Scalenohedron

A di-

n

-gonal scalenohedron is bounded by

2n

mutually congruent scalene triangles. As a convex polyhedron with

n=2

and

n=3

, it occurs as a crystal form of certain minerals.

Details

Consider a regular polygon with

2n

sides in a horizontal plane. Raise and lower its alternate vertices to get a regular skew polygon. Take its (vertical) axis of rotation, and, for some

h

, choose a pair of points on this axis at heights

h

and

-h

. Take each of those points as an apex connected to the vertices of the regular skew polygon by

4n

edges. These edges, together with the sides of the skew polygon, form the

4n

triangular faces of a general di-

n

-gonal scalenohedron. A general di-

n

-gonal scalenohedron is not necessarily a convex polyhedron. By moving the contols, you can see that in special cases bipyramids, streptohedra, rhombohedra, tetrahedra, and even the cube appear. In nature, the ditrigonal scalenohedron occurs as a well-known crystal form of calcspar (calcite). For

n=2

, the name tetragonal scalenohedron is preferred for "di-digonal scalenohedron" (a crystal form of chalcopyrite, the most frequent ore of copper).

External Links

Scalenohedron (Wolfram MathWorld)

Permanent Citation

Sándor Kabai, Gábor Gévay, Lajos Szilassi

"Scalenohedron"
http://demonstrations.wolfram.com/Scalenohedron/
Wolfram Demonstrations Project
Published: July 24, 2008