Rotunda (geometry)

Last updated
Set of rotundas
Pentagonal rotunda.png
Faces 1 n-gon
1 2n-gon
n pentagons
2n triangles
Edges 7n
Vertices 4n
Symmetry group Cnv, [n], (*nn), order 2n
Rotation group Cn, [n]+, (nn), order n
Properties convex

In geometry, a rotunda is any member of a family of dihedral-symmetric polyhedra. They are similar to a cupola but instead of alternating squares and triangles, it alternates pentagons and triangles around an axis. The pentagonal rotunda is a Johnson solid.

Contents

Other forms can be generated with dihedral symmetry and distorted equilateral pentagons. [ example needed ]

Examples

Rotundas
345678
Green triangular rotunda.svg
triangular rotunda
Green square rotunda.svg
square rotunda
Green pentagonal rotunda.svg
pentagonal rotunda
Green hexagonal rotunda.svg
hexagonal rotunda
Green heptagonal rotunda.svg
heptagonal rotunda
Green octagonal rotunda.svg
octagonal rotunda

Star-rotunda

Star-rotundas
57911
Pentagrammic rotunda.svg
Pentagrammic rotunda
Heptagrammic rotunda.svg
Heptagrammic rotunda
Enneagrammic rotunda.svg
Enneagrammic rotunda
Hendecagrammic rotunda.svg
Hendecagrammic rotunda

See also

Related Research Articles

In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.

<span class="mw-page-title-main">Johnson solid</span> 92 non-uniform convex polyhedra, with each face a regular polygon

In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides ; it has 1 square face and 4 triangular faces. Some authors require that the solid not be uniform before they refer to it as a "Johnson solid".

<span class="mw-page-title-main">Pentagonal bipyramid</span> 13th Johnson solid; two pentagonal pyramids joined at the bases

In geometry, the pentagonal bipyramid is third of the infinite set of face-transitive bipyramids, and the 13th Johnson solid. Each bipyramid is the dual of a uniform prism.

<span class="mw-page-title-main">Pentagonal rotunda</span> 6th Johnson solid (17 faces)

In geometry, the pentagonal rotunda is one of the Johnson solids. It can be seen as half of an icosidodecahedron, or as half of a pentagonal orthobirotunda. It has a total of 17 faces.

<span class="mw-page-title-main">Elongated pentagonal rotunda</span>

In geometry, the elongated pentagonal rotunda is one of the Johnson solids (J21). As the name suggests, it can be constructed by elongating a pentagonal rotunda (J6) by attaching a decagonal prism to its base. It can also be seen as an elongated pentagonal orthobirotunda (J42) with one pentagonal rotunda removed.

<span class="mw-page-title-main">Gyroelongated pentagonal rotunda</span>

In geometry, the gyroelongated pentagonal rotunda is one of the Johnson solids (J25). As the name suggests, it can be constructed by gyroelongating a pentagonal rotunda (J6) by attaching a decagonal antiprism to its base. It can also be seen as a gyroelongated pentagonal birotunda (J48) with one pentagonal rotunda removed.

<span class="mw-page-title-main">Pentagonal orthobirotunda</span> 34th Johnson solid; 2 pentagonal rotundae joined base-to-base

In geometry, the pentagonal orthobirotunda is one of the Johnson solids. It can be constructed by joining two pentagonal rotundae along their decagonal faces, matching like faces.

<span class="mw-page-title-main">Pentagonal cupola</span> 5th Johnson solid (12 faces)

In geometry, the pentagonal cupola is one of the Johnson solids. It can be obtained as a slice of the rhombicosidodecahedron. The pentagonal cupola consists of 5 equilateral triangles, 5 squares, 1 pentagon, and 1 decagon.

<span class="mw-page-title-main">Trigyrate rhombicosidodecahedron</span> 75th Johnson solid

In geometry, the trigyrate rhombicosidodecahedron is one of the Johnson solids. It contains 20 triangles, 30 squares and 12 pentagons. It is also a canonical polyhedron.

<span class="mw-page-title-main">Bilunabirotunda</span> 91st Johnson solid (14 faces)

In geometry, the bilunabirotunda is one of the Johnson solids. A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra. They were named by Norman Johnson, who first listed these polyhedra in 1966.

<span class="mw-page-title-main">Triangular hebesphenorotunda</span> 92nd Johnson solid (20 faces)

In geometry, the triangular hebesphenorotunda is one of the Johnson solids.

<span class="mw-page-title-main">Pentagonal orthocupolarotunda</span> 32nd Johnson solid; pentagonal cupola and rotunda joined base-to-base

In geometry, the pentagonal orthocupolarotunda is one of the Johnson solids. As the name suggests, it can be constructed by joining a pentagonal cupola and a pentagonal rotunda along their decagonal bases, matching the pentagonal faces. A 36-degree rotation of one of the halves before the joining yields a pentagonal gyrocupolarotunda.

<span class="mw-page-title-main">Pentagonal gyrocupolarotunda</span> 33rd Johnson solid; pentagonal cupola and rotunda joined base-to-base

In geometry, the pentagonal gyrocupolarotunda is one of the Johnson solids. Like the pentagonal orthocupolarotunda, it can be constructed by joining a pentagonal cupola and a pentagonal rotunda along their decagonal bases. The difference is that in this solid, the two halves are rotated 36 degrees with respect to one another.

<span class="mw-page-title-main">Elongated pentagonal gyrocupolarotunda</span> 41st Johnson solid

In geometry, the elongated pentagonal gyrocupolarotunda is one of the Johnson solids. As the name suggests, it can be constructed by elongating a pentagonal gyrocupolarotunda by inserting a decagonal prism between its halves. Rotating either the pentagonal cupola or the pentagonal rotunda through 36 degrees before inserting the prism yields an elongated pentagonal orthocupolarotunda.

<span class="mw-page-title-main">Elongated pentagonal orthocupolarotunda</span> 40th Johnson solid

In geometry, the elongated pentagonal orthocupolarotunda is one of the Johnson solids. As the name suggests, it can be constructed by elongating a pentagonal orthocupolarotunda by inserting a decagonal prism between its halves. Rotating either the cupola or the rotunda through 36 degrees before inserting the prism yields an elongated pentagonal gyrocupolarotunda.

<span class="mw-page-title-main">Uniform polyhedron</span> Isogonal polyhedron with regular faces

In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.

<span class="mw-page-title-main">Bicupola (geometry)</span> Solid made from 2 cupolae joined base-to-base

In geometry, a bicupola is a solid formed by connecting two cupolae on their bases.

<span class="mw-page-title-main">Birotunda</span> Solid made from 2 rotunda joined base-to-base

In geometry, a birotunda is any member of a family of dihedral-symmetric polyhedra, formed from two rotunda adjoined through the largest face. They are similar to a bicupola but instead of alternating squares and triangles, it alternates pentagons and triangles around an axis. There are two forms, ortho- and gyro-: an orthobirotunda has one of the two rotundas is placed as the mirror reflection of the other, while in a gyrobirotunda one rotunda is twisted relative to the other.

<span class="mw-page-title-main">Gyroelongated pyramid</span> Polyhedron formed by capping an antiprism with a pyramid

In geometry, the gyroelongated pyramids are an infinite set of polyhedra, constructed by adjoining an n-gonal pyramid to an n-gonal antiprism.

<span class="mw-page-title-main">Icosahedron</span> Polyhedron with 20 faces

In geometry, an icosahedron is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) 'twenty', and ἕδρα (hédra) 'seat'. The plural can be either "icosahedra" or "icosahedrons".

References