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 ...
, a trigonal trapezohedron is a
rhombohedron (a
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on th ...
with six
rhombus-shaped
faces) in which, additionally, all six faces are
congruent. Alternative names for the same shape are the ''trigonal deltohedron''
or ''
isohedral rhombohedron''.
Some sources just call them ''rhombohedra''.
Geometry
Six identical rhombic faces can construct two configurations of trigonal trapezohedra. The ''acute'' or ''prolate'' form has three acute angle corners of the rhombic faces meeting at the two polar axis vertices. The ''obtuse'' or ''oblate'' or ''flat'' form has three obtuse angle corners of the rhombic faces meeting at the two polar axis vertices.
More strongly than having all faces congruent, the trigonal trapezohedra are
isohedral figures, meaning that they have symmetries that take any face to any other face.
[
]
Special cases
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 ...
can be interpreted as a special case of a trigonal trapezohedron, with square rather than rhombic faces.
The two golden rhombohedra are the acute and obtuse form of the trigonal trapezohedron with golden rhombus faces.
Copies of these can be assembled to form other convex polyhedra with golden rhombus faces, including the Bilinski dodecahedron and rhombic triacontahedron.
Four oblate rhombohedra whose ratio of face diagonal lengths are the square root of two can be assembled to form a rhombic dodecahedron. The same rhombohedra also tile space in the trigonal trapezohedral honeycomb.
Related polyhedra
The trigonal trapezohedra are special cases of trapezohedra, polyhedra with an even number of congruent kite-shaped faces. When this number of faces is six, the kites degenerate to rhombi, and the result is a trigonal trapezohedron. As with the rhombohedra more generally, the trigonal trapezohedra are also special cases of parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidea ...
s, and are the only parallelepipeds with six congruent faces. Parallelepipeds are zonohedra
In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in ...
, and Evgraf Fedorov
Evgraf Stepanovich Fedorov (russian: Евгра́ф Степа́нович Фёдоров, – 21 May 1919) was a Russian mathematician, crystallographer and mineralogist.
Fedorov was born in the Russian city of Orenburg. His father was a topo ...
proved that the trigonal trapezohedra are the only infinite family of zonohedra whose faces are all congruent rhombi.[
]Dürer's solid
In geometry, the truncated triangular trapezohedron is the first in an infinite series of truncated trapezohedra. It has 6 pentagon and 2 triangle faces.
Geometry
This polyhedron can be constructed by Truncation (geometry), truncating two oppos ...
is generally presumed to be a truncated triangular trapezohedron
In geometry, the truncated triangular trapezohedron is the first in an infinite series of truncated trapezohedra. It has 6 pentagon and 2 triangle faces.
Geometry
This polyhedron can be constructed by truncating two opposite vertices of a cu ...
, a trigonal trapezohedron with two opposite vertices truncated, although its precise shape is still a matter for debate.[
]
See also
* Truncated triangular trapezohedron
In geometry, the truncated triangular trapezohedron is the first in an infinite series of truncated trapezohedra. It has 6 pentagon and 2 triangle faces.
Geometry
This polyhedron can be constructed by truncating two opposite vertices of a cu ...
References
External links
*
Space-filling polyhedra
Zonohedra
Polyhedra
{{Polyhedron-stub