The octahedral rotation group O with fundamental domain

Chiral and achiral octahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the largest symmetry groups compatible with translational symmetry. They are among the crystallographic point groups of the cubic crystal system. In mathematics, given a lattice Γ in a Lie group G, a fundamental domain is a set D of representatives for the cosets G/Γ, that is also a well-behaved set topologically, in a sense that can be made precise in one of several ways. ... A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ... The symmetry group of an object (e. ... In physics and mathematics, translational symmetry is the invariance of an object or a system of equations under the translations - operations that change the coordinates of all objects by a constant. ... In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ... In crystallography, the cubic crystal system (or isometric crystal system) is the most symmetric of the 7 crystal systems. ...

Chiral octahedral symmetry

O, 432, or of order 24, is chiral octahedral symmetry or rotational octahedral symmetry . This group is like chiral tetrahedral symmetry T, but the C₂ axes are now C₄ axes, and additionally there are 6 C₂ axes, through the midpoints of the edges of the cube. T_d and O are isomorphic as abstract groups: they both correspond to S₄, the symmetric group on 4 objects. T_d is the union of T and the set obtained by combining each element of O T with inversion. O is the rotation group of the cube and the regular octahedron. In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ... Jump to: navigation, search An octahedron (plural: octahedra) is a polyhedron with eight faces. ...

Achiral octahedral symmetry

• O_h (*432) of order 48 - achiral octahedral symmetry or full octahedral symmetry. This group has the same rotation axes as O, but with mirror planes, comprising both the mirror planes of T_d and T_h. This group is isomorphic to S₄ × C₂, and is the full symmetry group of the cube and octahedron. See also the isometries of the cube.

In the disdyakis dodecahedron one full face is a fundamental domain; other solids with the same symmetry can be obtained by adjusting the orientation of the faces, e.g. flattening selected subsets of faces to combine each subset into one face, or replacing each face by multiple faces, or a curved surface.

The octahedral group O_h with fundamental domain

With the 4-fold axes as coordinate axes, a fundamental domain of O_h is given by 0 ≤ x ≤ y ≤ z. An object with this symmetry is characterized by the part of the object in the fundamental domain, for example the cube is given by z = 1, and the octahedron by x + y + z = 1 (or the corresponding inequalities, to get the solid instead of the surface). ax + by + cz = 1 gives a polyhedron with 48 faces, e.g. the disdyakis dodecahedron. Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ... Jump to: navigation, search An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Cube may denote one of the following. ... Download high resolution version (765x802, 94 KB)Disdyakis dodecahedron, made by me using POV-Ray, see image:poly. ... Download high resolution version (765x802, 94 KB)Disdyakis dodecahedron, made by me using POV-Ray, see image:poly. ... A disdyakis dodecahedron is the Catalan solid whose Archimedean dual is the truncated cuboctahedron. ... Cube may denote one of the following. ... Jump to: navigation, search An octahedron (plural: octahedra) is a polyhedron with eight faces. ...

Faces are 8-by-8 combined to larger faces for a = b = 0 (cube) and 6-by-6 for a = b = c (octahedron).

Conjugacy classes

The conjugacy classes of O are: Jump to: navigation, search In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure. ...

• identity
• 6 × rotation by 90°
• 8 × rotation by 120°
• 3 × rotation by 180° about a 4-fold axis
• 6 × rotation by 180° about a 2-fold axis

Those of O_h include also each with inversion:

• inversion
• 6 × rotoreflection by 90°
• 8 × rotoreflection by 60°
• 3 × reflection in a plane perpendicular to a 4-fold axis
• 6 × reflection in a plane perpendicular to a 2-fold axis

The isometries of the cube

(To be integrated in the rest of the text.)

The cube has 48 isometries, forming the symmetry group O_h, isomorphic to S₄ × C₂. They can be categorized as follows: The symmetry group of an object (e. ... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ...

• O (the identity and 23 proper rotations) with the following conjugacy classes (in parentheses are given the permutations of the body diagonals and the unit quaternion representation):
  • identity (identity; 1)
  • rotation about an axis from the center of a face to the center of the opposite face by an angle of 90°: 3 axes, 2 per axis, together 6 ((1 2 3 4), etc.; ((1±i)/√2, etc.)
  • ditto by an angle of 180°: 3 axes, 1 per axis, together 3 ((1 2)(3 4), etc.; i,j,k)
  • rotation about an axis from the center of an edge to the center of the opposite edge by an angle of 180°: 6 axes, 1 per axis, together 6 ((1 2), etc.; ((i±j)/√2, etc.)
  • rotation about a body diagonal by an angle of 120°: 4 axes, 2 per axis, together 8 ((1 2 3), etc.; (1±i±j±k)/2)
• The same with inversion (x is mapped to −x) (also 24 isometries). Note that rotation by an angle of 180° about an axis combined with inversion is just reflection in the perpendicular plane. The combination of inversion and rotation about a body diagonal by an angle of 120° is rotation about the body diagonal by an angle of 60°, combined with reflection in the perpendicular plane (the rotation itself does not map the cube to itself; the intersection of the reflection plane with the cube is a regular hexagon).

An isometry of the cube can be identified in various ways: Jump to: navigation, search In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure. ... Quaternions are used in computer graphics and related fields because they allow for compact representations of rotations, or correspondingly, orientations, in 3D space: Consider the quaternions with modulus 1 (these are sometimes called versors). ... Inversion has different meanings in different fields of knowledge: Something that is inverted or the process by which an inverse is obtained. ... A regular hexagon A hexagon (also known as sexagon) is a polygon with six edges and six vertices. ...

• by the faces three given adjacent faces (say 1, 2, and 3 on a die) are mapped to
• by the image of a cube with on one face a non-symmetric marking: the face with the marking, whether it is normal or a mirror image, and the orientation
• by a permutation of the four body diagonals (each of the 24 permutations is possible), combined with a toggle for inversion of the cube, or not

For cubes with colors or markings (like dice have), the symmetry group is a subgroup of O_h. Examples: Rolling dice A die (Old French de, from Latin datum something given or played [1]) is a small polyhedral object (usually a cube) suitable as a gambling device (especially for craps or sic bo). ...

• C_4v: if one face has a different color (or two opposite faces have colors different from each other and from the other four), the cube has 8 isometries, like a square has in 2D.
• D_2h: if opposite faces have the same colors, different for each set of two, the cube has 8 isometries, like a cuboid.
• D_4h: if two opposite faces have the same color, and all other faces have one different color, the cube has 16 isometries, like a square prism (square box).
• C_2v:
  • if two adjacent faces have the same color, and all other faces have one different color, the cube has 4 isometries.
  • if three faces, of which two opposite to each other, have one color and the other three one other color, the cube has 4 isometries.
  • if two opposite faces have the same color, and two other opposite faces also, and the last two have different colors, the cube has 4 isometries, like a piece of blank paper with a shape with a mirror symmetry.
• C_s:
  • if two adjacent faces have colors different from each other, and the other four have a third color, the cube has 2 isometries.
  • if two opposite faces have the same color, and all other faces have different colors, the cube has 2 isometries, like an asymmetric piece of blank paper.
• C_3v: if three faces, of which none opposite to each other, have one color and the other three one other color, the cube has 6 isometries.

For some larger subgroups a cube with that group as symmetry group is not possible with just coloring whole faces. One has to draw some pattern on the faces. Examples: In anatomy, the cuboid bone is a bone in the foot. ... In geometry, a prism is a polyhedron made of two parallel copies of some polygonal base joined by faces that are rectangles or parallelograms. ...

• D_2d: if one face has a line segment dividing the face into two equal rectangles, and the opposite has the same in perpendicular direction, the cube has 8 isometries; there is a symmetry plane and 2-fold rotational symmetry with an axis at an angle of 45° to that plane, and, as a result, there is also another symmetry plane perpendicular to the first, and another axis of 2-fold rotational symmetry perpendicular to the first.
• T_h: if each face has a line segment dividing the face into two equal rectangles, such that the line segments of adjacent faces do not meet at the edge, the cube has 24 isometries: the even permutations of the body diagonals and the same combined with inversion (x is mapped to −x).
• T_d: if the cube consists of eight smaller cubes, four white and four black, put together alternatingly in all three standard directions, the cube has again 24 isometries: this time the even permutations of the body diagonals and the inverses of the other proper rotations.
• T: if each face has the same pattern with 2-fold rotational symmetry, say the letter S, such that at all edges a top of one S meets a side of the other S, the cube has 12 isometries: the even permutations of the body diagonals.

The full symmetry of the cube (O_h) is preserved iff all faces have the same pattern such that the full symmetry of the square is preserved, with for the square a symmetry group of order 8. ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...

The full symmetry of the cube under proper rotations (O) is preserved iff all faces have the same pattern with 4-fold rotational symmetry.

Apart from text and colors, this volleyball has symmetry T_h

Image File history File links Photograph of a beach volleyball in sand, by photographer Douglas Bishop of itsawebguy. ... Image File history File links Photograph of a beach volleyball in sand, by photographer Douglas Bishop of itsawebguy. ...

Chiral solids with octahedral rotational symmetry

Snub hexahedron (Ccw)

Pentagonal icositetrahedron (Ccw) - note that, not very clear in the image, at some vertices 4 faces meet (in the edge of the image)

Archimedean solids: Download high resolution version (837x817, 96 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Download high resolution version (837x817, 96 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... The snub cube, or snub cuboctahedron, is an Archimedean solid, usually regarded as a truncated polyhedron derived by truncating either a cube or an octahedron. ... Download high resolution version (829x813, 69 KB)Pentagonal icositetrahedron (Ccw), made by me using POV-Ray, see image:poly. ... Download high resolution version (829x813, 69 KB)Pentagonal icositetrahedron (Ccw), made by me using POV-Ray, see image:poly. ... A pentagonal icositetrahedron is a catalan solid which looks a bit like an octahedron undergoing cell division. ... 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. ...

Name	picture	Faces		Edges	Vertices	Vertex configuration
snub cube or snub cuboctahedron (2 chiral forms)	( Video) ( Video)	38	32 triangles 6 squares	60	24	3,3,3,3,4

Catalan solids: The snub cube, or snub cuboctahedron, is an Archimedean solid, usually regarded as a truncated polyhedron derived by truncating either a cube or an octahedron. ... Download high resolution version (837x817, 96 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning truncated tetrahedron, made by me using POV-Ray, see image:poly. ... Download high resolution version (843x833, 97 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning truncated tetrahedron, made by me using POV-Ray, see image:poly. ... A rhombic dodecahedron In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. ...

Name	picture	Dual Archimedean solid	Faces	Edges	Vertices	Face Polygon
pentagonal icositetrahedron	( Video)( Video)	snub cube	24	60	38	irregular pentagon

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, usually regarded as a truncated polyhedron derived by truncating either a cube or an octahedron. ... In geometry, a pentagon is any five-sided polygon. ...

Solids with full octahedral symmetry

Platonic solids

Name	Picture	Faces	Edges	Vertices	Edges per face	Faces meeting at each vertex
cube (hexahedron)	( Animation) 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 only on each of its vertices. ... Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ... Download high resolution version (742x826, 50 KB)Hexahedron, made by me using POV-Ray, see image:poly. ... Spinning hexahedron, made by me using POV-Ray, see image:poly. ...	6	12	8	4	3
octahedron	( Animation)	8	12	6	3	4

Jump to: navigation, search An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Download high resolution version (862x862, 41 KB)Octahedron, made by me using POV-Ray, see image:poly. ... Spinning octahedron, made by me using POV-Ray, see image:poly. ...

Archimedean solids

(semi-regular: vertex-uniform) 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. ...

Name	picture	Faces		Edges	Vertices	Vertex configuration
cuboctahedron (quasi-regular: vertex- and edge-uniform)	( Video)	14	8 triangles 6 squares	24	12	3,4,3,4
truncated cube or truncated hexahedron	( Video)	14	8 triangles 6 octagons	36	24	3,8,8
truncated octahedron	( Video)	14	6 squares 8 hexagons	36	24	4,6,6
rhombicuboctahedron or small rhombicuboctahedron	( Video)	26	8 triangles 18 squares	48	24	3,4,4,4
truncated cuboctahedron or great rhombicuboctahedron	( Video)	26	12 squares 8 hexagons 6 octagons	72	48	4,6,8

A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Download high resolution version (818x804, 71 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning cuboctahedron, made by me using POV-Ray, see image:poly. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... In plane geometry, a square is a polygon with four equal sides and equal angles. ... The truncated cube, or truncated hexahedron, is an Archimedean solid. ... Download high resolution version (819x855, 67 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning truncated hexahedron, made by me using POV-Ray, see :image:poly. ... Jump to: navigation, search One of the 8 semi-regular tessellations: octagons and squares An octagon is a polygon that has eight sides. ... The truncated octahedron, also known as a Mecon, is an Archimedean solid. ... Download high resolution version (857x789, 67 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning truncated tetrahedron, 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. ... Download high resolution version (823x836, 83 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning truncated tetrahedron, made by me using POV-Ray, see :image:poly. ... The truncated cuboctahedron, or great rhombicuboctahedron, is an Archimedean solid. ... Download high resolution version (855x868, 83 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Spinning truncated tetrahedron, made by me using POV-Ray, see image:poly. ...

Catalan solids

(semi-regular duals: face-uniform) A rhombic dodecahedron In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. ...

Name	picture	Dual Archimedean solid	Faces	Edges	Vertices	Face polygon
rhombic dodecahedron (quasi-regular dual: face- and edge-uniform)	( Video)	cuboctahedron	12	24	14	rhombus
triakis octahedron	( Video)	truncated cube	24	36	14	isosceles triangle
tetrakis hexahedron	( Video)	truncated octahedron	24	36	14	isosceles triangle
deltoidal icositetrahedron	( Video)	rhombicuboctahedron	24	48	26	kite
disdyakis dodecahedron or hexakis octahedron	( Video)	truncated cuboctahedron	48	72	26	scalene triangle

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. ... Jump to: navigation, search 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). ... Jump to: navigation, search 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 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). ...

Other

stella octangula

Wikipedia does not have an article with this exact name. ...

See also

• Polyhedral compound