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 n-dimensional generalization of Hill tetrahedra:
where vectors satisfy for all, and where. Hadwiger showed that all such simplices are scissor congruent to a hypercube.
