In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the Hill tetrahedra are a family of
space-filling tetrahedra
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
. They were discovered in 1896 by
M. J. M. Hill, a professor of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
at the
University College London
, mottoeng = Let all come who by merit deserve the most reward
, established =
, type = Public research university
, endowment = £143 million (2020)
, budget = ...
, who showed that they are
scissor-congruent to a
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. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
.
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, two with sides
and two with sides
.
Ludwig Schläfli
Ludwig Schläfli (15 January 1814 – 20 March 1895) was a Swiss mathematician, specialising in geometry and complex analysis (at the time called function theory) who was one of the key figures in developing the notion of higher-dimensional space ...
studied
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
.
* Every
can be
dissected
Dissection (from Latin ' "to cut to pieces"; also called anatomization) is the dismembering of the body of a deceased animal or plant to study its anatomical structure. Autopsy is used in pathology and forensic medicine to determine the cause ...
into three polytopes which can be reassembled into a
prism
Prism usually refers to:
* Prism (optics), a transparent optical component with flat surfaces that refract light
* Prism (geometry), a kind of polyhedron
Prism may also refer to:
Science and mathematics
* Prism (geology), a type of sedimentary ...
.
Generalizations
In 1951
Hugo Hadwiger
Hugo Hadwiger (23 December 1908 in Karlsruhe, Germany – 29 October 1981 in Bern, Switzerland) was a Swiss mathematician, known for his work in geometry, combinatorics, and cryptography.
Biography
Although born in Karlsruhe, Germany, Hadwi ...
found the following ''n''-dimensional generalization of Hill tetrahedra:
:
where vectors
satisfy
for all
, and where
. Hadwiger showed that all such
simplices
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
are scissor congruent to a
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, ...
.
References
* M. J. M. Hill, Determination of the volumes of certain species of tetrahedra without employment of the method of limits, ''Proc. London Math. Soc.'', 27 (1895–1896), 39–53.
*
H. Hadwiger, Hillsche Hypertetraeder, ''Gazeta Matemática (Lisboa)'', 12 (No. 50, 1951), 47–48.
*
H.S.M. CoxeterFrieze patterns ''Acta Arithmetica'' 18 (1971), 297–310.
* E. Hertel, Zwei Kennzeichnungen der Hillschen Tetraeder, ''J. Geom.'' 71 (2001), no. 1–2, 68–77.
* Greg N. Frederickson, ''Dissections: Plane and Fancy'', Cambridge University Press, 2003.
*
N.J.A. Sloane, V.A. Vaishampayan, ''Generalizations of Schobi’s Tetrahedral Dissection'', {{ArXiv, 0710.3857.
External links
Three piece dissection of a Hill tetrahedron into a triangular prismSpace-Filling Tetrahedra
Polyhedra
Space-filling polyhedra