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.
One rectangle and one cuboid are called “golden”, but there are another rectangle and another cuboid that are equally based on the Golden Ratio.

Golden Rectangle?

How about this one?
A (dynamic) rectangle with ratio 1 : square root of Phi and with diagonal Phi is not known as a “Golden Rectangle”, but it is quite "golden".
It divides – in a fractal way – by Phi in itself (major) and itself in 90 degrees rotation plus itself (minor).
See also: Ammann Chair.
Diagonal division makes a “Kepler triangle”, of which the edges are in geometrical progression, 1 : sqr(Phi) : Phi.
The Kepler triangle is sometimes called “golden triangle”, though there is another triangle that is more often called "golden triangle".

Golden Rectangle

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 Cuboid?

Another rectangular cuboid, with ratios 1 : sqr(Phi) : Phi, is not known as a “Golden Cuboid”, but it is quite "golden".
Two opposite faces are Phi rectangles (dark), two opposite faces are big sqr(Phi) rectangles, and two opposite faces are small sqr(Phi) rectangles (the yellow rectangles have equal ratio 1 : sqr(Phi) = sqr(Phi) : Phi). The (green) plane between the diagonals of the small rectangles is a square.

Golden Cuboid

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

Jan Maris, august 2009

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

Three well-known not-golden rectangles:

square

1 : 1
square

rectangle 1 : 2

1 : 2

1 : sqr(2)

diagonal plane of a cube
a dynamic rectangle,
first described by
G. C. Lichtenberg
ISO 216 (A4 paper size)

Golden Ratio

A line divided by the “Golden Ratio”
1 : 0.5(sqrt(5)-1)+1
(= 1.61803399... = Φ = Phi)
has ratios 1 : major = major : minor.

Golden Pyramid

A pyramid of which the apothem is Phi times half the base width (Kepler triangle) is sometimes called a “golden pyramid”.

Mark Reynolds: “The square root of the golden section, root-phi, is at times not given its due in discussions regarding the golden section. This is an unfortunate situation because its geometric properties provide us with a unique and rarely applied design tool.”