A Dissection of a Prolate Golden Rhombohedron
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This Demonstration shows a dissection of a prolate golden rhombohedron into smaller golden rhombohedra.
Contributed by: Izidor Hafner and Anja Komatar (April 2008)
Open content licensed under CC BY-NC-SA
"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." . 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, , , , , , , , … . 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:  W. W. Rouse Ball, H. S. M. Coxeter, Mathematical Recreations and Essays, 13th ed., New York: Dover Publications, 1987 p. 161.  M. Gardner, Aha! Gotcha: Paradoxes to Puzzle and Delight, San Francisco: W. H. Freeman, 1982 (Slovenian edition cited, 1992 p. 94).  I. Hafner, T. Zitko, "Introduction to Golden Rhombic Polyhedra," Visual Mathematics, 4(2), 2002.