Hill tetrahedron
# 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 $\alpha \in (0,2\pi /3)$, let $v_{1},v_{2},v_{3}\in \mathbb {R} ^{3}$ be three unit vectors with angle $\alpha$ between every two of them. Define the Hill tetrahedron $Q(\alpha )$ as follows:

    $Q(\alpha )\,=\,\{c_{1}v_{1}+c_{2}v_{2}+c_{3}v_{3}\mid 0\leq c_{1}\leq c_{2}\leq c_{3}\leq 1\}.$

A special case $Q=Q(\pi /2)$ is the tetrahedron having all sides right triangles, two with sides $(1,1,{\sqrt {2}})$ and two with sides $(1,{\sqrt {2}},{\sqrt {3}})$. 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

* A cube can be tiled with six copies of $Q$.
* Every $Q(\alpha )$ 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:

    $Q(w)\,=\,\{c_{1}v_{1}+\cdots +c_{n}v_{n}\mid 0\leq c_{1}\leq \cdots \leq c_{n}\leq 1\},$

where vectors $v_{1},\ldots ,v_{n}$ satisfy $(v_{i},v_{j})=w$ for all $1\leq i<j\leq n$, and where $-1/(n-1)<w<1$. Hadwiger showed that all such simplices are scissor congruent to a hypercube.

