Disphenoid
A disphenoid (also isosceles tetrahedron ) is a polyhedron with four congruent triangles as side faces . A disphenoid consists of two sphenoids (too ancient Greek σφήν "wedge"), these are open forms with two faces each ( dihedral ).
The term "isosceles tetrahedron" needs some explanation: A disphenoid is a tetrahedron in the general sense of the word, not necessarily a tetrahedron in the sense of the Platonic field of the same name . The adjective "isosceles" does not refer to its triangular surfaces, but to the property of the body that of its six edges, the opposite edges are the same length.
Characterization sets
According to Bang's theorem, a disphenoid is a three-dimensional simplex with one of the following equivalent characterizations:
- The opposite (unconnected) edges are the same length.
- The 4 triangles are congruent.
- The 4 triangles have the same perimeter.
- The 4 triangles have the same area.
Another set of characterization is the following:
- A tetrahedron is a disphenoid if and only if the incugel and the umkugel are concentric .
In full generality, the following characterization theorem applies:
- A tetrahedron is isosceles if and only if of the four points:
- at least two coincide. In this case, all four points even coincide.
Note:
The triangles all have the same orientation .
Special cases
If one of the triangles (and thus all of them) is isosceles , one speaks of a tetragonal disphenoid . Then 4 edges of the disphenoid are the same length and the other 2 are skewed perpendicular to each other.
If the sides of the triangle are different, the disphenoid is called rhombic .
(These terms come from crystallography .)
If a triangle (and therefore all of them) is equilateral , then the disphenoid is a regular tetrahedron .
Computation of any disphenoid
A disphenoid is determined by one of the 4 congruent triangles. Since a triangle is determined by 3 independent information about the size of its sides and / or angles, a disphenoid is also determined by 3 independent information.
Examples
Disphenoids occur in nature as a crystal form: They are the general planar form of crystal classes 222 (rhombic-disphenoidal) and 4 (tetragonal-disphenoidal class).
Dieder (sphenoid)
rhombic disphenoid with three unequal axes A, B, C
tetragonal disphenoid
Special case: regular tetrahedron with six equal edges
Web links
literature
- Nathan Altshiller-Court: Modern Pure Solid Geometry . 2nd Edition. Chelsea Publishing Company, Bronx, NY 1964, ISBN 0-8284-0147-0 .
- Adolf Schmidt, The Equilateral Tetrahedron , Journal for Mathematics and Physics XXIX, pp. 321–343. Teubner, Leipzig (1884).
Individual evidence
- ↑ Eric W. Weisstein : Isosceles Tetrahedron . In: MathWorld (English).
- ^ Ross Honsberger: Mathematical jewels. Vieweg Verlag, 1982, ISBN 3-528-08475-8 , p. 82.
- ^ A b N. Altshiller-Court: Modern Pure Solid Geometry . 1964, p. 105-108 .