240px, A tetradecahedron with D2d-symmetry, existing in the Weaire–Phelan structure
A tetradecahedron is a
polyhedron
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...
with 14
faces. There are numerous topologically distinct forms of a tetradecahedron, with many constructible entirely with
regular polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...
faces.
A tetradecahedron is sometimes called a tetrakaidecahedron. No difference in meaning is ascribed. The Greek word ''
kai'' means 'and'. There is evidence that mammalian
epidermal cells are shaped like flattened tetrakaidecahedra, an idea first suggested by
Lord Kelvin
William Thomson, 1st Baron Kelvin (26 June 182417 December 1907), was a British mathematician, Mathematical physics, mathematical physicist and engineer. Born in Belfast, he was the Professor of Natural Philosophy (Glasgow), professor of Natur ...
. The polyhedron can also be found in soap bubbles and in
sintered ceramics, due to its ability to
tesselate
A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensiona ...
in 3D space.
Convex
There are 1,496,225,352 topologically distinct ''convex'' tetradecahedra, excluding mirror images, having at least 9 vertices. (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.)
Examples
An incomplete list of forms includes:
Tetradecahedra having all
regular polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...
al faces (all exist in irregular-faced forms as well):
*
Archimedean solid
The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygon and are vertex-transitive, although they aren't face-transitive. The solids were named after Archimedes, although he did not claim credit for them. They ...
s:
**
Cuboctahedron
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertex (geometry), vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edge (geometry), edges, each separating a tr ...
(8
triangles
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensiona ...
, 6
squares
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
)
**
Truncated cube
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangle (geometry), triangular), 36 edges, and 24 vertices.
If the truncated cube has unit edge length, its dual triak ...
(8 triangles, 6
octagon
In geometry, an octagon () is an eight-sided polygon or 8-gon.
A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, which alternates two types of edges. A truncated octagon, t is a ...
s)
**
Truncated octahedron
In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...
(6 squares, 8
hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A regular hexagon is de ...
s)
*
Prisms and
antiprism
In geometry, an antiprism or is a polyhedron composed of two Parallel (geometry), parallel Euclidean group, direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway po ...
s:
**
Dodecagonal prism (12 squares, 2
dodecagon
In geometry, a dodecagon, or 12-gon, is any twelve-sided polygon.
Regular dodecagon
A regular polygon, regular dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry ...
s)
**
Hexagonal antiprism
In geometry, the hexagonal antiprism is the 4th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.
Antiprisms are similar to prism (geometry), prisms except the bases are twisted re ...
(12 triangles, 2 hexagons)
*
Johnson solid
In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two Solid geometry, s ...
s:
** J
18:
Elongated triangular cupola (4 triangles, 9 squares, 1 hexagon)
** J
27:
Triangular orthobicupola
In geometry, the triangular orthobicupola is one of the Johnson solids (). As the name suggests, it can be constructed by attaching two triangular cupolas () along their bases. It has an equal number of squares and triangles at each vertex; howe ...
(8 triangles, 6 squares)
** J
51:
Triaugmented triangular prism (14 triangles)
** J
55:
Parabiaugmented hexagonal prism (8 triangles, 4 squares, 2 hexagons)
** J
56:
Metabiaugmented hexagonal prism (8 triangles, 4 squares, 2 hexagons)
** J
65:
Augmented truncated tetrahedron (8 triangles, 3 squares, 3 hexagons)
** J
86:
Sphenocorona (12 triangles, 2 squares)
** J
91:
Bilunabirotunda (8 triangles, 2 squares, 4 pentagons)
Tetradecahedra having at least one irregular face:
*
Heptagonal bipyramid (14 triangles) (see ''
Dipyramid'')
*
Heptagonal trapezohedron (14
kites
A kite is a tethered heavier than air flight, heavier-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. Kites often have ...
) (see ''
Trapezohedron
In geometry, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, or -deltohedron Remarks: the faces of a deltohedron are deltoids; a (non-twisted) kite or deltoid can be Dissection (geometry), dissected into two isosceles triangle ...
'')
*
Tridecagonal pyramid (13 triangles, 1 regular
tridecagon) (see ''
Pyramid (geometry)
In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex (geometry), apex. Each base edge (geometry), edge and apex form a triangle, called a lateral face. A pyramid is a cone, conic solid with a po ...
'')
*
Dissected regular icosahedron (the vertex figure of the
grand antiprism
In geometry, the grand antiprism or pentagonal double antiprismoid is a uniform 4-polytope (4-dimensional uniform polytope) bounded by 320 cells: 20 pentagonal antiprisms, and 300 tetrahedra. It is an anomalous, non-Wythoffian uniform 4-polyto ...
) (12 equilateral triangles and 2
trapezoid
In geometry, a trapezoid () in North American English, or trapezium () in British English, is a quadrilateral that has at least one pair of parallel sides.
The parallel sides are called the ''bases'' of the trapezoid. The other two sides are ...
s)
*
Hexagonal truncated trapezohedron: (12
pentagon
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°.
A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
s, 2 hexagons)
Includes an optimal space-filling shape in foams (see ''
Weaire–Phelan structure'') and in the crystal structure of
clathrate hydrate
Clathrate hydrates, or gas hydrates, clathrates, or hydrates, are crystalline water-based solids physically resembling ice, in which small non-polar molecules (typically gases) or polar molecules with large hydrophobic moieties are trapped ins ...
(see illustration, next to label 5
126
2)
*
Hexagonal bifrustum (12 trapezoids, 2 hexagons)
* The
British £1 coin in circulation from 2017 – with twelve edges and two faces – is an irregular dodecagonal prism, when one disregards the edging and relief features.
See also
*
Császár polyhedron – A nonconvex tetradecahedron of all triangle faces
*
Steffen's polyhedron – A
flexible tetradecahedron
*
Permutohedron – A polyhedron that can be defined in any dimension and equals the truncated octahedron in three dimensions
References
* , with Greek Numerical Prefixes
External links
*
Self-dual tetradecahedra
{{Polyhedra
Polyhedra