"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 [
online],
4(2), 2002.
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