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]

References

1. Hill, M. J. M. (1895). "Determination of the volumes of certain species of tetrahedra without employment of the method of limits". Proceedings of the London Mathematical Society. 27: 39–53.
2. Sloane, N. J. A.; Vaishampayan, Vinay A. (2009). "Generalizations of Schöbi's Tetrahedral Dissection". Discrete & Computational Geometry. 41 (2): 232–248. arXiv:0710.3857. doi:10.1007/s00454-008-9086-6.
3. Coxeter, H.S.M. (1971). "Frieze patterns" (PDF). Acta Arithmetica. 18: 297–310. doi:10.4064/aa-18-1-297-310.
4. "Space-Filling Tetrahedra - Wolfram Demonstrations Project".
5. Hadwiger, Hugo (1951). "Hillsche Hypertetraeder". Gazeta Matemática (Lisboa). 12 (50): 47–48.

• Hertel, Eike (2001). "Zwei Kennzeichnungen der Hillschen Tetraeder". Journal of Geometry. 71 (1–2): 68–77. doi:10.1007/s00022-001-8553-5.
• Frederickson, Greg N. (2003). Dissections: Plane and Fancy. Cambridge University Press. pp. 235–236. ISBN 978-0-521-52582-4.

External links

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