A rhombic dodecahedron

In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan who first described them in 1865. Rhombic dodecahedron, rendered in povray 3. ... Rhombic dodecahedron, rendered in povray 3. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Inter. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ... In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ... Eugène Charles Catalan Eugène Charles Catalan (May 30, 1814 - February 14, 1894) was a Belgian mathematician. ... 1865 is a common year starting on Sunday. ...

The Catalan solids are all convex. They are face-uniform but not vertex-uniform. This is because the dual Archimedean solids are vertex-uniform and not face uniform. Note that unlike Platonic solids and Archimedean solids, the faces of Catalan solids are not regular polygons. However, the vertex figures of Catalan solids are regular, and they have constant dihedral angles. Additionally, two of the Catalan solids are edge-uniform: the rhombic dodecahedron and the rhombic triacontahedron. These are the duals of the two quasi-regular Archimedean solids. In solid geometry and some ancient physical theories, a Platonic solid is a convex polyhedron with all its faces being congruent regular polygons, and the same number of faces meeting at each of its vertices. ... In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ... A polygon (from the Greek poly, for many, and gonos, for angle) is a closed planar path composed of a finite number of sequential straight line segments. ... In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ... In Aerospace engineering, the dihedral is the angle that the two wings make with each other. ... The rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. ... The Rhombic triacontahedron is a convex polyhedron with 30 rhombic faces. ...

Just like their dual Archimedean partners there are two chiral Catalan solids: the pentagonal icositetrahedron and the pentagonal hexecontahedron. These each come in two enantiomorphs. Not counting the enantiomorphs there are a total of 13 Catalan solids. In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone. ... A pentagonal icositetrahedron is a catalan solid which looks a bit like an octahedron undergoing cell division. ... A pentagonal hexecontahedron is a Catalan solid which looks a bit like an icosahedron undergoing cell division. ... Chirality refers to several phenomena, all having to do with objects that differ from their mirror image. ...

Name	picture	Dual Archimedean solid	Faces	Edges	Vertices	Face Polygon	Symmetry
triakis tetrahedron	( Video)	truncated tetrahedron	12	18	8	isosceles triangle	Td
rhombic dodecahedron	( Video)	cuboctahedron	12	24	14	rhombus	Oh
triakis octahedron	( Video)	truncated cube	24	36	14	isosceles triangle	Oh
tetrakis hexahedron	( Video)	truncated octahedron	24	36	14	isosceles triangle	Oh
deltoidal icositetrahedron	( Video)	rhombicuboctahedron	24	48	26	kite	Oh
disdyakis dodecahedron or hexakis octahedron	( Video)	truncated cuboctahedron	48	72	26	scalene triangle	Oh
pentagonal icositetrahedron	( Video)( Video)	snub cube	24	60	38	irregular pentagon	O
rhombic triacontahedron	( Video)	icosidodecahedron	30	60	32	rhombus	Ih
triakis icosahedron	( Video)	truncated dodecahedron	60	90	32	isosceles triangle	Ih
pentakis dodecahedron	( Video)	truncated icosahedron	60	90	32	isosceles triangle	Ih
deltoidal hexecontahedron	( Video)	rhombicosidodecahedron	60	120	62	kite	Ih
disdyakis triacontahedron or hexakis icosahedron	( Video)	truncated icosidodecahedron	120	180	62	scalene triangle	Ih
pentagonal hexecontahedron	( Video)( Video)	snub dodecahedron	60	150	92	irregular pentagon	I

The symmetry group of an object (e. ... A triakis tetrahedron is a catalan solid which looks a bit like an overinflated tetrahedron. ... Download high resolution version (681x766, 51 KB)Triakis tetrahedron, made by me using POV-Ray, see image:poly. ... Spinning triakis tetrahedron, made by me using POV-Ray, see image:poly. ... The truncated tetrahedron is an Archimedean solid. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... The rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. ... Download high resolution version (849x754, 59 KB)Rhombic dodecahedron, made by me using POV-Ray, see image:poly. ... Spinning rhombic dodecahedron, made by me using POV-Ray, see image:poly. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... This shape is a rhombus In geometry, a rhombus (also known as a rhomb) is a quadrilateral in which all of the sides are of equal length. ... A triakis octahedron is a catalan solid which looks a bit like an overinflated octahedron. ... Download high resolution version (679x745, 67 KB)Triakis octahedron, made by me using POV-Ray, see image:poly. ... Spinning triakis octahedron, made by me using POV-Ray, see image:poly. ... The truncated cube, or truncated hexahedron, is an Archimedean solid. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... A tetrakis hexahedron is a Catalan solid which looks a bit like an overinflated cube. ... Download high resolution version (767x737, 72 KB)Tetrakis hexahedron, made by me using POV-Ray, see image:poly. ... Spinning tetrakis hexahedron, made by me using POV-Ray, see image:poly. ... The truncated octahedron, also known as a Mecon, is an Archimedean solid. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... A deltoidal icositetrahedron (or trapezoidal icositetrahedron) is a catalan solid which looks a bit like an overinflated cube. ... Download high resolution version (845x837, 71 KB)Deltoidal icositetrahedron, made by me using POV-Ray, see image:poly. ... Spinning deltoidal antifnordoid megaicositetrasuperhedron, made by me using POV-Ray, see image:poly. ... The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. ... The word kite has a number of possible meanings: A kite is a man-made, heavier-than-air object, designed to fly by opposing the force of the wind with the tension of a line held by the operator. ... A disdyakis dodecahedron, or hexakis octahedron, is the Catalan solid whose Archimedean dual is the truncated cuboctahedron. ... Download high resolution version (765x802, 94 KB)Disdyakis dodecahedron, made by me using POV-Ray, see image:poly. ... Spinning disdyakis dodecahedron, made by me using POV-Ray, see image:poly. ... The truncated cuboctahedron, or great rhombicuboctahedron, is an Archimedean solid. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... A pentagonal icositetrahedron is a catalan solid which looks a bit like an octahedron undergoing cell division. ... Download high resolution version (829x813, 69 KB)Pentagonal icositetrahedron (Ccw), made by me using POV-Ray, see image:poly. ... Download high resolution version (817x792, 71 KB)Pentagonal icositetrahedron (Cw), made by me using POV-Ray, see image:poly. ... Spinning pentagonal icositetrahedron (ccw), made by me using POV-Ray, see image:poly. ... Spinning pentagonal icositetrahedron (cw), made by me using POV-Ray, see image:poly. ... The snub cube, or snub cuboctahedron, is an Archimedean solid. ... In geometry, a pentagon is any five-sided polygon. ... The Rhombic triacontahedron is a convex polyhedron with 30 rhombic faces. ... Download high resolution version (780x859, 88 KB)Rhombic triacontahedron, made by me using POV-Ray, see image:poly. ... Spinning rhombic triacontahedron, made by me using POV-Ray, see image:poly. ... An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ... This shape is a rhombus In geometry, a rhombus (also known as a rhomb) is a quadrilateral in which all of the sides are of equal length. ... A triakis icosahedron is a catalan solid which looks a bit like an overinflated icosahedron. ... Download high resolution version (819x849, 107 KB)Triakis icosahedron, made by me using POV-Ray, see image:poly. ... Spinning triakis icosahedron, made by me using POV-Ray, see image:poly. ... The truncated dodecahedron is an Archimedean solid. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... A pentakis dodecahedron is a catalan solid which looks a bit like an overinflated dodecahedron. ... Download high resolution version (844x843, 106 KB)Pentakis dodecahedron, made by me using POV-Ray, see image:poly. ... Spinning pentakis dodecahedron, made by me using POV-Ray, see image:poly. ... The truncated icosahedron is an Archimedean solid. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... A deltoidal icositetrahedron is a catalan solid which looks a bit like an overinflated dodecahedron. ... Download high resolution version (854x843, 102 KB)Deltoidal hexecontahedron, made by me using POV-Ray, see image:poly. ... Spinning deltoidal hexecontahedron, made by me using POV-Ray, see image:poly. ... The rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid. ... The word kite has a number of possible meanings: A kite is a man-made, heavier-than-air object, designed to fly by opposing the force of the wind with the tension of a line held by the operator. ... A disdyakis triacontahedron, or hexakis icosahedron is the Catalan solid whose Archimedean dual is the truncated icosidodecahedron. ... Download high resolution version (812x847, 130 KB)Disdyakis triacontahedron, made by me using POV-Ray, see image:poly. ... Spinning disdyakis triacontahedron, made by me using POV-Ray, see image:poly. ... The truncated icosidodecahedron, or great rhombicosidodecahedron, is an Archimedean solid. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... A pentagonal hexecontahedron is a Catalan solid which looks a bit like an icosahedron undergoing cell division. ... Download high resolution version (834x851, 98 KB)Pentagonal hexecontahedron (Ccw), made by me using POV-Ray, see image:poly. ... Download high resolution version (846x841, 96 KB)Pentagonal hexecontahedron (Cw), made by me using POV-Ray, see image:poly. ... Spinning pentagonal hexecontahedron (ccw), made by me using POV-Ray, see image:poly. ... Spinning pentagonal hexecontahedron (cw), made by me using POV-Ray, see image:poly. ... The snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, usually regarded as a truncated polyhedron derived by truncating either a dodecahedron or an icosahedron. ... In geometry, a pentagon is any five-sided polygon. ...

External links

• Catalan Solid – MathWorld site
• Archimedean duals – at Virtual Reality Polyhedra
• Interactive Catalan Solid in Java