the nets of the non rectangular parallelepipeds

The opposite faces of these cobblestones are identical parallelograms. The twelve edges are divided into three groups of four edges parallel and of same length.
On each of the eight vertices the sum of the three angles must be less than 360°; if this condition is not satisfied faces overlap on the drawing here with and the parallelepiped doesn't exist !

We get these hexahedra by stretching or compressing a rectangular parallelepiped along one of its diagonals (so we get the rhombohedra starting from a cube).

There exists two kinds of non rectangular parallelepipeds:
one can always find two opposite vertices, one of which is represented by the yellow point on the two drawings below, where the three angles are either acute or obtuse.

Three acute angles on two opposite vertices (a, b, c around the yellow point); on each of the six other vertices only one angle is acute.
The condition of existence is   | b-c | < a < b+c 		Three obtuse angles on two opposite vertices (a, b, c around the yellow point); on each of the six other vertices only one angle is obtuse.
The condition of existence is simply   a+b+c < 360°

You may move the red points with your mouse to change the shape of the net.


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French 		convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects April 2002
updated 01-07-2002