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 $Q$ .^[1]
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.

References

^ "Space-Filling Tetrahedra - Wolfram Demonstrations Project".

M. J. M. Hill, Determination of the volumes of certain species of tetrahedra without employment of the method of limits, Proc. London Math. Soc., 27 (1895–1896), 39–53.
H. Hadwiger, Hillsche Hypertetraeder, Gazeta Matemática (Lisboa), 12 (No. 50, 1951), 47–48.
H.S.M. Coxeter, Frieze patterns, Acta Arithmetica 18 (1971), 297–310.
E. Hertel, Zwei Kennzeichnungen der Hillschen Tetraeder, J. Geom. 71 (2001), no. 1–2, 68–77.
Greg N. Frederickson, Dissections: Plane and Fancy, Cambridge University Press, 2003.
N.J.A. Sloane, V.A. Vaishampayan, Generalizations of Schobi’s Tetrahedral Dissection, arXiv:0710.3857.

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