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Pentagonal antiprism

 

 
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Pentagonal antiprism
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagon
Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
s joined to each other by a ring of 10 triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
s for a total of 12 faces. Hence, it is a non-regular dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
.

Geometry
If the faces of the pentagonal antiprism are all regular, it is a semiregular polyhedron
Semiregular polyhedron

A semiregular polyhedron is a polyhedron with regular polygon faces and a symmetry group which is transitive on its vertices. Or at least, that is what follows from Thorold Gosset's 1900 definition of the more general semiregular polytope....
. It can also be considered as a parabidiminished icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
.






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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagon
Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
s joined to each other by a ring of 10 triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
s for a total of 12 faces. Hence, it is a non-regular dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
.

Geometry


If the faces of the pentagonal antiprism are all regular, it is a semiregular polyhedron
Semiregular polyhedron

A semiregular polyhedron is a polyhedron with regular polygon faces and a symmetry group which is transitive on its vertices. Or at least, that is what follows from Thorold Gosset's 1900 definition of the more general semiregular polytope....
. It can also be considered as a parabidiminished icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
. The two pentagonal faces can be stellated to form the icosahedron.

Relation to polytopes


The pentagonal antiprism occurs as a constituent element in some higher-dimensional polytope
Polytope

In geometry, polytope is a generic term that can refer to a two-dimensional polygon, a three-dimensional polyhedron, or any of the various generalizations thereof, including generalizations to higher dimensions and other abstractions ....
s. Two rings of 10 pentagonal antiprisms each bound the hypersurface of the 4-dimensional grand antiprism
Grand antiprism

In geometry, the grand antiprism or pentagonal double antiprismoid is a uniform polychoron bounded by 320 cell : 20 pentagonal antiprisms, and 300 tetrahedron....
. If these antiprisms are stellated into pentagonal prism pyramids and linked with rings of 5 tetrahedra each, the 600-cell
600-cell

In geometry, the 600-cell is the convex regular 4-polytope, or polychoron, with Schl?fli symbol . Its boundary is composed of 600 tetrahedron cell with 20 meeting at each vertex....
 is obtained.

See also

  • Set of antiprisms
    Antiprism

    An n-sided antiprism is a polyhedron composed of 2 parallel copies of some particular n-sided polygon, connected by an alternating band of triangles....
  • Octahedron
    Octahedron

    An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
     Triangle-capped antiprism
  • Square antiprism
    Square antiprism

    In geometry, the square antiprism is the second in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps....
  • Hexagonal antiprism
    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 2 polygon caps....
  • Octagonal antiprism
    Octagonal antiprism

    In geometry, the octagonal antiprism is the 6th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps....


External links

  • www.georgehart.com: The Encyclopedia of Polyhedra
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