Hill tetrahedron
Encyclopedia
In geometry
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 Schläfli
Ludwig 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 orthoscheme
Schläfli 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 dissected
    Dissection (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 prism
    Prism (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 Hadwiger
Hugo 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 simplices
Simplex
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...

.

External links

The source of this article is wikipedia, the free encyclopedia.  The text of this article is licensed under the GFDL.
 
x
OK