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 [math]\displaystyle{ \alpha \in (0,2\pi/3) }[/math], let [math]\displaystyle{ v_1,v_2,v_3 \in \mathbb R^3 }[/math] be three unit vectors with angle [math]\displaystyle{ \alpha }[/math] between every two of them. Define the Hill tetrahedron [math]\displaystyle{ Q(\alpha) }[/math] as follows:
- [math]\displaystyle{ 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\}. }[/math]
A special case [math]\displaystyle{ Q=Q(\pi/2) }[/math] is the tetrahedron having all sides right triangles, two with sides [math]\displaystyle{ (1,1,\sqrt{2}) }[/math] and two with sides [math]\displaystyle{ (1,\sqrt{2},\sqrt{3}) }[/math]. Ludwig Schläfli studied [math]\displaystyle{ Q }[/math] 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 [math]\displaystyle{ Q }[/math].[1]
- Every [math]\displaystyle{ Q(\alpha) }[/math] 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:
- [math]\displaystyle{ Q(w) \, = \, \{c_1 v_1+\cdots +c_n v_n \mid 0 \le c_1 \le \cdots \le c_n \le 1\}, }[/math]
where vectors [math]\displaystyle{ v_1,\ldots,v_n }[/math] satisfy [math]\displaystyle{ (v_i,v_j) = w }[/math] for all [math]\displaystyle{ 1\le i\lt j\le n }[/math], and where [math]\displaystyle{ -1/(n-1)\lt w \lt 1 }[/math]. 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
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