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 $v_1,v_2,v_3 \in \Bbb R^3$ 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 with sides 1, $\sqrt{2}$ and $\sqrt{3}$. 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 6 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 $v_1,\ldots,v_n$ 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.

See also

• Schläfli orthoscheme

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