In
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 c ...
, the rhombicosidodecahedron is an
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 ...
, one of thirteen
convex isogonal nonprismatic solids constructed of two or more types of
regular polygon faces.
It has 20 regular
triangular faces, 30
square faces, 12 regular
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 ...
al faces, 60
vertices, and 120
edges
Edge or EDGE may refer to:
Technology Computing
* Edge computing, a network load-balancing system
* Edge device, an entry point to a computer network
* Adobe Edge, a graphical development application
* Microsoft Edge, a web browser developed by ...
.
Names
Johannes Kepler
Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...
in
Harmonices Mundi (1618) named this polyhedron a ''rhombicosidodecahedron'', being short for ''truncated icosidodecahedral rhombus'', with ''icosidodecahedral rhombus'' being his name for a
rhombic triacontahedron. There are different truncations of a rhombic triacontahedron into a
topological rhombicosidodecahedron: Prominently its
rectification
Rectification has the following technical meanings:
Mathematics
* Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points
* Rectifiable curve, in mathematics
* Recti ...
(left), the one that creates the uniform solid (center), and the rectification of the dual
icosidodecahedron (right), which is the core of the
dual compound.
It can also be called an ''
expanded'' or ''
cantellated'' dodecahedron or icosahedron, from truncation operations on either
uniform polyhedron
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent.
Uniform polyhedra may be regular (if also fa ...
.
Dimensions
For a rhombicosidodecahedron with edge length ''a'', its surface area and volume are:
:
Geometric relations
If you
expand an
icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
by moving the faces away from the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
the right amount, without changing the orientation or size of the faces, or do the same to its
dual
Dual or Duals may refer to:
Paired/two things
* Dual (mathematics), a notion of paired concepts that mirror one another
** Dual (category theory), a formalization of mathematical duality
*** see more cases in :Duality theories
* Dual (grammatical ...
dodecahedron, and patch the square holes in the result, you get a rhombicosidodecahedron. Therefore, it has the same number of triangles as an icosahedron and the same number of pentagons as a dodecahedron, with a square for each edge of either.
Alternatively, if you
expand each of five cubes by moving the faces away from the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
the right amount and rotating each of the five 72° around so they are equidistant from each other, without changing the orientation or size of the faces, and patch the pentagonal and triangular holes in the result, you get a rhombicosidodecahedron. Therefore, it has the same number of squares as five cubes.
Two clusters of faces of the
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 ...
, the ''lunes'' (each ''lune'' featuring two triangles adjacent to opposite sides of one square), can be aligned with a congruent patch of faces on the rhombicosidodecahedron. If two bilunabirotundae are aligned this way on opposite sides of the rhombicosidodecahedron, then a cube can be put between the bilunabirotundae at the very center of the rhombicosidodecahedron.
The rhombicosidodecahedron shares the vertex arrangement with the
small stellated truncated dodecahedron
In geometry, the small stellated truncated dodecahedron (or quasitruncated small stellated dodecahedron or small stellatruncated dodecahedron) is a nonconvex uniform polyhedron, indexed as U58. It has 24 faces (12 pentagons and 12 decagrams), 90 ...
, and with the uniform compounds of six or twelve
pentagrammic prism
In geometry, the pentagrammic prism is one of an infinite set of nonconvex prisms formed by square sides and two regular star polygon caps, in this case two pentagrams.
It is a special case of a right prism with a pentagram as base, which in ge ...
s.
The
Zometool
The term ''zome'' is used in several related senses. A zome in the original sense is a building using unusual geometries (different from the standard house or other building which is essentially one or a series of rectangular boxes).
The word "zome ...
kits for making
geodesic domes and other polyhedra use slotted balls as connectors. The balls are "expanded" rhombicosidodecahedra, with the squares replaced by rectangles. The expansion is chosen so that the resulting rectangles are
golden rectangles.
Twelve of the 92
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 are derived from the rhombicosidodecahedron, four of them by rotation of one or more
pentagonal cupola
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.
Formulae
The fol ...
e: the
gyrate,
parabigyrate,
metabigyrate, and
trigyrate rhombicosidodecahedron
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.
It can be constructed as a rhombicosidodecahedron with three pentag ...
. Eight more can be constructed by removing up to three cupolae, sometimes also rotating one or more of the other cupolae.
Cartesian coordinates
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
for the vertices of a rhombicosidodecahedron with an edge length of 2 centered at the origin are all
even permutation
In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total or ...
s of:
:(±1, ±1, ±''φ''
3),
:(±''φ''
2, ±''φ'', ±2''φ''),
:(±(2+''φ''), 0, ±''φ''
2),
where ''φ'' = is the
golden ratio. Therefore, the circumradius of this rhombicosidodecahedron is the common distance of these points from the origin, namely = for edge length 2. For unit edge length, R must be halved, giving
:''R'' = = ≈ 2.233.
Orthogonal projections
The ''rhombicosidodecahedron'' has six special
orthogonal projections, centered, on a vertex, on two types of edges, and three types of faces: triangles, squares, and pentagons. The last two correspond to the A
2 and H
2 Coxeter planes.
Spherical tiling
The rhombicosidodecahedron can also be represented as a
spherical tiling
In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most c ...
, and projected onto the plane via a
stereographic projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to ...
. This projection is
conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
Related polyhedra
Symmetry mutations
This polyhedron is topologically related as a part of a sequence of
cantellated polyhedra with vertex figure (3.4.n.4), which continues as tilings of the
hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
. These
vertex-transitive
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...
figures have (*n32) reflectional
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
.
Johnson solids
There are 12 related
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, 5 by diminishment, and 8 including gyrations:
Vertex arrangement
The rhombicosidodecahedron shares its
vertex arrangement with three
nonconvex uniform polyhedra: the
small stellated truncated dodecahedron
In geometry, the small stellated truncated dodecahedron (or quasitruncated small stellated dodecahedron or small stellatruncated dodecahedron) is a nonconvex uniform polyhedron, indexed as U58. It has 24 faces (12 pentagons and 12 decagrams), 90 ...
, the
small dodecicosidodecahedron
In geometry, the small dodecicosidodecahedron (or small dodekicosidodecahedron) is a nonconvex uniform polyhedron, indexed as U33. It has 44 faces (20 triangles, 12 pentagons, and 12 decagons), 120 edges, and 60 vertices. Its vertex figure is a ...
(having the triangular and pentagonal faces in common), and the
small rhombidodecahedron
In geometry, the small rhombidodecahedron is a nonconvex uniform polyhedron, indexed as U39. It has 42 faces (30 squares and 12 decagons), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral.
Related polyhedra
It shares i ...
(having the square faces in common).
It also shares its vertex arrangement with the
uniform compounds of
six
6 is a number, numeral, and glyph.
6 or six may also refer to:
* AD 6, the sixth year of the AD era
* 6 BC, the sixth year before the AD era
* The month of June
Science
* Carbon, the element with atomic number 6
* 6 Hebe, an asteroid
People ...
or
twelve pentagrammic prisms.
Rhombicosidodecahedral graph
In the
mathematical field of
graph theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...
, a rhombicosidodecahedral graph is the
graph of vertices and edges of the rhombicosidodecahedron, one of the
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. It has 60
vertices and 120 edges, and is a
quartic graph In the mathematical field of graph theory, a quartic graph is a graph where all vertices have degree 4. In other words, a quartic graph is a 4-regular graph.
Examples
Several well-known graphs are quartic. They include:
*The complete graph ''K'' ...
Archimedean graph
In the mathematical field of graph theory, an Archimedean graph is a graph that forms the skeleton of one of the Archimedean solids. There are 13 Archimedean graphs, and all of them are regular, polyhedral (and therefore by necessity also 3-vert ...
.
See also
*
Truncated rhombicosidodecahedron
In geometry, the truncated rhombicosidodecahedron is a polyhedron, constructed as a truncation (geometry), truncated rhombicosidodecahedron. It has 122 faces: 12 decagons, 30 octagons, 20 hexagons, and 60 squares.
Other names
*Truncated small rh ...
Notes
References
* (Section 3-9)
*
*''
The Big Bang Theory
''The Big Bang Theory'' is an American television sitcom created by Chuck Lorre and Bill Prady, both of whom served as executive producers on the series, along with Steven Molaro, all of whom also served as head writers. It premiered on CBS ...
Series 8 Episode 2 - The Junior Professor Solution'': features this solid as the answer to an impromptu science quiz the main four characters have in Leonard and Sheldon's apartment, and is also illustrated in
Chuck Lorre's Vanity Card #461 at the end of that episode.
External links
*
**
*
Editable printable net of a Rhombicosidodecahedron with interactive 3D viewThe Uniform PolyhedraThe Encyclopedia of Polyhedra
{{Polyhedron navigator
Uniform polyhedra
Archimedean solids