Rhombohedron with Variable Faces

tilt

x

create five

rotate 1

You can create various rhombohedra by tilting the four parallel edges of a cube or varying the face diagonals. This Demonstration focuses on the special case where the rhombohedron has four faces with diagonal ratio

2

and two faces with diagonal ratio

2

ϕ

, where

ϕ

is the golden ratio. Let the edge of a cube be

3

. Tilt four parallel edges by

arctan1

2

ϕ



degrees, then reduce the diagonal of two faces to

2/ϕ

. One interesting feature of this polyhedron is that the distance between the two thin rhombi is the golden ratio. The half-diagonals of the thin rhombus are

ϕ

and

1/ϕ

. This illustrates an interesting relationship:

2

ϕ

+1

2

ϕ

=

3

, that is,

2

ϕ

+1

2

ϕ

=3

.