Golden rectangle

Both rectangles - each with the aspect ratios a: b and (a + b): a - are each golden rectangles ( animated representation ).

A golden rectangle is a rectangle whose aspect ratio corresponds to the two sides and the golden ratio . ${\ displaystyle a}$ ${\ displaystyle b}$

The following applies to the aspect ratios - with equal a and equal b - ${\ displaystyle a}$ ${\ displaystyle b}$

{\ displaystyle a: b = (a + b): a}

.

A distinctive feature of this geometric figure is: If you remove a square section, a golden rectangle is created again .

Constructions and properties

Figure 1: Construction of a golden rectangle from a square

Image 2: Golden rectangle in a square with side length a

Probably the simplest construction is obtained by starting with a square (Fig. 1) and expanding it into a golden rectangle. To do this, you first select a parallel pair of sides of the square and construct its side centers. Then you lengthen the pair of sides on one side of the square and draw a circle around the center of the side , which goes through the corner points of the square opposite the center of the side. This circle intersects the extension of the side of the square at the corner of the golden rectangle. The second corner point is obtained by carrying out an analogous construction with the second center of the side or by creating a perpendicular to the first corner point of the golden rectangle , which intersects the second side extension of the square.

The sides of a square (Fig. 2) are divided in the golden ratio in such a way that only the shorter side sections rest on one opposite pair of corners and only the longer side sections on the other. The four dividing points on the sides of the square now form a golden rectangle.

Image 3: Construction of a golden rectangle from a pentagon

Fig. 4: Approximation of the golden spiral

In a regular pentagon (Fig. 3) the diagonals share a golden ratio. This property can also be used to construct a golden rectangle. First you construct a regular pentagon with side length including two of its intersecting diagonals . Now you take one of the diagonals as the base of the rectangle and set up a line of length perpendicular to it at each end, so you get a golden rectangle. ${\ displaystyle a}$ ${\ displaystyle a,}$

The fact that a golden rectangle is made up of a square and another golden rectangle can be used to divide a given golden rectangle in a spiral (Fig. 4) into an infinite series of squares. If you draw adjacent quarter circles in these squares, you get a flat spiral composed of ever smaller quarter circles . If the starting rectangle has the side lengths 1 and this spiral forms a relatively exact approximation of the golden spiral ${\ displaystyle \ varphi = {\ tfrac {1 + {\ sqrt {5}}} {2}}}$

literature

Alexey Stakhov: Golden Rectangle and Golden Brick . In: Alexey Stakhov, Alekseĭ Petrovich Stakhov, Scott Anthony Olsen: The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science . Word Scientific 2009, ISBN 978-981-277-582-5 , pp. 20-23 ( excerpt (Google) )
Albrecht Beutelspacher, Bernhard Petri: The golden ratio. Spektrum, Heidelberg, Berlin, Oxford 1988. ISBN 3-411-03155-7 , pp. 47-56
Edward B. Burger, Michael P. Starbird: The Heart of Mathematics: An Invitation to Effective Thinking . Springer 2005, ISBN 1-931914-41-9 , pp. 232–248 ( excerpt (Google) )

Web links

Commons : Golden Rectangle - collection of images, videos and audio files

Eric W. Weisstein : Golden Rectangle . In: MathWorld (English).
Golden Rectangle - University website