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# Hill tetrahedron

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 Title: Hill tetrahedron Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### 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.

## Contents

• Construction 1
• Properties 2
• Generalizations 3
• References 5

## Construction

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

• 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

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.