geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a

point group In geometry, a point group is a group (mathematics), mathematical group of symmetry operations (isometry, isometries in a Euclidean space) that have a Fixed point (mathematics), fixed point in common. The Origin (mathematics), coordinate origin o ...

in four dimensions is an

isometry group In mathematics, the isometry group of a metric space is the set of all bijective isometries (that is, bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element ...

in four dimensions that leaves the origin fixed, or correspondingly, an isometry group of a

3-sphere In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere, ''n''-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point. The interior o ...

History on four-dimensional groups

* 1889

Édouard Goursat Édouard Jean-Baptiste Goursat (21 May 1858 – 25 November 1936) was a French mathematician, now remembered principally as an expositor for his ''Cours d'analyse mathématique'', which appeared in the first decade of the twentieth century. It s ...

, ''Sur les substitutions orthogonales et les divisions régulières de l'espace'', Annales Scientifiques de l'École Normale Supérieure, Sér. 3, 6, (pp. 9–102, pp. 80–81 tetrahedra),

Goursat tetrahedron In geometry, a Goursat tetrahedron is a tetrahedron, tetrahedral fundamental domain of a Wythoff construction. Each tetrahedral face represents a reflection hyperplane on 3-dimensional surfaces: the 3-sphere, Euclidean 3-space, and hyperbolic 3-spa ...

* 1951, A. C. Hurley, ''Finite rotation groups and crystal classes in four dimensions'', Proceedings of the Cambridge Philosophical Society, vol. 47, issue 04, p. 650 * 1962 A. L. MacKay ''Bravais Lattices in Four-dimensional Space'' * 1964

Patrick du Val Patrick du Val (March 26, 1903 – January 22, 1987) was a British mathematician, known for his work on algebraic geometry, differential geometry, and general relativity. The concept of Du Val singularity of an algebraic surface is named af ...

, ''Homographies, quaternions and rotations'',

quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...

-based 4D point groups * 1975 Jan Mozrzymas, Andrzej Solecki, ''R4 point groups'', Reports on Mathematical Physics, Volume 7, Issue 3, p. 363-394 * 1978 H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus, ''Crystallographic Groups of Four-Dimensional Space.'' * 1982 N. P. Warner, ''The symmetry groups of the regular tessellations of S2 and S3'' * 1985 E. J. W. Whittaker, ''An atlas of hyperstereograms of the four-dimensional crystal classes'' * 1985 H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', Coxeter notation for 4D point groups * 2003 John Conway and Smith, ''On Quaternions and Octonions'', Completed

-based 4D point groups * 2018 N. W. Johnson ''Geometries and Transformations'', Chapter 11,12,13, Full polychoric groups, p. 249, duoprismatic groups p. 269

Isometries of 4D point symmetry

There are four basic isometries of 4-dimensional point symmetry:

reflection symmetry In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a Reflection (mathematics), reflection. That is, a figure which does not change upon undergoing a reflection has reflecti ...

rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape (geometry), shape has when it looks the same after some rotation (mathematics), rotation by a partial turn (angle), turn. An object's degree of rotational s ...

rotoreflection In geometry, an improper rotation. (also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion) is an isometry in Euclidean space that is a combination of a Rotation (geometry), rotation about an axis and a reflection ( ...

, and

double rotation In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4). The name comes from the fact that it is the special orthogonal group of order 4. In this article ''rotation'' means ''rotational dis ...

Notation for groups

Point groups in this article are given in

Coxeter notation In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, ...

, which are based on

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...

s, with markups for extended groups and subgroups. Coxeter notation has a direct correspondence the Coxeter diagram like ,3,3 ,3,3 ^1,1,1 ,4,3 ,3,3 and ,2,q These groups bound the

into identical hyperspherical tetrahedral domains. The number of domains is the order of the group. The number of mirrors for an irreducible group is ''nh/2'', where ''h'' is the Coxeter group's Coxeter number, ''n'' is the dimension (4). For cross-referencing, also given here are

based notations by

(1964) and John Conway (2003). Conway's notation allows the order of the group to be computed as a product of elements with chiral polyhedral group orders: (T=12, O=24, I=60). In Conway's notation, a (±) prefix implies

central inversion In geometry, a point reflection (also called a point inversion or central inversion) is a geometric transformation of affine space in which every point (geometry), point is reflected across a designated inversion center, which remains Fixed p ...

, and a suffix (.2) implies mirror symmetry. Similarly Du Val's notation has an asterisk (*) superscript for mirror symmetry.

Involution groups

There are five

involution Involution may refer to: Mathematics * Involution (mathematics), a function that is its own inverse * Involution algebra, a *-algebra: a type of algebraic structure * Involute, a construction in the differential geometry of curves * Exponentiati ...

al groups: no symmetry nbsp;sup>+,

nbsp; 2-fold

sup>+, 2-fold

⁺,2⁺ and central point symmetry ⁺,2⁺,2⁺as a 2-fold

Rank 4 Coxeter groups

A polychoric group is one of five

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 of the 4-dimensional

regular polytope In mathematics, a regular polytope is a polytope whose symmetry group acts transitive group action, transitively on its flag (geometry), flags, thus giving it the highest degree of symmetry. In particular, all its elements or -faces (for all , w ...

s. There are also three polyhedral prismatic groups, and an infinite set of duoprismatic groups. Each group defined by a

fundamental domain Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each ...

bounded by mirror planes. The dihedral angles between the mirrors determine order of

dihedral symmetry In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, g ...

. The

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 ...

is a graph where nodes represent mirror planes, and edges are called branches, and labeled by their dihedral angle order between the mirrors. The term ''polychoron'' (plural ''polychora'', adjective ''polychoric''), from the

Greek Greek may refer to: Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group *Greek language, a branch of the Indo-European language family **Proto-Greek language, the assumed last common ancestor of all kno ...

roots ''poly'' ("many") and ''choros'' ("room" or "space") and was advocated by Norman Johnson and George Olshevsky in the context of uniform polychora (4-polytopes), and their related 4-dimensional symmetry groups. Rank 4

s allow a set of 4 mirrors to span 4-space, and divides the

into tetrahedral fundamental domains. Lower rank Coxeter groups can only bound

hosohedron In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular -gonal hosohedron has Schläfli symbol with each spherical lune ha ...

or hosotope fundamental domains on the 3-sphere. Like the 3D

polyhedral group In geometry, the polyhedral groups are the symmetry groups of the Platonic solids. Groups There are three polyhedral groups: *The Tetrahedral symmetry, tetrahedral group of order 12, rotational symmetry group of the tetrahedron, regular tetrahe ...

s, the names of the 4D polychoric groups given are constructed by the Greek prefixes of the cell counts of the corresponding triangle-faced regular polytopes. Extended symmetries exist in uniform polychora with symmetric ring-patterns within the

Coxeter diagram Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated ...

construct. Chiral symmetries exist in alternated uniform polychora. Only irreducible groups have Coxeter numbers, but duoprismatic groups ,2,pcan be doubled to p,2,p by adding a 2-fold gyration to the fundamental domain, and this gives an effective Coxeter number of 2''p'', for example the ,2,4and its full symmetry B₄, ,3,3group with Coxeter number 8. The symmetry order is equal to the number of cells of the regular polychoron times the symmetry of its cells. The omnitruncated dual polychora have cells that match the fundamental domains of the symmetry group.

Chiral subgroups

Direct subgroups of the reflective 4-dimensional point groups are:

Pentachoric symmetry

* Pentachoric group – A₄, ,3,3 (), order 120, (Du Val #51' (I^†/C₁;I/C₁)^†*, Conway +¹/₆₀ ×I2₁), named for the

5-cell In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional space, four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, hypertetrahedron, pentachoron, pentatope, pe ...

(pentachoron), given by ringed

. It is also sometimes called the hyper-tetrahedral group for extending the tetrahedral group ,3 There are 10 mirror hyperplanes in this group. It is isomorphic to the abstract

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 ...

, S₅. **The extended pentachoric group, Aut(A₄), , (The doubling can be hinted by a folded diagram, ), order 240, (Du Val #51 (I^†*/C₂;I/C₂)^†*, Conway ±¹/₆₀ ×2). It is isomorphic to the direct product of abstract groups: S₅×C₂. *** The chiral extended pentachoric group is ⁺, (), order 120, (Du Val #32 (I^†/C₂;I/C₂)^†, Conway ±¹/₆₀ x. This group represents the construction of the omnisnub 5-cell, , although it can not be made uniform. It is isomorphic to the direct product of abstract groups: A₅×C₂. **The chiral pentachoric group is ,3,3sup>+, (), order 60, (Du Val #32' (I^†/C₁;I/C₁)^†, Conway +¹/₆₀ ×. It is isomorphic to the abstract

alternating group In mathematics, an alternating group is the Group (mathematics), group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted ...

, A₅. *** The extended chiral pentachoric group is ,3,3sup>+">,3,3sup>+ order 120, (Du Val #51" (I^†/C₁;I/C₁)_–^†*, Conway +¹/₆₀ xI2₃). Coxeter relates this group to the abstract group (4,6, 2,3).Coxeter, ''The abstract groups G^m;n;p'', (1939) It is also isomorphic to the abstract

, S₅.

Hexadecachoric symmetry

* Hexadecachoric group – B₄, ,3,3 (), order 384, (Du Val #47 (O/V;O/V)^*, Conway ±¹/₆ ×O2), named for the

16-cell In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the ...

(hexadecachoron), . There are 16 mirror hyperplanes in this group, which can be identified in 2 orthogonal sets: 12 from a ^1,1,1subgroup, and 4 from a ,2,2subgroup. It is also called a hyper-octahedral group for extending the 3D

octahedral group 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 polyhedr ...

,3 and the tesseractic group for the

tesseract In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. Just as the perimeter of the square consists of four edges and the surface of the cube consists of six ...

, . ** The chiral hexadecachoric group is ,3,3sup>+, (), order 192, (Du Val #27 (O/V;O/V), Conway ±¹/₆ ×O. This group represents the construction of an omnisnub tesseract, , although it can not be made uniform. ** The ionic diminished hexadecachoric group is ,(3,3)⁺ (), order 192, (Du Val #41 (T/V;T/V)^*, Conway ±¹/₃ ×T2). This group leads to the

snub 24-cell In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular Tetrahedron, tetrahedral and 24 Regular icosahedron, icosahedral cell (mathematics), cells. Five tetrahedra and three icosahedra meet ...

with construction . ** The half hexadecachoric group is ⁺,4,3,3 ( = ), order 192, and same as the #demitesseractic symmetry: ^1,1,1 This group is expressed in the

alternated construction of the

, = . *** The group ⁺,4,(3,3)⁺ ( = ), order 96, and same as the chiral demitesseractic group ^1,1,1sup>+ and also is the

commutator subgroup In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal ...

of ,3,3 ** A high-index reflective subgroup is the ''prismatic octahedral symmetry'', ,3,2(), order 96, subgroup index 4, (Du Val #44 (O/C₂;O/C₂)^*, Conway ±¹/₂₄ ×O2). The truncated cubic prism has this symmetry with Coxeter diagram and the cubic prism is a lower symmetry construction of the

, as . *** Its chiral subgroup is ,3,2sup>+, (), order 48, (Du Val #26 (O/C₂;O/C₂), Conway ±¹/₂₄ ×O. An example is the snub cubic antiprism, , although it can not be made uniform. *** The ionic subgroups are: **** 3,4)⁺,2 (), order 48, (Du Val #44b' (O/C₁;O/C₁)₋^*, Conway +¹/₂₄ ×O2₁). The snub cubic prism has this symmetry with Coxeter diagram . ***** 3,4)⁺,2⁺ (), order 24, (Du Val #44' (T/C₂;T/C₂)₋^*, Conway +¹/₁₂ ×T2₁). **** ,3⁺,2 (), order 48, (Du Val #39 (T/C₂;T/C₂)_c^*, Conway ±¹/₁₂ ×T2). ***** ,3⁺,2,1⁺= ,3⁺,1= ,3⁺ ( = ), order 24, (Du Val #44" (T/C₂;T/C₂)^*, Conway +¹/₁₂ ×T2₃). This is the 3D '' pyritohedral group'', ,3⁺ ***** ⁺,4,2⁺ (), order 24, (Du Val #21 (T/C₂;T/C₂), Conway ±¹/₁₂ ×T. **** ,4,2⁺ (), order 48, (Du Val #39' (T/C₂;T/C₂)₋^*, Conway ±¹/₁₂ ×2). **** ,(3,2)⁺ (), order 48, (Du Val #40b' (O/C₁;O/C₁)₋^*, Conway +¹/₂₄ ×2₁). *** A half subgroup ,3,2,1⁺= ,3,1= ,3 ( = ), order 48 (Du Val #44b" (O/C₁;O/C₁)_c^*, Conway +¹/₂₄ ×O2₃). It is called the octahedral pyramidal group and is 3D ''

octahedral symmetry 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 polyhedr ...

'', ,3 A

cubic pyramid In 4-dimensional geometry, the cubic pyramid is bounded by one cube on the base and 6 square pyramid cell (mathematics), cells which meet at the Apex (geometry), apex. Since a cube has a circumradius divided by edge length less than one, the squ ...

can have this symmetry, with

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

: ( ) ∨ .

**** A chiral half subgroup 4,3)⁺,2,1⁺= ,3,1sup>+ = ,3sup>+, ( = ), order 24 (Du Val #26b' (O/C₁;O/C₁), Conway +¹/₂₄ ×O. This is the 3D ''chiral octahedral group'', ,3sup>+. A snub cubic pyramid can have this symmetry, with Schläfli symbol: ( ) ∨ sr. ** Another high-index reflective subgroup is the ''prismatic tetrahedral symmetry'', ,3,2 (), order 48, subgroup index 8, (Du Val #40b" (O/C₁;O/C₁)^*, Conway +¹/₂₄ ×2₃). *** The chiral subgroup is ,3,2sup>+, (), order 24, (Du Val #26b" (O/C₁;O/C₁), Conway +¹/₂₄ ×. An example is the snub tetrahedral antiprism, , although it can not be made uniform. *** The ionic subgroup is 3,3)⁺,2 (), order 24, (Du Val #39b' (T/C₁;T/C₁)_c^*, Conway +¹/₁₂ ×2₃). An example is the snub tetrahedral prism, . *** The half subgroup is ,3,2,1⁺= ,3,1= ,3 ( = ), order 24, (Du Val #39b" (T/C₁;T/C₁)₋^*, Conway +¹/₁₂ ×2₁). It is called the tetrahedral pyramidal group and is the 3D '' tetrahedral group'', ,3 A regular tetrahedral pyramid can have this symmetry, with Schläfli symbol: ( ) ∨ . Sphere symmetry group td

**** The chiral half subgroup 3,3)⁺,2,1⁺= ,3sup>+( = ), order 12, (Du Val #21b' (T/C₁;T/C₁), Conway +¹/₁₂ ×T. This is the 3D ''chiral tetrahedral group'', ,3sup>+. A snub tetrahedral pyramid can have this symmetry, with Schläfli symbol: ( ) ∨ sr. ** Another high-index radial reflective subgroup is ,(3,3)^* index 24, removes mirrors with order-3 dihedral angles, creating ,2,2(), order 16. Others are ,2,4(), ,2,2(), with subgroup indices 6 and 12, order 64 and 32. These groups are lower symmetries of the

: (), (), and (). These groups are #duoprismatic symmetry.

Icositetrachoric symmetry

* Icositetrachoric group – F₄, ,4,3 (), order 1152, (Du Val #45 (O/T;O/T)^*, Conway ±¹/₂ xO2), named for the

24-cell In four-dimensional space, four-dimensional geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octa ...

(icositetrachoron), . There are 24 mirror planes in this symmetry, which can be decomposed into two orthogonal sets of 12 mirrors in demitesseractic symmetry ^1,1,1subgroups, as ^*,4,3and ,4,3^* as index 6 subgroups. **The extended icositetrachoric group, Aut(F₄), , () has order 2304, (Du Val #48 (O/O;O/O)^*, Conway ± ×O2). *** The chiral extended icositetrachoric group, ⁺, () has order 1152, (Du Val #25 (O/O;O/O), Conway ± xO. This group represents the construction of the omnisnub 24-cell, , although it can not be made uniform. **The ionic diminished icositetrachoric groups, ⁺,4,3and ,4,3⁺ ( or ), have order 576, (Du Val #43 (T/T;T/T)^*, Conway ± ×T2). This group leads to the

with construction or . ***The double diminished icositetrachoric group, ⁺,4,3⁺(the double diminishing can be shown by a gap in the diagram 4-branch: ), order 288, (Du Val #20 (T/T;T/T), Conway ± ×T is the

of ,4,3 **** It can be extended as 3⁺,4,3⁺, () order 576, (Du Val #23 (T/T;O/O), Conway ± xT. ** The chiral icositetrachoric group is ,4,3sup>+, (), order 576, (Du Val #28 (O/T;O/T), Conway ±¹/₂ ×O. *** The extended chiral icositetrachoric group, ,4,3sup>+">,4,3sup>+has order 1152, (Du Val #46 (O/T;O/T)₋^*, Conway ±¹/₂ xO). Coxeter relates this group to the abstract group (4,8, 2,3).

Demitesseractic symmetry

* Demitesseractic group – D₄, ^1,1,1 ,3^1,1or ,3,4,1⁺ ( = ), order 192, (Du Val #42 (T/V;T/V)₋^*, Conway ±¹/₃ ×2), named for the (demitesseract) 4-demicube construction of the 16-cell, or . There are 12 mirrors in this symmetry group. **There are two types of extended symmetries by adding mirrors: < ,3^1,1 which becomes ,3,3by bisecting the fundamental domain by a mirror, with 3 orientations possible; and the full extended group [3^1,1,1 becomes ,4,3 ** The chiral demitesseractic group is ^1,1,1sup>+ or ⁺,4,(3,3)⁺ ( = ), order 96, (Du Val #22 (T/V;T/V), Conway ±¹/₃ ×T. This group leads to the

with construction = .

Hexacosichoric symmetry

* Hexacosichoric group – H₄, ,3,3 (), order 14400, (Du Val #50 (I/I;I/I)^*, Conway ± ×I2), named for the 600-cell (hexacosichoron), . It is also sometimes called the hyper-icosahedral group for extending the 3D

icosahedral group 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 of th ...

,3 and hecatonicosachoric group or dodecacontachoric group from the

120-cell In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hec ...

, . ** The chiral hexacosichoric group is ,3,3sup>+, (), order 7200, (Du Val #30 (I/I;I/I), Conway ± ×I. This group represents the construction of the snub 120-cell, , although it can not be made uniform. ** A high-index reflective subgroup is the ''prismatic icosahedral symmetry'', ,3,2 (), order 240, subgroup index 60, (Du Val #49 (I/C₂;I/C₂)^*, Conway ±¹/₆₀ xI2). *** Its chiral subgroup is ,3,2sup>+, (), order 120, (Du Val #31 (I/C₂;I/C₂), Conway ±¹/₆₀ xI. This group represents the construction of the snub dodecahedral antiprism, , although it can't be made uniform. *** An ionic subgroup is 5,3)⁺,2 (), order 120, (Du Val #49' (I/C₁;I/C₁)^*, Conway +¹/₆₀ xI2₁). This group represents the construction of the snub dodecahedral prism, . *** A half subgroup is ,3,2,1⁺= ,3,1= ,3 ( = ), order 120, (Du Val #49" (I/C₁;I/C₁)₋^*, Conway +¹/₆₀ xI2₃). It is called the icosahedral pyramidal group and is the 3D ''

'', ,3 A regular dodecahedral pyramid can have this symmetry, with

: ( ) ∨ . **** A chiral half subgroup is 5,3)⁺,2,1⁺= ,3,1sup>+ = ,3sup>+, ( = ), order 60, (Du Val #31' (I/C₁;I/C₁), Conway +¹/₆₀ xI. This is the 3D ''chiral icosahedral group'', ,3sup>+. A snub dodecahedral pyramid can have this symmetry, with

: ( ) ∨ sr.

Duoprismatic symmetry

* Duoprismatic groups – ,2,q (), order 4''pq'', exist for all 2 ≤ ''p'',''q'' < ∞. There are p+q mirrors in this symmetry, which are trivially decomposed into two orthogonal sets of p and q mirrors of

: and ** The chiral subgroup is ,2,psup>+,(), order 2''pq''. It can be doubled as 2p,2,2psup>+]. ** If p and q are equal, ,2,p (), the symmetry can be doubled as , (). *** Doublings: 2⁺,2,p⁺, (), 2p,2⁺,2p, 2p⁺,2⁺,2p⁺. ** ,2,∞ (), it represents a line groups in 3-space, ** ��,2,∞ () it represents the Euclidean plane symmetry with two sets of parallel mirrors and a rectangular domain (

orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space that is locally a finite group quotient of a Euclidean space. D ...

*2222). ** Subgroups include: ⁺,2,q (), ,2,q⁺ (), ⁺,2,q⁺ (). ** And for even values: p,2⁺,2q (), p,2⁺,2q⁺ (), p,2)⁺,2q (), p,(2,q)⁺ (), p,2)⁺,2q⁺ (), p⁺,(2,q)⁺ (), p⁺,2⁺,2q⁺ (), and communtator subgroup, index 16, p⁺,2⁺,2q⁺sup>+, (). * Digonal duoprismatic group – ,2,2 (), order 16. ** The chiral subgroup is ,2,2sup>+, (), order 8. ** Extended , (), order 32. The 4-4 duoprism has this extended symmetry, . *** The chiral extended group is ⁺, order 16. *** Extended chiral subgroup is 2,2,2]⁺], order 16, with

generators. It is isomorphic to the abstract group (4,4, 2,2). ** Other extended 3,3)[2,2,2= ,3,3 order 384, #Hexadecachoric symmetry">,2,2.html" ;"title="3,3)[2,2,2">3,3)[2,2,2= ,3,3 order 384, #Hexadecachoric symmetry. The

has this symmetry, as or . ** Ionic diminished subgroups is [2⁺,2,2], order 8. *** The double diminished subgroup is [2⁺,2,2⁺], order 4. **** Extended as 2⁺,2,2⁺, order 8. *** The rotoreflection subgroups are [2⁺,2⁺,2], ,2⁺,2⁺ ⁺,(2,2)⁺ 2,2)⁺,2⁺order 4. *** The triple diminished subgroup is ⁺,2⁺,2⁺ (), order 2. It is a 2-fold

and a 4D

. ** Half subgroup is ⁺,2,2,2 ,2,2 order 8. * Triangular duoprismatic group – ,2,3 , order 36. ** The chiral subgroup is ,2,3sup>+, order 18. ** Extended , order 72. The 3-3 duoprism has this extended symmetry, . *** The chiral extended group is ⁺, order 36. *** Extended chiral subgroup is 3,2,3]⁺], order 36, with

generators. It is isomorphic to the abstract group (4,4, 2,3). ** Other extended 3],2,3], ,2,[3, order 72, and are isomorphic to [6,2,3">.html" ;"title=",2,[3">,2,[3, order 72, and are isomorphic to [6,2,3and [3,2,6]. ** And 3,2,3, order 144, and is isomorphic to [6,2,6]. ** And [3,2,[3], order 288, isomorphic to . The duoprism, 6–6 duoprism has this symmetry, as or . ** Ionic diminished subgroups are ⁺,2,3 ,2,3⁺ order 18. *** The double diminished subgroup is ⁺,2,3⁺ order 9. **** Extended as 3⁺,2,3⁺, order 18. ** A high index subgroup is ,2, order 12, index 3, which is isomorphic to the

dihedral symmetry in three dimensions In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as an abstract group is a dihedral group Dih''n'' (for ''n'' ≥ 2). Types Ther ...

group, ,2 D_3h. *** ,2sup>+, order 6 * Square duoprismatic group – ,2,4 , order 64. ** The chiral subgroup is ,2,4sup>+, order 32. ** Extended , order 128. The 4–4 duoprism has this extended symmetry, . *** The chiral extended group is ⁺, order 64. *** Extended chiral subgroup is 4,2,4]⁺], order 64, with

generators. It is isomorphic to the abstract group (4,4, 2,4). ** Other extended 4],2,4], ,2,[4, order 128, and are isomorphic to [8,2,4">.html" ;"title=",2,[4">,2,[4, order 128, and are isomorphic to [8,2,4and [4,2,8]. The duoprism, 4–8 duoprism has this symmetry, as or . ** And 4,2,4, order 256, and is isomorphic to [8,2,8]. ** And [4,2,[4] order 512, isomorphic to . The duoprism, 8–8 duoprism has this symmetry, as or . ** Ionic diminished subgroups are ⁺,2,4 ,2,4⁺ order 32. *** The double diminished subgroup is ⁺,2,4⁺ order 16. **** Extended as 4⁺,2,4⁺, order 32. *** The rotoreflection subgroups are ⁺,2⁺,4 ,2⁺,4⁺ ⁺,(2,4)⁺ 4,2)⁺,4⁺ (, , , ) order 16. *** The triple diminished subgroup is ⁺,2⁺,4⁺ (), order 8. ** Half subgroups are ⁺,4,2,4 ,2,4 (), ,2,4,1⁺ ,2,2 (), order 32. *** ⁺,4,2,4sup>+= ,2,4sup>+, (), ,2,4,1⁺sup>+= ,2,2sup>+, (), order 16. ** Half again subgroup is ⁺,4,2,4,1⁺ ,2,2 (), order 16. *** ⁺,4,2,4,1⁺sup>+ = ⁺,4,2⁺,4,1⁺= ,2,2sup>+, () order 8

Summary of some 4-dimensional point groups

This is a summary of 4-dimensional

s in

. 227 of them are crystallographic point groups (for particular values of p and q). (nc) is given for non-crystallographic groups. Some crystallographic group have their orders indexed (order.index) by their abstract group structure.Coxeter, ''Regular and Semi-Regular Polytopes II'' (1985)

References

* H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973 *
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, ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', ath. Zeit. 46 (1940) 380–407, MR 2,10** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', ath. Zeit. 188 (1985) 559–591** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3–45* H.S.M. Coxeter and W. O. J. Moser. ''Generators and Relations for Discrete Groups'' 4th ed, Springer-Verlag. New York. 1980 p92, p122. * John .H. Conway and M.J.T. Guy: ''Four-Dimensional Archimedean Polytopes'', Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965 * N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966 * N.W. Johnson: ''Geometries and Transformations'', (2018) Chapter 11: ''Finite Symmetry Groups'', 11.5 ''Spherical Coxeter groups'', p. 249 * John H. Conway and Derek A. Smith, ''On Quaternions and Octonions'', 2003, * John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, (Chapter 26)

External links

* * {{KlitzingPolytopes, polychora.htm, 4D, uniform polytopes 4-polytopes