Disphenoid

A Disphenoid (also isosceles tetrahedra ) is a polyhedron with four congruent triangles as faces. A Disphenoid consists of two Sphenoiden ( to Greek σφήν " wedge "), which are open forms, each with two faces ( dihedral ).

Characterization sets

By the theorem of Bang a Disphenoid is a three-dimensional simplex with one of the following equivalent characterizations:

• Each pair of opposite ( unconnected ) edges have the same length.
• The four triangles are congruent.
• The four triangles have the same perimeter.
• The four triangles have the same area.

One other characterization theorem is the following:

In full generality even following characterization theorem:

Comment: The triangles all have the same orientation.

Special cases

Is one of the triangles (and thus all ) isosceles, then one speaks of a tetragonal Disphenoid. Then four edges of the Disphenoids are equally long and the other 2 are skew orthogonal.

Are the sides of the triangle are different, the Disphenoid is called rhombic.

( These conceptions come from crystallography. )

Is a triangle (and all ) equal each other, then the Disphenoid is a regular tetrahedron.

Calculation of any Disphenoids

A Disphenoid is determined by one of the four congruent triangles. Since a triangle is determined by three independent information about the size of its sides and / or angle, a Disphenoid independent information is also determined by 3 from each other.

Examples

Disphenoide occur in nature as a crystal form: they are the general surface shape of the crystal classes 222 ( rhombic- disphenoidische ) and 4 (tetragonal - disphenoidische class).

Rhombic with three unequal axes Disphenoid A, B, C

Tetragonal Disphenoid

Special case: regular tetrahedron with six equal edges