Complex polytope

In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one. A complex polytope may be understood as a collection of complex points, lines, planes, and so on, where every point is the junction of multiple lines, every line of multiple planes, and so on. Precise definitions exist only for the regular complex polytopes, which are configurations. The regular complex polytopes have been completely characterized, and can be described using a symbolic notation developed by Coxeter. Some complex polytopes which are not fully regular have also been described.

In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one. A complex polytope may be understood as a collection of complex points, lines, planes, and so on, where every point is the junction of multiple lines, every line of multiple planes, and so on. Precise definitions exist only for the regular complex polytopes, which are configurations. The regular complex polytopes have been completely characterized, and can be described using a symbolic notation developed by Coxeter. Some complex polytopes which are not fully regular have also been described. In a regular complex polytope the vertices incident on the edge are arranged symmetrically about their centroid, which is often used as the origin of the edge's coordinate system (in the real case the centroid is just the midpoint of the edge). The symmetry arises from a complex reflection about the centroid; this reflection will leave the magnitude of any vertex unchanged, but change its argument by a fixed amount, moving it to the coordinates of the next vertex in order. So we may assume (after a suitable choice of scale) that the vertices on the edge satisfy the equation x^p - 1 = 0 where p is the number of incident vertices. Thus, in the Argand diagram of the edge, the vertex points lie at the vertices of a regular polygon centered on the origin. Three real projections of regular complex polygon 4{4}2 are illustrated above, with edges a, b, c, d, e, f, g, h. It has 16 vertices, which for clarity have not been individually marked. Each edge has four vertices and each vertex lies on two edges, hence each edge meets four other edges. In the first diagram, each edge is represented by a square. The sides of the square are not parts of the polygon but are drawn purely to help visually relate the four vertices. The edges are laid out symmetrically. (Note that the diagram looks similar to the B 4 Coxeter plane projection of the tesseract, but it is structurally different). The middle diagram abandons octagonal symmetry in favour of clarity. Each edge is shown as a real line, and each meeting point of two lines is a vertex. The connectivity between the various edges is clear to see. The last diagram gives a flavour of the structure projected into three dimensions: the two cubes of vertices are in fact the same size but are seen in perspective at different distances away in the fourth dimension. as regular polygons for p = 2, 3, 4, 5, and 6, with black vertices. The centroid of the p vertices is shown seen in red. The sides of the polygons represent one application of the symmetry generator, mapping each vertex to the next counterclockwise copy. These polygonal sides are not edge elements of the polytope, as a complex 1-polytope can have no edges (it often is a complex edge) and only contains vertex elements.]] A real 1-dimensional polytope exists as a closed segment in the real line \mathbb{R}^1 , defined by its two end points or vertices in the line. Its Schläfli symbol is {} . Analogously, a complex 1-polytope exists as a set of p vertex points in the complex line \mathbb{C}^1 . These may be represented as a set of points in an Argand diagram (x,y)=x+iy. A regular complex 1-dimensional polytope p {} has p (p ≥ 2) vertex points arranged to form a convex regular polygon {p} in the Argand plane. Unlike points on the real line, points on the complex line have no natural ordering. Thus, unlike real polytopes, no interior can be defined. Despite this, complex 1-polytopes are often drawn, as here, as a bounded regular polygon in the Argand plane. A regular real 1-dimensional polytope is represented by an empty Schläfli symbol {}, or Coxeter-Dynkin diagram . The dot or node of the Coxeter-Dynkin diagram itself represents a reflection generator while the circle around the node means the generator point is not on the reflection, so its reflective image is a distinct point from itself. By extension, a regular complex 1-dimensional polytope in \mathbb{C}^1 has Coxeter-Dynkin diagram , for any positive integer p, 2 or greater, containing p vertices. p can be suppressed if it is 2. It can also be represented by an empty Schläfli symbol p {}, } p {, {} p , or p {2} 1 . The 1 is a notational placeholder, representing a nonexistent reflection, or a period 1 identity generator. (A 0-polytope, real or complex is a point, and is represented as } {, or 1 {2} 1 .) The symmetry is denoted by the Coxeter diagram , and can alternatively be described in Coxeter notation as p [], [] p or ] p [, p [2] 1 or p [1] p . The symmetry is isomorphic to the cyclic group, order p. The subgroups of p [] are any whole divisor d, d [], where d≥2. A unitary operator generator for is seen as a rotation by 2π/p radians counter clockwise, and a edge is created by sequential applications of a single unitary reflection. A unitary reflection generator for a 1-polytope with p vertices is . When p = 2, the generator is e πi = –1, the same as a point reflection in the real plane. In higher complex polytopes, 1-polytopes form p-edges. A 2-edge is similar to an ordinary real edge, in that it contains two vertices, but need not exist on a real line. The complex polygon illustrated above has eight square edges (p 1 =4) and sixteen vertices (p 2 =2). From this we can work out that g = 32, giving the modified Schläfli symbol 4(32)2. The symmetry group p 1 [q] p 2 is represented by 2 generators R 1 , R 2 , where: R 1 p 1 = R 2 p 2 = I. If q is even, (R 2 R 1 ) q/2 = (R 1 R 2 ) q/2 . If q is odd, (R 2 R 1 ) (q−1)/2 R 2 = (R 1 R 2 ) (q−1)/2 R 1 . When q is odd, p 1 =p 2 . For 4 [4] 2 has R 1 4 = R 2 2 = I, (R 2 R 1 ) 2 = (R 1 R 2 ) 2 . For 3 [5] 3 has R 1 3 = R 2 3 = I, (R 2 R 1 ) 2 R 2 = (R 1 R 2 ) 2 R 1 . [[File:Rank2_shephard_subgroups2_series.png|thumb| p [4] 2 subgroups: p=2,3,4... p [4] 2 → [p], index p p [4] 2 → p []× p [], index 2]] Coxeter enumerated this list of regular complex polygons in \mathbb{C}^2 . A regular complex polygon, p {q} r or , has p-edges, and r-gonal vertex figures. p {q} r is a finite polytope if (p+r)q>pr(q-2). Its symmetry is written as p [q] r , called a Shephard group, analogous to a Coxeter group, while also allowing unitary reflections. For nonstarry groups, the order of the group p [q] r can be computed as g = 8/q \cdot (1/p+2/q+1/r-1)^{-2} . The Coxeter number for p [q] r is h = 2/(1/p+2/q+1/r-1) , so the group order can also be computed as g = 2h^2/q . A regular complex polygon can be drawn in orthogonal projection with h-gonal symmetry. The rank 2 solutions that generate complex polygons are: Excluded solutions with odd q and unequal p and r are: 6 [3] 2 , 6 [3] 3 , 9 [3] 3 , 12 [3] 3 , ..., 5 [5] 2 , 6 [5] 2 , 8 [5] 2 , 9 [5] 2 , 4 [7] 2 , 9 [5] 2 , 3 [9] 2 , and 3 [11] 2 . Other whole q with unequal p and r, create starry groups with overlapping fundamental domains: , , , , , and . The dual polygon of p {q} r is r {q} p . A polygon of the form p {q} p is self-dual. Groups of the form p [2q] 2 have a half symmetry p [q] p , so a regular polygon is the same as quasiregular . As well, regular polygon with the same node orders, , have an alternated construction , allowing adjacent edges to be two different colors. The group order, g, is used to compute the total number of vertices and edges. It will have g/r vertices, and g/p edges. When p=r, the number of vertices and edges are equal. This condition is required when q is odd. k = \sqrt \frac{ \cos(\frac{\pi}{p}-\frac{\pi}{r})+\cos(\frac{2\pi}{q}) }{2\sin\frac{\pi}{p}\sin\frac{\pi}{r} } Examples | | |- valign=top | | | |} Complex bipartite graph square.svg| 2 {4} 2 , , with 4 vertices, and 4 edges Complex polygon 2-4-3-bipartite graph.png| 2 {4} 3 , , with 6 vertices, and 9 edges Complex polygon 2-4-4 bipartite graph.png| 2 {4} 4 , , with 8 vertices, and 16 edges Complex polygon 2-4-5-bipartite graph.png| 2 {4} 5 , , with 10 vertices, and 25 edges 6-generalized-2-orthoplex.svg| 2 {4} 6 , , with 12 vertices, and 36 edges 7-generalized-2-orthoplex.svg| 2 {4} 7 , , with 14 vertices, and 49 edges 8-generalized-2-orthoplex.svg| 2 {4} 8 , , with 16 vertices, and 64 edges 9-generalized-2-orthoplex.svg| 2 {4} 9 , , with 18 vertices, and 81 edges 10-generalized-2-orthoplex.svg| 2 {4} 10 , , with 20 vertices, and 100 edges Complex polygons p {4} 2 : Polygons of the form p {4} 2 are called generalized hypercubes (squares for polygons). They share vertices with the 4D p-p duoprisms, vertices connected by p-edges. Vertices are drawn in green, and p-edges are drawn in alternate colors, red and blue. The perspective is distorted slightly for odd dimensions to move overlapping vertices from the center. 2-generalized-2-cube.svg| 2 {4} 2 , or , with 4 vertices, and 4 2-edges 3-generalized-2-cube_skew.svg| 3 {4} 2 , or , with 9 vertices, and 6 (triangular) 3-edges 4-generalized-2-cube.svg| 4 {4} 2 , or , with 16 vertices, and 8 (square) 4-edges 5-generalized-2-cube_skew.svg| 5 {4} 2 , or , with 25 vertices, and 10 (pentagonal) 5-edges 6-generalized-2-cube.svg| 6 {4} 2 , or , with 36 vertices, and 12 (hexagonal) 6-edges 7-generalized-2-cube_skew.svg| 7 {4} 2 , or , with 49 vertices, and 14 (heptagonal)7-edges 8-generalized-2-cube.svg| 8 {4} 2 , or , with 64 vertices, and 16 (octagonal) 8-edges 9-generalized-2-cube_skew.svg| 9 {4} 2 , or , with 81 vertices, and 18 (enneagonal) 9-edges 10-generalized-2-cube.svg| 10 {4} 2 , or , with 100 vertices, and 20 (decagonal) 10-edges 3D perspective projections of complex polygons p {4} 2 . The duals 2 {4} p : are seen by adding vertices inside the edges, and adding edges in place of vertices. Complex polygon 3-4-2-stereographic3.svg| 3 {4} 2 , or with 9 vertices, 6 3-edges in 2 sets of colors Complex polygon 2-4-3-stereographic0.png| 2 {4} 3 , with 6 vertices, 9 edges in 3 sets Complex polygon 4-4-2-stereographic3.svg| 4 {4} 2 , or with 16 vertices, 8 4-edges in 2 sets of colors and filled square 4-edges Complex_polygon_5-4-2-stereographic3.svg| 5 {4} 2 , or with 25 vertices, 10 5-edges in 2 sets of colors Other Complex polygons p {r} 2 : Complex_polygon_3-6-2.svg| 3 {6} 2 , or , with 24 vertices in black, and 16 3-edges colored in 2 sets of 3-edges in red and blue Complex_polygon_3-8-2.svg| 3 {8} 2 , or , with 72 vertices in black, and 48 3-edges colored in 2 sets of 3-edges in red and blue 2D orthogonal projections of complex polygons, p {r} p : Polygons of the form p {r} p have equal number of vertices and edges. They are also self-dual. Complex polygon 3-3-3.svg| 3 {3} 3 , or , with 8 vertices in black, and 8 3-edges colored in 2 sets of 3-edges in red and blue Complex_polygon_3-4-3-fill1.png| 3 {4} 3 , or , with 24 vertices and 24 3-edges shown in 3 sets of colors, one set filled Complex polygon 4-3-4.png| 4 {3} 4 , or , with 24 vertices and 24 4-edges shown in 4 sets of colors Complex polygon 3-5-3.png| 3 {5} 3 , or , with 120 vertices and 120 3-edges Complex polygon 5-3-5.png| 5 {3} 5 , or , with 120 vertices and 120 5-edges There are infinite families of regular complex polytopes that occur in all dimensions, generalizing the hypercubes and cross polytopes in real space. Shephard's "generalized orthotope" generalizes the hypercube; it has symbol given by γ = p {4} 2 {3} 2 ... 2 {3} 2 and diagram ... . Its symmetry group has diagram p [4] 2 [3] 2 ... 2 [3] 2 ; in the Shephard–Todd classification, this is the group G(p, 1, n) generalizing the signed permutation matrices. Its dual regular polytope, the "generalized cross polytope", is represented by the symbol β = 2 {3} 2 {3} 2 ... 2 {4} p and diagram ... . A 1-dimensional regular complex polytope in \mathbb{C}^1 is represented as , having p vertices, with its real representation a regular polygon, {p}. Coxeter also gives it symbol γ or β as 1-dimensional generalized hypercube or cross polytope. Its symmetry is p [] or , a cyclic group of order p. In a higher polytope, p {} or represents a p-edge element, with a 2-edge, {} or , representing an ordinary real edge between two vertices. A dual complex polytope is constructed by exchanging k and (n-1-k)-elements of an n-polytope. For example, a dual complex polygon has vertices centered on each edge, and new edges are centered at the old vertices. A v-valence vertex creates a new v-edge, and e-edges become e-valence vertices. The dual of a regular complex polytope has a reversed symbol. Regular complex polytopes with symmetric symbols, i.e. p {q} p , p {q} r {q} p , p {q} r {s} r {q} p , etc. are self dual. A complex regular polyhedron p {n 1 } q {n 2 } r requires both g 1 = order( p [n 1 ] q ) and g 2 = order( q [n 2 ] r ) be finite. Given g = order( p [n 1 ] q [n 2 ] r ), the number of vertices is g/g 2 , and the number of faces is g/g 1 . The number of edges is g/pr. For each dimension there are 12 apeirotopes symbolized as δ exists in any dimensions \mathbb{C}^n , or \mathbb{R}^n if p=q=2. Coxeter calls these generalized cubic honeycombs for n>2. Each has proportional element counts given as: k-faces = {n \choose k}p^{n-k}r^k , where {n \choose m}=\frac{n!}{m!\,(n-m)!} and n! denotes the factorial of n. Rank 2 complex apeirogons have symmetry p [q] r , where 1/p + 2/q + 1/r = 1. Coxeter expresses them as δ where q is constrained to satisfy . There are 8 solutions: There are two excluded solutions odd q and unequal p and r: 10 [5] 2 and 12 [3] 4 , or and . A regular complex apeirogon p {q} r has p-edges and r-gonal vertex figures. The dual apeirogon of p {q} r is r {q} p . An apeirogon of the form p {q} p is self-dual. Groups of the form p [2q] 2 have a half symmetry p [q] p , so a regular apeirogon is the same as quasiregular . Apeirogons can be represented on the Argand plane share four different vertex arrangements. Apeirogons of the form 2 {q} r have a vertex arrangement as {q/2,p}. The form p {q} 2 have vertex arrangement as r{p,q/2}. Apeirogons of the form p {4} r have vertex arrangements {p,r}. Including affine nodes, and \mathbb{C}^2 , there are 3 more infinite solutions: ∞ [2] ∞ , ∞ [4] 2 , ∞ [3] 3 , and , , and . The first is an index 2 subgroup of the second. The vertices of these apeirogons exist in \mathbb{C}^1 . There are 22 regular complex apeirohedra in \mathbb{C}^2 of the form p {a} q {b} r . 8 are self-dual (p=r and a=b), while 14 exist as dual polytope pairs. Three are entirely real (p=q=r=2). Coxeter symbolizes 12 of them as δ or p {4} 2 {4} r is the regular form of the product apeirotope δ × δ or p {q} r × p {q} r , where q is determined from p and r. is the same as , as well as , for p,r=2,3,4,6. Also = . There are 16 regular complex apeirotopes in \mathbb{C}^3 . Coxeter expresses 12 of them by δ where q is constrained to satisfy . These can also be decomposed as product apeirotopes: = . The first case is the \mathbb{R}^3 cubic honeycomb. There are 15 regular complex apeirotopes in \mathbb{C}^4 . Coxeter expresses 12 of them by δ where q is constrained to satisfy . These can also be decomposed as product apeirotopes: = . The first case is the \mathbb{R}^4 tesseractic honeycomb. The 16-cell honeycomb and 24-cell honeycomb are real solutions. The last solution is generated has Witting polytope elements. There are only 12 regular complex apeirotopes in \mathbb{C}^5 or higher expressed δ where q is constrained to satisfy . These can also be decomposed a product of n apeirogons: ... = ... . The first case is the real \mathbb{R}^n hypercube honeycomb. There are 7 quasiregular complex apeirogons which alternate edges of a regular apeirogon and its regular dual. The vertex arrangements of these apeirogon have real representations with the regular and uniform tilings of the Euclidean plane. The last column for the 6{3}6 apeirogon is not only self-dual, but the dual coincides with itself with overlapping hexagonal edges, thus their quasiregular form also has overlapping hexagonal edges, so it can't be drawn with two alternating colors like the others. The symmetry of the self-dual families can be doubled, so creating an identical geometry as the regular forms: = The group is defined by 3 unitary reflections, R 1 , R 2 , R 3 , all order 2: R 1 2 = R 1 2 = R 3 2 = (R 1 R 2 ) 3 = (R 2 R 3 ) 3 = (R 3 R 1 ) 3 = (R 1 R 2 R 3 R 1 ) p = 1. The period p can be seen as a double rotation in real \mathbb{R}^4 . As with all Wythoff constructions, polytopes generated by reflections, the number of vertices of a single-ringed Coxeter diagram polytope is equal to the order of the group divided by the order of the subgroup where the ringed node is removed. For example, a real cube has Coxeter diagram , with octahedral symmetry order 48, and subgroup dihedral symmetry order 6, so the number of vertices of a cube is 48/6=8. Facets are constructed by removing one node furthest from the ringed node, for example for the cube. Vertex figures are generated by removing a ringed node and ringing one or more connected nodes, and for the cube. Coxeter represents these groups by the following symbols. Some groups have the same order, but a different structure, defining the same vertex arrangement in complex polytopes, but different edges and higher elements, like and with p≠3. Coxeter calls some of these complex polyhedra almost regular because they have regular facets and vertex figures. The first is a lower symmetry form of the generalized cross-polytope in \mathbb{C}^3 . The second is a fractional generalized cube, reducing p-edges into single vertices leaving ordinary 2-edges. Three of them are related to the finite regular skew polyhedron in \mathbb{R}^4 . Coxeter defines other groups with anti-unitary constructions, for example these three. The first was discovered and drawn by Peter McMullen in 1966.

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