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Golden rectangle

 Golden rectangle

Indicating German Der goldene schnitt, das rechteck im goldenen schnitt

Indicating French 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
Two rectangles with the classic proportion and the ISO proportion 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.

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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

Geometric construction for the golden rectangle proportion

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).

Spiral line formed by means of quarter circles. The radius decreases by the factor of the golden rectangle from quarter to quarter. The ratio is about 0.86

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