Hill tetrahedron

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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]

Contents

Construction

For every , let be three unit vectors with angle between every two of them. Define the Hill tetrahedron as follows:

A special case is the tetrahedron having all sides right triangles, two with sides and two with sides . Ludwig Schläfli studied as a special case of the orthoscheme, and H. S. M. Coxeter called it the characteristic tetrahedron of the cubic spacefilling. [3]

Properties

Generalizations

In 1951, Hugo Hadwiger found the following -dimensional generalization of Hill tetrahedra: where vectors satisfy for all , and where . 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. 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.