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.
