# Golden rectangle

Golden rectangle

Der goldene schnitt, das rechteck im goldenen schnitt

Proportion d'or, coupe d'or ???

A rectangle with proportions that from classical Greek times has been thought optically pleasing.

The ration of the two sides is approxiamtely 8 : 13 or 1 : 1.625. The exact value of the proportion factor is determined by the formula (sqrt(5) - 1) / 2, giving 1.618033989..

ISO standard 1 : 1.414 |
Golden rectangle 1 : 1.618 |

Belletristic books very often are created in these proportions. You can also see this proportion in classical buildings and statues.

### Background

The *rule* to get the proportion (as stated in ancient times) is:

The relation between the shorter and longer side of a golden rectangle is the same as the relation between the larger side to the sum of both lengthes: a : b = b : (a+b)

If this proportion factor is *p*, then the following formulas demonstrate
the very nature of this factor:

p = 1 + 1/p; p^{2} = p + 1

The value *p* is the limes of quotients of two Fibonacci numbers.
Fibonacci numbers *f* are built up be summing the current number with
the predecessor: 1, 1, 2, 3, 5, 8, ... or in a formula *f*_{n+1}
= *f*_{n} + *f*_{n-1} with *f*_{0}
= *f*_{1} = 1.

The proportion factor *p* = lim *f*_{n} / *f*_{n-1}
(with n going to infinity):

1/2, 2/3, 3/5, 5/8, 8/13, 13/21, 21/34, 34/55, 55/89, 89/144, 144/233, 233/377, 377/610, 610/987, 987/1597, 1597/2584 2584/4181 4181/6765 ... (See also Fibonacci calculator).

### Geometric construction

### Proportion compas

A special drawing tool to get the proportion immediately may be still in use by artists.

### Sideeffects of the Golden Section

Inscribing a square in a *golden rectangle* leaves another *golden
rectangle*. Setting up quarter circles in each of the squares create very
nice spirals (approximations of hyperbolic spirals).

## Sources

- Albrecht Beutelspacher und Bernhard, Petri: Der Goldene Schnitt. Spektrum Akademischer Verlag, Heidelberg 1996.
- Spektrum der Wissenschaft, November 1997: Mathematische Unterhaltung.

## Further links

- Artist's rendering of the golden rectangle by Tontyn
Hopman in Die
Ordnung der Schöpfung in Zahl und Geometrie:

- Encyclopedia Britannica
- www.mcs.surrey.ac.uk