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.

Construction
For every $$\alpha \in (0,2\pi/3)$$, let $$v_1,v_2,v_3 \in \mathbb R^3$$ 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 $$Q$$.
 * 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.