geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the Hill tetrahedra are a family of space-filling

tetrahedra In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

. They were discovered in 1896 by M. J. M. Hill, a professor of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

at the

University College London University College London (Trade name, branded as UCL) is a Public university, public research university in London, England. It is a Member institutions of the University of London, member institution of the Federal university, federal Uni ...

, who showed that they are scissor-congruent to a

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

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) \, = \, \.

A special case

Q=Q(\pi/2)

is the tetrahedron having all sides right triangles, two with sides

(1,1,\sqrt)

and two with sides

(1,\sqrt,\sqrt)

. 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 PRISM is a code name for a program under which the United States National Security Agency (NSA) collects internet communications from various U.S. internet companies. The program is also known by the SIGAD . PRISM collects stored internet ...

Generalizations

In 1951

Hugo Hadwiger Hugo Hadwiger (23 December 1908 in Karlsruhe, Germany – 29 October 1981 in Bern, Switzerland) was a Swiss people, Swiss mathematician, known for his work in geometry, combinatorics, and cryptography. Biography Although born in Karlsruhe, Ge ...

found the following ''n''-dimensional generalization of Hill tetrahedra: :

Q(w) \, = \, \,

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 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 ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...

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. Coxeter
Frieze 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 prism

Space-Filling Tetrahedra
Tetrahedra Space-filling polyhedra