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.

α ∈ ( 0 , 2 π

be three unit vectors with angle

α

Define the Hill tetrahedron

( α )

as follows: A special case

( π

is the tetrahedron having all sides right triangles, 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.

In 1951 Hugo Hadwiger found the following n-dimensional generalization of Hill tetrahedra: where vectors

Hadwiger showed that all such simplices are scissor congruent to a hypercube.