Hill tetrahedron explained

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 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, two with sides ( 1 , 1 ,\sqrt{ 2 }) and two with sides ( 1 ,\sqrt{ 2 },\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

• A cube can be tiled with six copies of

Q .^[1]

• 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) = \{c _{1} v _{1+ …} + c _{n} v _{n} \mid 0 \le c _{1} \le … \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.

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, .

External links

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

Notes and References

1. Web site: Space-Filling Tetrahedra - Wolfram Demonstrations Project.