Rectangle whose side lengths are in the golden ratio
This article is about the geometrical figure. For the Indian highway project, see
Golden Quadrilateral.
In
geometry, a golden rectangle is a
rectangle whose side lengths are in the
golden ratio, $1:{\tfrac {1+{\sqrt {5}}}{2}}$, which is $1:\varphi$ (the Greek letter phi), where $\varphi$ is approximately 1.618.
Golden rectangles exhibit a special form of
self-similarity: All rectangles created by adding or removing a square from an end are golden rectangles as well.
Drawing a line from the midpoint of one side of the square to an opposite corner
Using that line as the radius to draw an arc that defines the height of the rectangle
Completing the golden rectangle
A distinctive feature of this shape is that when a
square section is added—or removed—the product is another golden rectangle, having the same
aspect ratio as the first. Square addition or removal can be repeated infinitely, in which case corresponding corners of the squares form an infinite sequence of points on the
golden spiral, the unique
logarithmic spiral with this property. Diagonal lines drawn between the first two orders of embedded golden rectangles will define the intersection point of the diagonals of all the embedded golden rectangles;
Clifford A. Pickover referred to this point as "the Eye of God".^{
[2]}
According to Livio, since the publication of
Luca Pacioli's Divina proportione in 1509, "the Golden Ratio started to become available to artists in theoretical treatises that were not overly mathematical, that they could actually use."^{
[6]}
Euclid gives an alternative construction of the golden rectangle using three polygons
circumscribed by congruent circles: a regular
decagon,
hexagon, and
pentagon. The respective lengths a, b, and c of the sides of these three polygons satisfy the equation a^{2} + b^{2} = c^{2}, so line segments with these lengths form a
right triangle (by the converse of the
Pythagorean theorem). The ratio of the side length of the hexagon to the decagon is the golden ratio, so this triangle forms half of a golden rectangle.^{
[8]}
The
convex hull of two opposite edges of a regular
icosahedron forms a golden rectangle. The twelve vertices of the icosahedron can be decomposed in this way into three mutually-perpendicular golden rectangles, whose boundaries are linked in the pattern of the
Borromean rings.^{
[9]}
See also
Fibonacci number – Numbers obtained by adding the two previous onesPages displaying short descriptions of redirect targets
Golden angle -- Circle with sectors in golden ratio
Golden rhombus – Rhombus with diagonals in the golden ratio
Kepler triangle – Right triangle related to the golden ratio
^Olsen, Scott (2006). The Golden Section: Nature's Greatest Secret. Glastonbury: Wooden Books. p.
3.
ISBN978-1-904263-47-0.
^Van Mersbergen, Audrey M., Rhetorical Prototypes in Architecture: Measuring the Acropolis with a Philosophical Polemic, Communication Quarterly, Vol. 46, 1998 ("a 'Golden Rectangle' has a ratio of the length of its sides equal to 1:1.61803+. The Parthenon is of these dimensions.")
^Le Corbusier, The Modulor, p. 35, as cited in Padovan, Richard, Proportion: Science, Philosophy, Architecture (1999), p. 320. Taylor & Francis.
ISBN0-419-22780-6: "Both the paintings and the architectural designs make use of the golden section".