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.

References

• 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.

External links

• Three piece dissection of a Hill tetrahedron into a triangular prism
• Space-Filling Tetrahedra