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:
:$Q(\alpha) \, = \, \.$
A special case $Q=Q(\pi/2)$ is the tetrahedron having all sides right triangles, two with sides $(1,1,\sqrt)$ and two with sides $(1,\sqrt,\sqrt)$. 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:

: $Q(w) \, = \, \,$

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.

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

Polyhedra
Space-filling polyhedra