Hill tetrahedron

In geometry, the Hill tetrahedra are a family of space-filling tetrahedra. They were discovered in 1896 by M. J. M. Hill, a professor of mathematics at the University College London, who showed that they are scissor-congruent to a cube.

Construction

For every , let be three unit vectors with angle between every two of them. Define the Hill tetrahedron as follows:

A special case is the tetrahedron having all sides right triangles, two with sides and two with sides . Ludwig Schläfli studied as a special case of the orthoscheme, and H. S. M. Coxeter called it the characteristic tetrahedron of the cubic spacefilling.

Properties

• A cube can be tiled with six copies of .
• Every can be dissected into three polytopes which can be reassembled into a prism.

Generalizations

In 1951, Hugo Hadwiger found the following dimensional generalization of Hill tetrahedra:

where vectors satisfy for all , and where . Hadwiger showed that all such simplices are scissor congruent to a hypercube.

References

External links

• Three piece dissection of a Hill tetrahedron into a triangular prism
• Space-Filling Tetrahedra