Hill tetrahedron

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

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[edit]

  • 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[edit]

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[edit]

References[edit]

  • 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[edit]