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 \alpha between every two of them. Define the Hill tetrahedron Q(\alpha) as follows: A special case Q=Q(\pi/2) is the tetrahedron having all sides right triangles, two with sides and two with sides. Ludwig Schläfli studied Q as a special case of the orthoscheme, and H. S. M. Coxeter called it the characteristic tetrahedron of the cubic spacefilling.

Properties

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.