geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a

point group In geometry, a point group is a mathematical group of symmetry operations ( isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every ...

in four dimensions is an

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

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

3-sphere In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimens ...

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 se ...

, ''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 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-space. Coxeter n ...

* 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 Alan Lindsay Mackay Fellow of the Royal Society, FRS (born 6 September 1926) is a British crystallographer, born in Wolverhampton. He spent his scientific career at Birkbeck College, founded by George Birkbeck, one of the Colleges of the Univer ...

''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 a ...

, ''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. Hamilton defined a quat ...

-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 Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

, ''Regular and Semi-Regular Polytopes II'', Coxeter notation for 4D point groups * 2003

John Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...

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 In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is inv ...

reflection symmetry In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D the ...

rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...

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 about an axis and a reflection in a plane perpendicul ...

, 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 d ...

Notation for groups

Point groups in this article are given in Coxeter notation, 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 In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there a ...

, ''n'' is the dimension (4). For cross-referencing, also given here are

based notations by

(1964) and

(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 (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...

, 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: * Involute, a construction in the differential geometry of curves * ''Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour input ...

al groups: no symmetry

nbsp; In word processing and digital typesetting, a non-breaking space, , also called NBSP, required space, hard space, or fixed space (though it is not of fixed width), is a space character that prevents an automatic line break at its position. In ...

sup>+,

2-fold

sup>+, 2-fold

⁺,2⁺ and central

⁺,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 transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...

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 o ...

bounded by mirror planes. The

dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the un ...

s 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, ...

. The

Coxeter–Dynkin diagram In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes). It describes ...

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: Greece 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 ...

roots ''poly'' ("many") and ''choros'' ("room" or "space") and is advocated by Norman Johnson and George Olshevsky in the context of

uniform polychora In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedron, uniform polyhedra, and faces are regular polygons. There are 47 non-Prism (geometry), prism ...

(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 ...

hosotope 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 havin ...

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

polyhedral group In geometry, the polyhedral group is any of the symmetry groups of the Platonic solids. Groups There are three polyhedral groups: *The tetrahedral group of order 12, rotational symmetry group of the regular tetrahedron. It is isomorphic to ''A' ...

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 and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

construct. Chiral symmetries exist in alternated uniform polychora. Only irreducible groups have Coxeter numbers, but duoprismatic groups

,2,p The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...

can 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

,2,4 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...

and 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 object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is ...

(pentachoron), given by ringed

. It is also sometimes called the hyper-tetrahedral group for extending the

tetrahedral group 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 combine a reflection ...

,3 There are 10 mirror hyperplanes in this group. It is isomorphic to the

abstract Abstract may refer to: * ''Abstract'' (album), 1962 album by Joe Harriott * Abstract of title a summary of the documents affecting title to parcel of land * Abstract (law), a summary of a legal document * Abstract (summary), in academic publishi ...

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 group ...

, S₅. **The extended pentachoric group,

Aut AUT may refer to the following. Locations *Austria (ISO 3166-1 country code) * Agongointo-Zoungoudo Underground Town, Benin *Aktio–Preveza Undersea Tunnel, Greece *Airstrip on Atauro Island, East Timor (IATA airport code) Organizations * Arr ...

(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

alternating group In mathematics, an alternating group is the 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 by or Basic pr ...

, 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

, 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 m ...

(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

,2,2 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...

subgroup. It is also called a

hyper-octahedral group In mathematics, a hyperoctahedral group is an important type of group that can be realized as the group of symmetries of a hypercube or of a cross-polytope. It was named by Alfred Young in 1930. Groups of this type are identified by a paramet ...

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 is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of ei ...

, . ** 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 In four-dimensional geometry, a runcinated tesseract (or ''runcinated 16-cell'') is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular tesseract. There are 4 variations of runcinations of the tesseract includ ...

, , 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 tetrahedral and 24 icosahedral cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular faces, ...

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 In geometry, a truncated cubic prism is a convex uniform polychoron (four-dimensional polytope). It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids or Archimedean solids in p ...

has this symmetry with Coxeter diagram and the

cubic prism In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eig ...

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 In geometry, a snub cubic prism or snub cuboctahedral prism is a convex uniform polychoron (four-dimensional polytope). It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids or ...

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 polyhed ...

'', ,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. Since a cube has a circumradius divided by edge length less than one, the square pyramids can be ...

can have this symmetry, with

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mor ...

: ( ) ∨ .

**** 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 In geometry, a pyramid () is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a ''lateral face''. It is a conic solid with polygonal base. A pyramid with an bas ...

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 In geometry, an icosahedral prism is a convex uniform 4-polytope (four-dimensional polytope). This 4-polytope has 22 polyhedral cells: 2 icosahedra connected by 20 triangular prisms. It has 70 faces: 30 squares and 40 triangles. It has 72 edges ...

, . *** 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 ''

'', ,3 A regular

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

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

(), order 16. Others are

(),

(), 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 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 "octahedral complex"), icosatetrahedroid, oc ...

(icositetrachoron), . There are 24 mirror planes in this symmetry, which can be decomposed into two orthogonal sets of 12 mirrors in

demitesseractic symmetry In geometry, a point group in four dimensions is an isometry group in four dimensions that leaves the origin fixed, or correspondingly, an isometry group of a 3-sphere. History on four-dimensional groups * 1889 Édouard Goursat, ''Sur les subst ...

^1,1,1subgroups, as ^*,4,3and ,4,3^* as index 6 subgroups. **The extended icositetrachoric group,

(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 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 ...

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, heca ...

, . ** 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 In geometry, a truncated icosidodecahedral prism or great rhombicosidodecahedral prism is a convex uniform 4-polytope (four-dimensional polytope). It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs ...

, , 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 In geometry, a snub dodecahedral prism or snub icosidodecahedral prism is a convex uniform polychoron (four-dimensional polytope). It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic s ...

, . *** 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 In 4-dimensional geometry, the dodecahedral pyramid is bounded by one dodecahedron on the base and 12 pentagonal pyramid cells which meet at the apex. Since a dodecahedron's circumradius is greater than its edge length, sqrt 9+3 sqrt(5))/8≒ 1.401 ...

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 In geometry, a pyramid () is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a ''lateral face''. It is a conic solid with polygonal base. A pyramid with an base ...

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

sup>+,(), order 2''pq''. It can be doubled as 2p,2,2psup>+]. ** If p and q are equal,

(), the symmetry can be doubled as , (). *** Doublings: p⁺,2,p⁺, (), 2p,2⁺,2p, 2p⁺,2⁺,2p⁺. **

,2,∞ The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...

(), it represents a

line group A line group is a mathematical way of describing symmetries associated with moving along a line. These symmetries include repeating along that line, making that line a one-dimensional lattice. However, line groups may have more than one dimension, a ...

s 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 which 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 –

(), order 16. ** The chiral subgroup is

sup>+, (), order 8. ** Extended , (), order 32. The

4-4 duoprism In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eigh ...

has this extended symmetry, . *** The chiral extended group is ⁺, order 16. *** Extended chiral subgroup is [

sup>+], 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

order 8. * Triangular duoprismatic group –

,2,3 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...

, order 36. ** The chiral subgroup is

sup>+, order 18. ** Extended , order 72. The

3-3 duoprism In the geometry of 4 dimensions, the 3-3 duoprism or triangular duoprism is a four-dimensional convex polytope. It can be constructed as the Cartesian product of two triangles and is the simplest of an infinite family of four-dimensional polytopes ...

has this extended symmetry, . *** The chiral extended group is ⁺, order 36. *** Extended chiral subgroup is [

sup>+], order 36, with

generators. It is isomorphic to the abstract group (4,4, 2,3). ** Other extended 32,3], ,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 The ...

group, ,2 D_3h. *** ,2sup>+, order 6 * Square duoprismatic group –

, order 64. ** The chiral subgroup is

sup>+, order 32. ** Extended , order 128. The 4–4 duoprism has this extended symmetry, . *** The chiral extended group is ⁺, order 64. *** Extended chiral subgroup is [

sup>+], order 64, with

generators. It is isomorphic to the abstract group (4,4, 2,4). ** Other extended 42,4], ,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

(), ,2,4,1⁺

(), order 32. *** ⁺,4,2,4sup>+=

sup>+, (), ,2,4,1⁺sup>+=

sup>+, (), order 16. ** Half again subgroup is ⁺,4,2,4,1⁺

(), order 16. *** ⁺,4,2,4,1⁺sup>+ = ⁺,4,2⁺,4,1⁺=

sup>+, () order 8

Summary of some 4-dimensional point groups

This is a summary of 4-dimensional

s in Coxeter notation. 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

History on four-dimensional groups

Isometries of 4D point symmetry

Notation for groups

Involution groups

Rank 4 Coxeter groups

Chiral subgroups

Pentachoric symmetry

Hexadecachoric symmetry

Icositetrachoric symmetry

Demitesseractic symmetry

Hexacosichoric symmetry

Duoprismatic symmetry

Summary of some 4-dimensional point groups

See also

References

External links