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,^[1] who showed that they are scissor-congruent to a cube.^[2]

Construction

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

A special case ${\displaystyle Q=Q(\pi /2)}$ is the tetrahedron having all sides right triangles, two with sides ${\displaystyle (1,1,{\sqrt {2}})}$ and two with sides ${\displaystyle (1,{\sqrt {2}},{\sqrt {3}})}$. Ludwig Schläfli studied ${\displaystyle Q}$ as a special case of the orthoscheme, and H. S. M. Coxeter called it the characteristic tetrahedron of the cubic spacefilling.^[3]

Properties

• A cube can be tiled with six copies of ${\displaystyle Q}$.^[4]
• Every ${\displaystyle Q(\alpha )}$ can be dissected into three polytopes which can be reassembled into a prism.

Generalizations

In 1951, Hugo Hadwiger found the following ${\displaystyle n}$-dimensional generalization of Hill tetrahedra: $${\displaystyle Q(w)=\{c_{1}v_{1}+\cdots +c_{n}v_{n}\mid 0\leq c_{1}\leq \cdots \leq c_{n}\leq 1\},}$$ where vectors ${\displaystyle v_{1},\ldots ,v_{n}}$ satisfy ${\displaystyle (v_{i},v_{j})=w}$ for all ${\displaystyle 1\leq i<j\leq n}$, and where ${\displaystyle -1/(n-1)<w<1}$. Hadwiger showed that all such simplices are scissor congruent to a hypercube.^[5]