Meta tags:

Headings (most frequently used words):

octahedral, symmetry, the, of, full, solids, with, chiral, contents, details, isometries, cube, bolza, surface, see, also, references, further, reading, external, links, rotation, matrices, subgroups,

Text of the page (most frequently used words):
the (192), #symmetry (59), and (55), cube (42), with (35), octahedral (34), group (32), displaystyle (26), faces (25), has (24), fold (21), have (20), #rotation (20), other (18), isometries (18), edit (17), full (17), phantom (17), octahedron (15), two (15), axis (15), same (14), one (14), smallmatrix (14), for (13), are (13), about (12), each (12), color (12), this (11), face (11), three (11), inversion (11), opposite (11), subgroups (11), axes (11), all (10), reflection (10), from (9), rotational (9), symmetries (9), chiral (9), that (9), subgroup (8), 432 (8), body (8), four (8), plane (8), angle (8), different (8), matrices (8), mirrors (8), wikipedia (7), groups (7), also (7), dual (7), order (7), permutations (7), perpendicular (7), like (7), left (7), begin (7), end (7), right (7), solid (6), surface (6), diagonals (6), colors (6), can (6), etc (6), elements (6), rotoreflection (6), projection (6), page (5), coxeter (5), tetrahedral (5), regular (5), bolza (5), set (5), square (5), 332 (5), together (5), combined (5), there (5), mirror (5), 180 (5), identity (5), dihedral (5), article (5), toggle (4), contents (4), search (4), citations (4), product (4), links (4), see (4), dodecahedron (4), picture (4), name (4), edges (4), solids (4), which (4), rotations (4), pattern (4), such (4), adjacent (4), not (4), 222 (4), symmetric (4), given (4), 120 (4), per (4), center (4), fundamental (4), domain (4), but (4), mathrm (4), hide (4), move (4), sidebar (4), view (3), text (3), may (3), image (3), articles (3), 978 (3), isbn (3), references (3), wikiversity (3), hyperoctahedral (3), polyhedron (3), disdyakis (3), truncated (3), vertices (3), among (3), proper (3), preserved (3), top (3), even (3), line (3), into (3), edge (3), mapped (3), examples (3), 422 (3), permutation (3), orientation (3), diagonal (3), isomorphic (3), they (3), cycle (3), these (3), stereographic (3), gives (3), point (3), details (3), tools (3), main (3), languages (2), table (2), contact (2), privacy (2), policy (2), under (2), terms (2), using (2), non (2), wikimedia (2), commons (2), was (2), last (2), edited (2), categories (2), pages (2), lacking (2), 2013 (2), short (2), description (2), wikidata (2), finite (2), index (2), direct (2), external (2), spherical (2), transformations (2), peter (2), wiley (2), selected (2), john (2), conway (2), things (2), further (2), reading (2), related (2), icosahedral (2), self (2), compound (2), cuboctahedron (2), icositetrahedron (2), catalan (2), archimedean (2), class (2), obtained (2), cover (2), riemann (2), sphere (2), its (2), includes (2), hyperelliptic (2), involution (2), only (2), say (2), cubes (2), black (2), segment (2), dividing (2), equal (2), rectangles (2), another (2), first (2), some (2), larger (2), possible (2), just (2), piece (2), blank (2), paper (2), marking (2), combination (2), itself (2), conjugacy (2), classes (2), notation (2), 224 (2), mmm (2), reflective (2), diagram (2), generators (2), reflections (2), sign (2), determinant (2), correspond (2), orthographic (2), orthogonal (2), object (2), corresponding (2), get (2), planes (2), both (2), achiral (2), objects (2), circ (2), applied (2), represented (2), identify (2), inversions (2), list (2), preserving (2), since (2), here (2), learn (2), help (2), general (2), appearance (2), upload (2), file (2), changes (2), history (2), read (2), log (2), create (2), account (2), donate (2), menu (2), add, topic, mobile, cookie, statement, statistics, developers, code, conduct, legal, safety, contacts, disclaimers, available, additional, apply, site, you, agree, registered, trademark, profit, organization, foundation, inc, use, creative, attribution, sharealike, license, rendered, parsoid, september, 2026, utc, hidden, multiple, manual, scaled, images, matches, retrieved, https, org, php, title, octahedral_symmetry, oldid, 1375167150, groupprops, mathworld, weisstein, eric, 2018, chapter, 107, 10340, geometries, johnson, arthur, sherk, mcmullen, anthony, thompson, asia, ivić, weiss, interscience, publication, 1995, 471, 01003, com, kaleidoscopes, writings, heidi, burgiel, chaim, goodman, strauss, 2008, 56881, 220, cromwell, 1997, 295, polyhedra, fig, 280, learning, materials, great, circles, binary, stellated, deltoidal, rhombicuboctahedron, tetrakis, hexahedron, triakis, rhombic, platonic, pentagonal, snub, theory, sometimes, called, curve, ramified, double, ramification, locus, inscribed, automorphism, flips, sheets, quotient, generated, yields, precisely, many, remarkable, properties, fact, maximizes, genus, hyperbolic, surfaces, systole, dih, letter, meets, side, consists, eight, smaller, white, put, alternatingly, standard, directions, again, time, inverses, segments, meet, direction, result, coloring, whole, draw, none, asymmetric, third, shape, box, prism, cuboid, markings, dice, whether, normal, die, isometry, identified, various, ways, note, does, map, intersection, hexagon, following, parentheses, ditto, unit, quaternion, representation, forming, categorized, follows, 322, 223, mm2, 4mm, cyc, structure, orb, schoenflies, containing, ordered, hasse, graphs, matrix, reflectional, generator, needed, represent, produce, dynkin, take, assign, combinations, total, giving, lines, divided, drawn, purple, red, representing, subsymmetries, doubled, restoring, orientations, coordinate, characterized, part, example, inequalities, instead, comprising, m3m, now, additionally, through, midpoints, abstract, union, combining, element, gyration, illustration, rotoreflections, dimension, natural, way, pairs, simply, their, simeq, rtimes, wreath, complete, found, equivalently, largest, compatible, cubic, crystal, system, crystallographic, translational, discrete, exactly, altogether, include, combine, hexagonal, cycles, knot, common, hexagons, graph, redirects, mathematical, pertaining, specifically, 3x3x3, twisty, puzzle, rubik, 532, n32, polyhedral, n22, cyclic, involutional, dimensions, how, when, remove, message, please, more, precise, introducing, improve, lacks, sufficient, inline, free, encyclopedia, item, projects, printable, version, download, pdf, print, export, switch, legacy, parser, shortened, url, cite, information, permanent, link, what, actions, english, talk, slovenščina, română, 한국어, bahasa, indonesia, suomi, español, esperanto, deutsch, subsection, personal, special, recent, community, portal, contribute, random, current, events, navigation, jump, content,

Text of the page (random words):
n n dihedral symmetry d nh n22 n 2 polyhedral group n 3 n32 tetrahedral symmetry t d 332 3 3 octahedral symmetry o h 432 4 3 icosahedral symmetry i h 532 5 3 cube group redirects here for the mathematical group pertaining specifically to the 3x3x3 twisty puzzle cube see rubik s cube group cycle graph the four hexagonal cycles have the inversion the black knot on top in common the hexagons are symmetric so e g 3 and 4 are in the same cycle a regular octahedron has 24 rotational or orientation preserving symmetries and 48 symmetries altogether these include transformations that combine a reflection and a rotation a cube 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 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 edit chiral and full or 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 conjugacy classes elements of o inversions of elements of o identity 0 inversion 0 3 rotation by 180 about a 4 fold axis 7 16 23 3 reflection in a plane perpendicular to a 4 fold axis 7 16 23 8 rotation by 120 about a 3 fold axis 3 4 8 11 12 15 19 20 8 rotoreflection by 60 3 4 8 11 12 15 19 20 6 rotation by 180 about a 2 fold axis 1 2 5 6 14 21 6 reflection in a plane perpendicular to a 2 fold axis 1 2 5 6 14 21 6 rotation by 90 about a 4 fold axis 9 10 13 17 18 22 6 rotoreflection by 90 9 10 13 17 18 22 examples 0 0 0 displaystyle 0 0 0 3 0 7 displaystyle 3 0 7 0 3 3 displaystyle 0 3 3 7 1 1 displaystyle 7 1 1 4 5 9 displaystyle 4 5 9 7 0 0 displaystyle 7 0 0 4 0 7 displaystyle 4 0 7 7 3 3 displaystyle 7 3 3 0 1 1 displaystyle 0 1 1 3 5 9 displaystyle 3 5 9 a complete list can be found in the wikiversity article as the hyperoctahedral group of dimension 3 the full octahedral group is the wreath product s 2 s 3 s 2 3 s 3 displaystyle mathrm s _ 2 wr mathrm s _ 3 simeq mathrm s _ 2 3 rtimes mathrm s _ 3 and a natural way to identify its elements is as pairs m n with m 0 2 3 displaystyle m in 0 2 3 and n 0 3 displaystyle n in 0 3 but as it is also the direct product s 4 s 2 one can simply identify the elements of tetrahedral subgroup t d as a 0 4 displaystyle a in 0 4 and their inversions as a displaystyle a so e g the identity 0 0 is represented as 0 and the inversion 7 0 as 0 3 1 is represented as 6 and 4 1 as 6 a rotoreflection is a combination of rotation and reflection illustration of rotoreflections the reflection 7 applied on the 120 rotation 4 gives the 60 rotoreflection 8 7 4 8 displaystyle 7 circ 4 8 the reflection 7 applied on the 90 rotation 22 gives the 90 rotoreflection 17 7 22 17 displaystyle 7 circ 22 17 chiral octahedral symmetry edit gyration axes c 4 c 3 c 2 3 4 6 o 432 or 4 3 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 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 chiral octahedral symmetry orthogonal projection stereographic projection 2 fold 4 fold 3 fold 2 fold full octahedral symmetry edit o h 432 4 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 and octahedron it is the hyperoctahedral group for n 3 see also the isometries of the cube compound of cube and octahedron each face of the disdyakis dodecahedron is a fundamental domain 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 faces are 8 by 8 combined to larger faces for a b 0 cube and 6 by 6 for a b c 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 d 2h and t d d 2h symmetry can be doubled to d 4h by restoring 2 mirrors from one of three orientations octahedral symmetry and reflective subgroups orthographic projection stereographic projection 4 fold 3 fold 2 fold o h 4 3 full octahedral symmetry 3 6 mirrors t d 3 3 1 4 3 tetrahedral subgroup 6 mirrors c 3v 3 dihedral subgroup 3 mirrors orthographic projection stereographic projection 4 fold 3 fold 2 fold d 4h 4 2 dihedral subgroup 1 2 2 mirrors d 2h 2 2 4 3 dihedral subgroup 1 1 1 mirrors c 4v 4 dihedral subgroup 2 2 mirrors rotation matrices edit take the set of all 3 3 permutation matrices and assign a or sign to each of the three 1s there are 3 6 displaystyle 3 6 permutations and 2 3 8 displaystyle 2 3 8 sign combinations for a total of 48 matrices giving the full octahedral group 24 of these matrices have a determinant 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 the product of the reflections produce 3 rotational generators 4 3 reflections rotations rotoreflection generators r 0 r 1 r 2 r 0 r 1 r 1 r 2 r 0 r 2 r 0 r 1 r 2 group order 2 2 2 4 3 2 6 matrix 1 0 0 0 1 0 0 0 1 displaystyle left begin smallmatrix 1 phantom 0 phantom 0 0 phantom 1 phantom 0 0 phantom 0 1 end smallmatrix right 1 0 0 0 0 1 0 1 0 displaystyle left begin smallmatrix 1 0 0 0 0 1 0 1 0 end smallmatrix right 0 1 0 1 0 0 0 0 1 displaystyle left begin smallmatrix 0 1 0 1 0 0 0 0 1 end smallmatrix right 1 0 0 0 0 1 0 1 0 displaystyle left begin smallmatrix 1 phantom 0 phantom 0 0 phantom 0 phantom 1 0 1 phantom 0 end smallmatrix right 0 1 0 0 0 1 1 0 0 displaystyle left begin smallmatrix 0 1 0 0 0 1 1 0 0 end smallmatrix right 0 1 0 1 0 0 0 0 1 displaystyle left begin smallmatrix 0 phantom 1 phantom 0 1 phantom 0 phantom 0 0 phantom 0 1 end smallmatrix right 0 1 0 0 0 1 1 0 0 displaystyle left begin smallmatrix phantom 0 1 0 phantom 0 0 1 1 0 0 end smallmatrix right subgroups of full octahedral symmetry edit o t d t h cycle graphs of subgroups of order 24 subgroups ordered in a hasse diagram rotational subgroups reflective subgroups subgroups containing inversion schoenflies notation coxeter orb h m structure cyc order index o h 4 3 432 m 3 m s 4 s 2 48 1 t d 3 3 332 4 3m s 4 24 2 d 4h 2 4 224 4 mmm d 2 d 8 16 3 d 2h 2 2 222 mmm d 3 2 d 2 d 4 8 6 c 4v 4 44 4mm d 8 8 6 c 3v 3 33 3m d 6 s 3 6 8 c 2v 2 22 mm2 d 2 2 d 4 4 12 c s c 1v 2 or m d 2 2 24 t h 3 4 3 2 m 3 a 4 s 2 24 2 c 4h 4 2 4 4 m z 4 d 2 8 6 d 3d 2 6 2 3 3 m d 12 z 2 d 6 12 4 d 2d 2 4 2 2 4 2m d 8 8 6 c 2h d 1d 2 2 2 2 m z 2 d 2 4 12 s 6 2 6 3 3 z 6 z 2 z 3 6 8 s 4 2 4 2 4 z 4 4 12 s 2 2 2 1 s 2 2 24 o 4 3 432 432 s 4 24 2 t 3 3 332 23 a 4 12 4 d 4 2 4 224 422 d 8 8 6 d 3 2 3 223 322 d 6 s 3 6 8 d 2 2 2 222 222 d 4 z 2 2 4 12 c 4 4 44 4 z 4 4 12 c 3 3 33 3 z 3 a 3 3 16 c 2 2 22 2 z 2 2 24 c 1 11 1 z 1 1 48 octahedral subgroups in coxeter notation 1 the isometries of the cube edit 48 symmetry elements of a cube the cube has 48 isometries symmetry elements forming the symmetry group o h isomorphic to s 4 z 2 they can be categorized as follows 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 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 c 4v 4 422 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 2 222 if opposite faces have the same colors different for each set of two the cube has 8 isometries like a cuboid d 4h 4 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 2 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 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 3 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 2 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 t h 3 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 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 3 3 332 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 4 3 432 is preserved if and only if all faces have the same pattern such that the full symmetry of the square is preserved with for the square a symmetry group dih 4 4 of order 8 the full symmetry of the cube under proper rotations o 4 3 432 is preserved if and only if all faces have the same pattern with 4 fold rotational symmetry z 4 4 octahedral symmetry of the bolza surface edit in riemann surface 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 among all genus 2 hyperbolic surfaces solids with octahedral chiral symmetry edit class name picture faces edges vertices dual name picture archimedean solid dual catalan solid snub cube 38 60 24 pentagonal icositetrahedron solids with full octahedral symmetry edit class name picture faces edges vertices dual name picture platonic solid cube 6 12 8 octahedron archimedean solid dual catalan solid cuboctahedron 14 24 12 rhombic dodecahedron truncated cube 14 36 24 triakis octahedron truncated octahedron 14 36 24 tetrakis hexahedron rhombicuboctahedron 26 48 24 deltoidal icositetrahedron truncated cuboctahedron 26 72 48 disdyakis dodecahedron regular compound polyhedron stellated octahedron 8 12 8 self dual cube and octahedron 14 24 14 self dual see also edit tetrahedral symmetry icosahedral symmetry binary octahedral group hyperoctahedral group 25 great circles of the spherical octahedron learning materials related to full octahedral group at wikiversity references edit john conway the symmetries of things fig 20 8 p 280 further reading edit peter r cromwell 1997 polyhedra p 295 john h conway heidi burgiel chaim goodman strauss 2008 the symmetries of things isbn 978 1 56881 220 5 kaleidoscopes selected writings of h s m coxeter edited by f arthur sherk peter mcmullen anthony c thompson asia ivić weiss wiley interscience publication 1995 wiley com isbn 978 0 471 01003 6 n w johnson geometries and transformations 2018 chapter 11 finite symmetry groups 11 5 spherical c...