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A regular
octahedron
In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of i ...
has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A
cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
has the same set of symmetries, since it is the polyhedron that is
dual to an octahedron.
The group of orientation-preserving symmetries is S
4, the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
or the group of permutations of four objects, since there is exactly one such symmetry for each permutation of the four diagonals of the cube.
Details
Chiral and full (or achiral) octahedral symmetry are the
discrete point symmetries (or equivalently,
symmetries on the sphere) with the largest
symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
s compatible with
translational symmetry
In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation (without rotation). Discrete translational symmetry is invariant under discrete translation.
Analogously, an operato ...
. They are among the
crystallographic point groups of the
cubic crystal system
In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals.
There are three main varieties o ...
.
As the
hyperoctahedral group
A hyperoctahedral group is a type of mathematical Group (mathematics), group that arises as the symmetry group, group of symmetries of a hypercube or of a cross-polytope. It was named by Alfred Young (mathematician), Alfred Young in 1930. Groups ...
of dimension 3 the full octahedral group is the
wreath product
In group theory, the wreath product is a special combination of two groups based on the semidirect product. It is formed by the action of one group on many copies of another group, somewhat analogous to exponentiation. Wreath products are used ...
, and a natural way to identify its elements is as pairs with
is a combination of rotation and reflection.
Chiral octahedral symmetry
O, 432, or
,3sup>+ of order 24, is chiral octahedral symmetry or rotational octahedral symmetry . This group is like chiral tetrahedral symmetry T, but the C
2 axes are now C
4 axes, and additionally there are 6 C
2 axes, through the midpoints of the edges of the cube. T
d and O are isomorphic as abstract groups: they both correspond to S
4, the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
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
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
and the regular
octahedron
In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of i ...
.
Full octahedral symmetry
O
h, *432,
,3 or m3m 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 S
4.C
2, and is the full symmetry group of the
cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
and
octahedron
In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of i ...
. It is the
hyperoctahedral group
A hyperoctahedral group is a type of mathematical Group (mathematics), group that arises as the symmetry group, group of symmetries of a hypercube or of a cross-polytope. It was named by Alfred Young (mathematician), Alfred Young in 1930. Groups ...
for . See also
the isometries of the cube.
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
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
is given by , and the
octahedron
In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of i ...
by (or the corresponding inequalities, to get the solid instead of the surface).
gives a polyhedron with 48 faces, e.g. the disdyakis dodecahedron.
Faces are 8-by-8 combined to larger faces for (cube) and 6-by-6 for (octahedron).
The 9 mirror lines of full octahedral symmetry can be divided into two subgroups of 3 and 6 (drawn in purple and red), representing in two orthogonal subsymmetries:
D2h, and
Td. D
2h symmetry can be doubled to D
4h by restoring 2 mirrors from one of three orientations.
Rotation matrices
Take the set of all 3×3
permutation matrices
In mathematics, particularly in Matrix (mathematics), matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column with all other entries 0. An permutation matrix can represent a permu ...
and assign a + or − sign to each of the three 1s. There are
permutations and
sign combinations for a total of 48 matrices, giving the full octahedral group. 24 of these matrices have a
determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
of +1; these are the rotation matrices of the chiral octahedral group. The other 24 matrices have a determinant of −1 and correspond to a reflection or inversion.
Three reflectional generator matrices are needed for octahedral symmetry, which represent the three mirrors of a
Coxeter–Dynkin diagram
In geometry, a Harold Scott MacDonald Coxeter, Coxeter–Eugene Dynkin, Dynkin diagram (or Coxeter diagram, Coxeter graph) is a Graph (discrete mathematics), graph with numerically labeled edges (called branches) representing a Coxeter group or ...
. The product of the reflections produce 3 rotational generators.
Subgroups of full octahedral symmetry
The isometries of the cube
The cube has 48 isometries (symmetry elements), forming the
symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
O
h, isomorphic to
S4 × Z
2. They can be categorized as follows:
* O (the identity and 23 proper rotations) with the following
conjugacy class
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other ...
es (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'')/, 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'')/, 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
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A regular hexagon is de ...
).
An isometry of the cube can be identified in various ways:
* 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
A die (: dice, sometimes also used as ) is a small, throwable object with marked sides that can rest in multiple positions. Dice are used for generating random values, commonly as part of tabletop games, including dice games, board games, ro ...
have), the symmetry group is a subgroup of O
h.
Examples:
* ''C''
4v,
(*422): 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,
,2 (*222): if opposite faces have the same colors, different for each set of two, the cube has 8 isometries, like a
cuboid
In geometry, a cuboid is a hexahedron with quadrilateral faces, meaning it is a polyhedron with six Face (geometry), faces; it has eight Vertex (geometry), vertices and twelve Edge (geometry), edges. A ''rectangular cuboid'' (sometimes also calle ...
.
* ''D''
4h,
,2 (*422): 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,
(*22):
** 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,
nbsp; (*):
** 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,
(*33): 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:
* ''D''
2d,
+,4 (2*2): 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.
*
Th,
+,4 (3*2): 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,
,3 (*332): 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,
,3sup>+, (332): 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,
,3 (*432), is preserved
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
all faces have the same pattern such that the full symmetry of the
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 ...
is preserved, with for the square a symmetry group,
Dih4,
of order 8.
The full symmetry of the cube under proper rotations, O,
,3sup>+, (432), is preserved if and only if all faces have the same pattern with
4-fold rotational symmetry, Z
4,
sup>+.
Octahedral symmetry of the Bolza surface
In
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
theory, the
Bolza surface, sometimes called the Bolza curve, is obtained as the ramified double cover of the Riemann sphere, with ramification locus at the set of vertices of the regular inscribed octahedron. Its automorphism group includes the hyperelliptic involution which flips the two sheets of the cover. The quotient by the order 2 subgroup generated by the hyperelliptic involution yields precisely the group of symmetries of the octahedron. Among the many remarkable properties of the Bolza surface is the fact that it maximizes the
systole
Systole ( ) is the part of the cardiac cycle during which some chambers of the heart contract after refilling with blood. Its contrasting phase is diastole, the relaxed phase of the cardiac cycle when the chambers of the heart are refilling ...
among all genus 2 hyperbolic surfaces.
Solids with octahedral chiral symmetry
Solids with full octahedral symmetry
See also
*
Tetrahedral symmetry
image:tetrahedron.svg, 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry
A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that co ...
*
Icosahedral symmetry
In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual polyhedr ...
*
Binary octahedral group
*
Hyperoctahedral group
A hyperoctahedral group is a type of mathematical Group (mathematics), group that arises as the symmetry group, group of symmetries of a hypercube or of a cross-polytope. It was named by Alfred Young (mathematician), Alfred Young in 1930. Groups ...
*
References
Further reading
* Peter R. Cromwell, ''Polyhedra'' (1997), p. 295
* ''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss,
* ''Kaleidoscopes: Selected Writings of
H.S.M. Coxeter'', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
*
Norman Johnson (mathematician), N.W. Johnson: ''Geometries and Transformations'', (2018) Chapter 11: ''Finite symmetry groups'', 11.5 Spherical Coxeter groups
External links
*
* Groupprops
Direct product of S4 and Z2
{{DEFAULTSORT:Octahedral Symmetry
Finite groups
Rotational symmetry