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 $v_1,v_2,v_3\in {ℝ}^3$ be three unit vectors with angle $\alpha$ between every two of them. Define the Hill tetrahedron $Q(\alpha )$ as follows:

$Q(\alpha )=\{c_1{v}_1+c_2{v}_2+c_3{v}_3\mid 0\le c_1\le c_2\le c_3\le 1\}.$

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:

$Q(w)=\{c_1{v}_1+\cdots +c_n{v}_n\mid 0\le c_1\le \cdots \le c_n\le 1\},$

where vectors $v_1,\dots ,v_n$ satisfy $(v_i,v_j)=w$ for all $1\le i<j\le n$, 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, Template:ArXiv.

External links

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