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.
For every α ∈ ( 0 , 2 π / 3 ) , let v 1 , v 2 , v 3 ∈ R 3 be three unit vectors with angle α between every two of them. Define the Hill tetrahedron Q ( α ) as follows:
Q ( α ) = { c 1 v 1 + c 2 v 2 + c 3 v 3 ∣ 0 ≤ c 1 ≤ c 2 ≤ c 3 ≤ 1 } . A special case Q = Q ( π / 2 ) is the tetrahedron having all sides right triangles, two with sides (1,1, 2 ) and two with sides (1, 2 and 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.
A cube can be tiled with 6 copies of Q .Every Q ( α ) can be dissected into three polytopes which can be reassembled into a prism.In 1951 Hugo Hadwiger found the following n-dimensional generalization of Hill tetrahedra:
Q ( w ) = { c 1 v 1 + ⋯ + c n v n ∣ 0 ≤ c 1 ≤ ⋯ ≤ c n ≤ 1 } , where vectors v 1 , … , v n satisfy ( v i , v j ) = w for all 1 ≤ i < j ≤ n , and where − 1 / ( n − 1 ) < w < 1 . Hadwiger showed that all such simplices are scissor congruent to a hypercube.
