240px, A tetradecahedron with ''D2d'' symmetry, existing in the ">Weaire–Phelan structure
A tetradecahedron is 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 14
faces
The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...
. 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 p ...
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
The epidermis is the outermost of the three layers that comprise the skin, the inner layers being the dermis and hypodermis. The epidermis layer provides a barrier to infection from environmental pathogens and regulates the amount of water relea ...
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. Professor of Natural Philosophy (Glasgow), Professor of Natural Philoso ...
. 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, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of g ...
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 p ...
al faces (all exist in irregular-faced forms as well):
*
Archimedean solid
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...
s:
**
Cuboctahedron
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...
(8
triangles
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non-collinear ...
, 6
squares
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
)
**
Truncated cube
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices.
If the truncated cube has unit edge length, its dual triakis octahedron has edg ...
(8 triangles, 6
octagon
In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") 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, whi ...
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 Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°.
Regular hexa ...
s)
*
Prisms and
antiprism
In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation .
Antiprisms are a subclass o ...
s:
**
Dodecagonal prism
In geometry, the dodecagonal prism is the tenth in an infinite set of prisms, formed by square sides and two regular dodecagon caps.
If faces are all regular, it is a uniform polyhedron.
Use
It is used in the construction of two prismatic uni ...
(12 squares, 2
dodecagon
In geometry, a dodecagon or 12-gon is any twelve-sided polygon.
Regular dodecagon
A 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 and rotational sym ...
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 prisms except the bases are twisted relative to each oth ...
(12 triangles, 2 hexagons)
*
Johnson solid
In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...
s:
**J
18:
Elongated triangular cupola
In geometry, the elongated triangular cupola is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a triangular cupola () by attaching a hexagonal prism to its base.
Formulae
The following formulae for vol ...
(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; howev ...
(8 triangles, 6 squares)
**J
51:
Triaugmented triangular prism
The triaugmented triangular prism, in geometry, is a convex polyhedron with 14 equilateral triangles as its faces. It can be constructed from a triangular prism by attaching equilateral square pyramids to each of its three square faces. The same ...
(14 triangles)
**J
55:
Parabiaugmented hexagonal prism
In geometry, the parabiaugmented hexagonal prism is one of the Johnson solids (). As the name suggests, it can be constructed by doubly augmenting a hexagonal prism by attaching square pyramids () to two of its nonadjacent, parallel (opposite) eq ...
(8 triangles, 4 squares, 2 hexagons)
**J
56:
Metabiaugmented hexagonal prism
In geometry, the metabiaugmented hexagonal prism is one of the Johnson solids (). As the name suggests, it can be constructed by doubly augmenting a hexagonal prism by attaching square pyramids () to two of its nonadjacent, nonparallel equatorial ...
(8 triangles, 4 squares, 2 hexagons)
**J
65:
Augmented truncated tetrahedron
In geometry, the augmented truncated tetrahedron is one of the Johnson solids (). It is created by attaching a triangular cupola () to one hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a s ...
(8 triangles, 3 squares, 3 hexagons)
**J
86:
Sphenocorona
In geometry, the sphenocorona is one of the Johnson solids (). It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids.
Johnson uses the prefix ''spheno-'' to ref ...
(12 triangles, 2 squares)
**J
91:
Bilunabirotunda
In geometry, the bilunabirotunda is one of the Johnson solids ().
Geometry
It is one of the elementary Johnson solids, which do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids.
However, it does have a str ...
(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 or lighter-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. ...
) (see
Trapezohedron
In geometry, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, or -deltohedron is the dual polyhedron of an antiprism. The faces of an are congruent and symmetrically staggered; they are called ''twisted kites''. With a hi ...
)
*
Tridecagonal pyramid (13 triangles, 1 regular
tridecagon
In geometry, a tridecagon or triskaidecagon or 13-gon is a thirteen-sided polygon.
Regular tridecagon
A '' regular tridecagon'' is represented by Schläfli symbol .
The measure of each internal angle of a regular tridecagon is approximately 1 ...
) (see
Pyramid (geometry)
In geometry, a pyramid () is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a ''lateral face''. It is a conic solid with polygonal base. A pyramid with an base ...
)
*
Dissected regular icosahedron
In geometry, an edge-contracted icosahedron is a polyhedron with 18 triangular faces, 27 edges, and 11 vertices.
Construction
It can be constructed from the regular icosahedron, with one edge contraction, removing one vertex, 3 edges, and ...
(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-polytope ...
) (12 equilateral triangles and 2
trapezoid
A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium ().
A trapezoid is necessarily a Convex polygon, convex quadri ...
s)
*
Hexagonal truncated trapezohedron
In geometry, the truncated hexagonal trapezohedron is the fourth in an infinite series of truncated trapezohedra. It has 12 pentagon and 2 hexagon faces. It can be constructed by taking a hexagonal trapezohedron and truncating the polar axis v ...
: (12
pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ...
s, 2 hexagons)
Includes an optimal space-filling shape in foams (see
Weaire–Phelan structure
In geometry, the Weaire–Phelan structure is a three-dimensional structure representing an idealised foam of equal-sized bubbles, with two different shapes. In 1993, Denis Weaire and Robert Phelan found that this structure was a better solution ...
) and in the crystal structure of
Clathrate hydrate
Clathrate hydrates, or gas hydrates, clathrates, hydrates, etc., 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
The hexagonal bifrustum or truncated hexagonal bipyramid is the fourth in an infinite series of bifrustum polyhedra. It has 12 trapezoid and 2 hexagonal faces.
This polyhedron can be constructed by taking a hexagonal dipyramid and truncating the ...
(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
In geometry, the Császár polyhedron () is a nonconvex toroidal polyhedron with 14 triangular faces.
This polyhedron has no diagonals; every pair of vertices is connected by an edge. The seven vertices and 21 edges of the Császár polyhedron ...
– A nonconvex tetradecahedron of all triangle faces
*
Steffen's polyhedron – A
flexible
Flexible may refer to:
Science and technology
* Power cord, a flexible electrical cable.
** Flexible cable, an Electrical cable as used on electrical appliances
* Flexible electronics
* Flexible response
* Flexible-fuel vehicle
* Flexible rake re ...
tetradecahedron
*
Permutohedron
In mathematics, the permutohedron of order ''n'' is an (''n'' − 1)-dimensional polytope embedded in an ''n''-dimensional space. Its vertex coordinates (labels) are the permutations of the first ''n'' natural numbers. The edges ident ...
– 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