"A rhombohedron is a parallelepiped bounded by six congruent rhombs. It has two opposite vertices at which the three face-angles are equal; it is said to be acute or obtuse according to the nature of these angles. A golden rhombohedron has faces whose diagonals are in the golden ratio

:1." [1]. The volumes of A6 (the acute rhombohedron) and O6 (the obtuse rhombohedron) are also in the ratio

:1.

The Fibonacci sequence 1,

, 1+

, 1+2

, 2+3

, 3+5

, 5+8

, 8+13

, … is equal to the geometric sequence 1,

,

,

,

,

,

,

, … [2]. Taking the obtuse rhombohedron of volume 1, the obtuse rhombohedron whose edges are

as long has volume

. So there exists a dissection of

O6 to one O6 and two A6.

On the other hand, the volume of

A6 is

, so there exists a dissection of the solid to two O6 and three A6. But

so there is a dissection of

A6 to

O6, O6, and A6.

Finally, it is possible to dissect O6 to 1/

O6 and two 1/

A6. This Demonstration illustrates a dissection of

A6 to

O6, A6, 1/

O6, and two 1/

A6. There is a three-piece dissection of a parallelepiped to A6. There is a four-piece dissection of the bottom parallelepiped to

O6. The blue and the red parallelepipeds are 1/

A6. The parallelepiped with the magenta part is similar to the bottom parallelepiped, so there exists a four-piece dissection of the parallelepiped to 1/

O6.

References:

[1] W. W. Rouse Ball, H. S. M. Coxeter,

*Mathematical Recreations and Essays*, 13th ed., New York: Dover Publications, 1987 p. 161.

[2] M. Gardner,

*Aha! Gotcha: Paradoxes to Puzzle and Delight*, San Francisco: W. H. Freeman, 1982 (Slovenian edition cited, 1992 p. 94).

[3] I. Hafner, T. Zitko, "Introduction to Golden Rhombic Polyhedra,"

*Visual Mathematics*,

**4**(2), 2002.