Reference

1. Livio, Mario. *The Golden Ratio: The Story of Phi, the World's
Most Astonishing Number*. New York: Broadway Books, 2002.