Pentagonal Cupola
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The pentagonal
cupola In architecture, a cupola () is a relatively small, usually dome-like structure on top of a building often crowning a larger roof or dome. Cupolas often serve as a roof lantern to admit light and air or as a lookout. The word derives, via Ital ...
's faces are five
equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
s, five
square 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 ...
s, one
regular 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 ...
, and one regular decagon. It has the property of convexity and
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, from which it is classified as the fifth
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 ...
. This cupola produces two or more regular polyhedrons by slicing it with a plane, an
elementary polyhedron In geometry, a composite polyhedron is a convex polyhedron that produces other polyhedrons when sliced by a plane. Examples can be found in Johnson solids. Definition and examples A convex polyhedron is said to be ''composite'' if there exists ...
's example. The following
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...
e for circumradius R , and height h ,
surface area The surface area (symbol ''A'') of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the d ...
A , and
volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...
V may be applied if all
faces The face is the front of the 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 affect the ...
are
regular Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, ...
with edge length a : \begin h &= \sqrta &\approx 0.526a, \\ R &= \fraca &\approx 2.233a, \\ A &= \fraca^2 &\approx 16.580a^2, \\ V &= \fraca^3 &\approx 2.324a^3. \end It has an
axis of symmetry An axis (: axes) may refer to: Mathematics *A specific line (often a directed line) that plays an important role in some contexts. In particular: ** Coordinate axis of a coordinate system *** ''x''-axis, ''y''-axis, ''z''-axis, common names f ...
passing through the center of both top and base, which is symmetrical by rotating around it at one-, two-, three-, and four-fifth of a full-turn angle. It is also mirror-symmetric relative to any perpendicular plane passing through a bisector of the hexagonal base. Therefore, it has
pyramidal symmetry In three dimensional geometry, there are four infinite series of point groups in three dimensions (''n''≥1) with ''n''-fold rotational or reflectional symmetry about one axis (by an angle of 360°/''n'') that does not change the object. They are ...
, the
cyclic group In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
C_ of order ten.


Related polyhedron

The pentagonal cupola can be applied to construct 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 ...
. A construction that involves the attachment of its base to another polyhedron is known as
augmentation Augment or augmentation may refer to: Language *Augment (Indo-European), a syllable added to the beginning of the word in certain Indo-European languages *Augment (Bantu languages), a morpheme that is prefixed to the noun class prefix of nouns i ...
; attaching it to prisms or
antiprisms 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 ...
is known as elongation or
gyroelongation 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 solids with such ...
. Some of the Johnson solids with such constructions are:
elongated pentagonal cupola In geometry, the elongated pentagonal cupola is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a pentagonal cupola () by attaching a decagonal prism to its base. The solid can also be seen as an elongated ...
J_ ,
gyroelongated pentagonal cupola In geometry, the gyroelongated pentagonal cupola is one of the Johnson solids (''J''24). As the name suggests, it can be constructed by gyroelongating a pentagonal cupola (''J''5) by attaching a decagonal antiprism to its base. It can also be see ...
J_ ,
pentagonal orthobicupola In geometry, the pentagonal orthobicupola is one of the Johnson solids (). As the name suggests, it can be constructed by joining two pentagonal cupolae () along their decagonal bases, matching like faces. A 36-degree rotation of one cupola befor ...
J_ ,
pentagonal gyrobicupola The pentagonal gyrobicupola is a polyhedron that is constructed by attaching two pentagonal cupolas base-to-base, each of its cupolas is twisted at 36°. It is an example of a Johnson solid and a composite polyhedron. Construction The pentagon ...
J_ ,
pentagonal orthocupolarotunda 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 face ...
J_ ,
pentagonal gyrocupolarotunda 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 differ ...
J_ ,
elongated pentagonal orthobicupola In geometry, the elongated pentagonal orthobicupola or cantellated pentagonal prism is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a pentagonal orthobicupola () by inserting a decagonal prism between it ...
J_ ,
elongated pentagonal gyrobicupola In geometry, the elongated pentagonal gyrobicupola is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a pentagonal gyrobicupola () by inserting a decagonal prism between its congruent halves. Rotating one o ...
J_ ,
elongated pentagonal orthocupolarotunda 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 eit ...
J_ ,
gyroelongated pentagonal bicupola In geometry, the gyroelongated pentagonal bicupola is one of the Johnson solids (). As the name suggests, it can be constructed by gyroelongating a pentagonal bicupola ( or ) by inserting a decagonal antiprism between its congruent halves. The g ...
J_ ,
gyroelongated pentagonal cupolarotunda In geometry, the gyroelongated pentagonal cupolarotunda is one of the Johnson solids (). As the name suggests, it can be constructed by gyroelongating a pentagonal cupolarotunda ( or ) by inserting a decagonal antiprism between its two halves. ...
J_ ,
augmented truncated dodecahedron In geometry, the augmented truncated dodecahedron is one of the Johnson solids (). As its name suggests, it is created by attaching a pentagonal cupola () onto one decagonal face of a truncated dodecahedron In geometry, the truncated dodecahed ...
J_ , parabiaugmented truncated dodecahedron J_ ,
metabiaugmented truncated dodecahedron In geometry, the metabiaugmented truncated dodecahedron is one of the Johnson solids (). As its name suggests, it is created by attaching two pentagonal cupolas () onto two nonadjacent, nonparallel decagonal faces of a truncated dodecahedron I ...
J_ ,
triaugmented truncated dodecahedron In geometry, the triaugmented truncated dodecahedron is one of the Johnson solids (); of them, it has the greatest volume in proportion to the cube of the side length, as well as the greatest number of edges. As its name suggests, it is created b ...
J_ ,
gyrate rhombicosidodecahedron In geometry, the gyrate rhombicosidodecahedron is one of the Johnson solids (). It is also a Midsphere#Canonical polyhedron, canonical polyhedron. Construction The gyrate rhombicosidodecahedron can be constructed similarly as rhombicosidodec ...
J_ ,
parabigyrate rhombicosidodecahedron In geometry, the parabigyrate rhombicosidodecahedron is one of the Johnson solids (). It can be constructed as a rhombicosidodecahedron with two opposing pentagonal cupolae rotated through 36 degrees. It is also a canonical polyhedron. Altern ...
J_ ,
metabigyrate rhombicosidodecahedron In geometry, the metabigyrate rhombicosidodecahedron is one of the Johnson solids (). It can be constructed as a rhombicosidodecahedron with two non-opposing pentagonal cupolae rotated through 36 degrees. It is also a canonical polyhedron. Alt ...
J_ , and trigyrate rhombicosidodecahedron J_ . Relatedly, a construction from polyhedra by removing one or more pentagonal cupolas is known as
diminishment Diminishment is the legal process by which the United States Congress can reduce the size of an Indian reservation. History In 1984, the United States Supreme Court The Supreme Court of the United States (SCOTUS) is the highest court in the ...
: diminished rhombicosidodecahedron J_ , paragyrate diminished rhombicosidodecahedron J_ ,
metagyrate diminished rhombicosidodecahedron In geometry, the metagyrate diminished rhombicosidodecahedron is one of the Johnson solids (). It can be constructed as a rhombicosidodecahedron In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal n ...
J_ ,
bigyrate diminished rhombicosidodecahedron In geometry, the bigyrate diminished rhombicosidodecahedron is one of the Johnson solids (). It can be constructed as a rhombicosidodecahedron In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal non ...
J_ ,
parabidiminished rhombicosidodecahedron In geometry, the parabidiminished rhombicosidodecahedron is one of the Johnson solids (). It is also a canonical polyhedron. It can be constructed as a rhombicosidodecahedron In geometry, the rhombicosidodecahedron is an Archimedean solid, o ...
J_ , metabidiminished rhombicosidodecahedron J_ , gyrate bidiminished rhombicosidodecahedron J_ , and tridiminished rhombicosidodecahedron J_ .


References


External links

* {{Johnson solids navigator Elementary polyhedron Prismatoid polyhedra Johnson solids