The other Golden Rectangle and Golden Cuboid

In mathematics the word “golden” is used for several shapes and forms that are based on the Golden Ratio, for instance one specific rectangle and one specific cuboid. But another rectangle and another cuboid are also based on the Golden Ratio.

Golden Rectangle 1

A (dynamic) rectangle with the ratio of 1 : Phi is known as “Golden Rectangle”.
It divides by Phi in a square (major) and itself in a quarter turn (minor).
It divides in a fractal way in combination with the square.
This 1 : Phi rectangle is often used as an aesthetical standard.

Golden Rectangle 2

A (dynamic) rectangle with ratio 1 : square root of Phi and with diagonal Phi is not known as a “Golden Rectangle”.
It can be divided by Phi such that the major part is another golden rectangle and the minor part are two golden rectangles. This can be repeated ad infinitum (fractalic): every next rectangle is smaller and rotated 90 degrees.
See also: Ammann Chair.
Diagonal division makes a “Kepler triangle” (and here), of which the edges are in geometrical progression, 1 : √Phi : Phi. The Kepler triangle is sometimes called “golden triangle”, but there is also another golden triangle, sometimes called “sublime triangle”.

Golden Cuboid 1

A cuboid with ratio 1 : Phi : Phi² is sometimes called a “Golden Cuboid”, e.g. listening room for audiophiles. Four opposite faces have ratio 1 : Phi, and two opposite faces have ratio 1 : Phi²

Golden Cuboid 2

Another rectangular cuboid, with ratios 1 : √Phi : Phi, not known as a “Golden Cuboid”.
Two opposite faces are Phi rectangles (dark), two opposite faces are big √Phi rectangles, and two opposite faces are small √Phi rectangles.
The yellow rectangles have equal ratio 1 : √Phi = √Phi : Phi.
The diagonal (green) plane is a square.


3D wiggle stereo


in three different positions

Jan Maris, august 2009

(august 2009 basic content. lay-out changes, small corrections and additions later)