Hill tetrahedron
Encyclopedia
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
.
, let 
be three unit vectors with angle
between every two of them.
Define the Hill tetrahedron
as follows:
A special case
is the tetrahedron having all sides right triangles with sides 1, 
and 
. Ludwig Schläfli
studied
as a special case of the orthoscheme
, and H.S.M. Coxeter called it the characteristic tetrahedron of the cubic spacefilling.
found the following n-dimensional generalization of Hill tetrahedra:
where vectors
satisfy 
for all 
, and where 
. Hadwiger showed that all such simplices
are scissor congruent to a hypercube
.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, the Hill tetrahedra are a family of space-filling tetrahedra
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
. They were discovered in 1896 by M.J.M. Hill, a professor of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
at the University College London
University College London
University College London is a public research university located in London, United Kingdom and the oldest and largest constituent college of the federal University of London...
, who showed that they are scissor-congruent
Hilbert's third problem
The third on Hilbert's list of mathematical problems, presented in 1900, is the easiest one. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the...
to a cube
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
.
Construction
For every, let be three unit vectors with angle
between every two of them.Define the Hill tetrahedron
as follows:A special case
is the tetrahedron having all sides right triangles with sides 1, and . Ludwig SchläfliLudwig Schläfli
Ludwig Schläfli was a Swiss geometer and complex analyst who was one of the key figures in developing the notion of higher dimensional spaces. The concept of multidimensionality has since come to play a pivotal role in physics, and is a common element in science fiction...
studied
as a special case of the orthoschemeSchläfli orthoscheme
In geometry, Schläfli orthoscheme is a type of simplex. They are defined by a sequence of edges , , \dots, \, that are mutually orthogonal. These were introduced by Ludwig Schläfli, who called them orthoschemes and studied their volume in the Euclidean, Lobachevsky and the spherical geometry. ...
, and H.S.M. Coxeter called it the characteristic tetrahedron of the cubic spacefilling.
Properties
- A cube can be tiled with 6 copies of
. - Every
can be dissectedDissection (geometry)In geometry, a dissection problem is the problem of partitioning a geometric figure into smaller pieces that may be rearranged into a new figure of equal content. In this context, the partitioning is called simply a dissection...
into three polytopes which can be reassembled into a prismPrism (geometry)In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...
.
Generalizations
In 1951 Hugo HadwigerHugo Hadwiger
Hugo Hadwiger was a Swiss mathematician, known for his work in geometry, combinatorics, and cryptography.-Biography:...
found the following n-dimensional generalization of Hill tetrahedra:
where vectors
satisfy for all , and where . Hadwiger showed that all such simplicesSimplex
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
are scissor congruent to a hypercube
Hypercube
In geometry, a hypercube is an n-dimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.An...
.