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 ...

, an truncated trapezohedron 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 ...

formed by a

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 ...

with

pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilat ...

s truncated from its two polar axis vertices. If the polar vertices are completely truncated (diminished), a trapezohedron becomes an

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 ...

. The vertices exist as 4 in four parallel planes, with alternating orientation in the middle creating the

pentagons In geometry, a pentagon (from the Greek language, 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 polygon, simple pentagon is ...

. The

regular dodecahedron A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 ed ...

is the most common polyhedron in this class, being a

Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...

, with 12

congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...

pentagonal faces. A truncated trapezohedron has all vertices with 3 faces. This means that the

dual polyhedra In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. ...

, the set of gyroelongated dipyramids, have all triangular faces. For example, the icosahedron is the dual of the

dodecahedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...

Forms

* Triangular truncated trapezohedron ( Dürer's solid) – 6 pentagons, 2 triangles, dual gyroelongated triangular dipyramid * Truncated square trapezohedron – 8 pentagons, 2 squares, dual

gyroelongated square dipyramid In geometry, the gyroelongated square bipyramid, heccaidecadeltahedron, or tetrakis square antiprism is one of the Johnson solids (). As the name suggests, it can be constructed by gyroelongating an octahedron (square bipyramid) by inserting a ...

*''Truncated pentagonal trapezohedron'' or

– 12 pentagonal faces, dual icosahedron * Truncated hexagonal trapezohedron – 12 pentagons, 2 hexagons, dual gyroelongated hexagonal dipyramid *... *Truncated ''n''-gonal trapezohedron – 2''n'' pentagons, 2 ''n''-gons, dual gyroelongated dipyramids

External links

Conway Notation for Polyhedra
Try: "tndAn", where n=4,5,6... example "t5dA5" is a dodecahedron. Polyhedra Truncated tilings {{polyhedron-stub

Forms

See also

External links