  # 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; p2 = 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 fn+1 = fn + fn-1 with f0 = f1 = 1.

The proportion factor p = lim fn / fn-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  URL: Created: 1997-11-06 Updated: © Docu+Design Daube, Zürich