Complex polytope
Encyclopedia
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.
On a real line, two points bound a segment. This defines an edge with two bounding vertices. For a real polytope it is not possible to have third vertex because one would then lie in between the other two. On the complex line, which may be represented as an Argand diagram, points are not ordered and so more than two vertex points may be allowed.
Also, a real polygon has just two sides at each vertex, such that the boundary forms a closed loop. A real polyhedron has two faces at each edge such that the boundary forms a closed surface. A polychoron has two cells at each wall, and so on. These loops and surfaces have no analogy in complex spaces, for example a set of complex lines and points may form a closed chain of connections, but this chain does not bound a polygon. Thus, more than two elements meeting in one place may be allowed.
Since bounding does not occur, we cannot think of a complex edge as a line segment, but as the whole line. Similarly, we cannot think of a bounded polygonal face but must accept the whole plane.
Thus, a complex polytope may be understood as an arrangement
of connected points, lines, planes and so on, where every point is the junction of multiple lines, every line of multiple planes, and so on. Likewise, each line must contain multiple points, each plane multiple lines, and so on.
ones. Shephard (1952) discovered them, and Coxeter (1974) developed the idea extensively. Shephard treated his figures as configurations
from the start, while Coxeter only found it necessary to do so from Chapter 12 onwards.
In the Argand diagram, of the edge of a regular complex polytope, the vertex points lie at the vertices of a regular polygon
centered on the origin. Given the general point x + iy in the complex plane, for an edge having p vertices, these lie at the p roots of the equation:
(For p = 2 these are the real points +1 and − 1, and the edge is real).
Two real projections of the same regular complex octagon with edges a,b,c,d,e,f,g,h are illustrated. It has 16 vertices, which for clarity have not been individually marked. Each edge has four vertices at which it meets another edge, 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 - this is important to understand - but are drawn in purely to help visually relate the four vertices. The edges are laid out symmetrically (coincidentally the diagram looks the same as a common projection of the hypercube
, but in the case of the complex octagon the diamond shapes which can be traced are not parts of the structure). The second diagram abandons octagonal symmetry in favour of clarity. Each edge is shown as a line, and each meeting point on the line is a vertex on that edge. The connectivity between the various edges is clear to see.
Shephard originally devised a modified form of Schläfli's notation for regular polytopes. For a polygon bounded by p1-edges, with a p2-set as vertex figure and overall symmetry group of order g, we denote the polygon as p1(g)p2.
The number of vertices V is then g/p2 and the number of edges E is g/p1.
The complex octagon illustrated has eight 4-edges (p1=4) and sixteen 2-vertices (p2=2). From this we can work out that g = 32, giving the modified Schläfli symbol 4(32)2.
Coxeter's notation
The modern notation p1{q}p2 is due to Coxeter, and is based on group theory. The nodes p1 and p2 represent mirrors producing p1 and p2 images in the plane. In group theory, this might be represented (for the example left) as AAAA = BB = 1. q represents the number of alternate reflections in the two mirrors that become equal to its opposite, ie for q=4, ABAB = BABA. When q is odd, then p1 = p2, e.g. 3{5}3 means AAA = BBB = 1; ABABA = BABAB.
The example octagon is represented as 4{4}2, which belongs to symmetry group AAAA = BB = 1, ABAB = BABA.
To obtain the real conjugate, we discard the imaginary part of any coordinate. For example the complex point (a + ib) has real conjugate a.
The real conjugate of a complex edge is a line with the vertex points distributed along it (not generally evenly spaced). The second of the two octagon projections above shows the real conjugates of the sides.
Polytope
In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
in real space
Real space
Position space denominates the space of possible locations of an object in classical physics.The real space coordinates specify the position of an object. For instance,...
to an analogous structure in a complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
, where each real dimension is accompanied by an imaginary
Imaginary number
An imaginary number is any number whose square is a real number less than zero. When any real number is squared, the result is never negative, but the square of an imaginary number is always negative...
one.
On a real line, two points bound a segment. This defines an edge with two bounding vertices. For a real polytope it is not possible to have third vertex because one would then lie in between the other two. On the complex line, which may be represented as an Argand diagram, points are not ordered and so more than two vertex points may be allowed.
Also, a real polygon has just two sides at each vertex, such that the boundary forms a closed loop. A real polyhedron has two faces at each edge such that the boundary forms a closed surface. A polychoron has two cells at each wall, and so on. These loops and surfaces have no analogy in complex spaces, for example a set of complex lines and points may form a closed chain of connections, but this chain does not bound a polygon. Thus, more than two elements meeting in one place may be allowed.
Since bounding does not occur, we cannot think of a complex edge as a line segment, but as the whole line. Similarly, we cannot think of a bounded polygonal face but must accept the whole plane.
Thus, a complex polytope may be understood as an arrangement
Arrangement of lines
In geometry an arrangement of lines is the partition of the plane formed by a collection of lines. Bounds on the complexity of arrangements have been studied in discrete geometry, and computational geometers have found algorithms for the efficient construction of arrangements.-Definition:For any...
of connected points, lines, planes and so on, where every point is the junction of multiple lines, every line of multiple planes, and so on. Likewise, each line must contain multiple points, each plane multiple lines, and so on.
Regular complex polytopes
The only complex polytopes to have been systematically studied are the regularRegular polytope
In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...
ones. Shephard (1952) discovered them, and Coxeter (1974) developed the idea extensively. Shephard treated his figures as configurations
Configuration (geometry)
In mathematics, specifically projective geometry, a configuration in the plane consists of a finite set of points, and a finite arrangement of lines, such that each point is incident to the same number of lines and each line is incident to the same number of points.Although certain specific...
from the start, while Coxeter only found it necessary to do so from Chapter 12 onwards.
In the Argand diagram, of the edge of a regular complex polytope, the vertex points lie at the vertices of a regular polygon
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...
centered on the origin. Given the general point x + iy in the complex plane, for an edge having p vertices, these lie at the p roots of the equation:
(For p = 2 these are the real points +1 and − 1, and the edge is real).
Two real projections of the same regular complex octagon with edges a,b,c,d,e,f,g,h are illustrated. It has 16 vertices, which for clarity have not been individually marked. Each edge has four vertices at which it meets another edge, 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 - this is important to understand - but are drawn in purely to help visually relate the four vertices. The edges are laid out symmetrically (coincidentally the diagram looks the same as a common projection of the hypercube
Hypercube
In geometry, a hypercube is an n-dimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.An...
, but in the case of the complex octagon the diamond shapes which can be traced are not parts of the structure). The second diagram abandons octagonal symmetry in favour of clarity. Each edge is shown as a line, and each meeting point on the line is a vertex on that edge. The connectivity between the various edges is clear to see.
Modified Schläfli notation
Shephard's notationShephard originally devised a modified form of Schläfli's notation for regular polytopes. For a polygon bounded by p1-edges, with a p2-set as vertex figure and overall symmetry group of order g, we denote the polygon as p1(g)p2.
The number of vertices V is then g/p2 and the number of edges E is g/p1.
The complex octagon illustrated has eight 4-edges (p1=4) and sixteen 2-vertices (p2=2). From this we can work out that g = 32, giving the modified Schläfli symbol 4(32)2.
Coxeter's notation
The modern notation p1{q}p2 is due to Coxeter, and is based on group theory. The nodes p1 and p2 represent mirrors producing p1 and p2 images in the plane. In group theory, this might be represented (for the example left) as AAAA = BB = 1. q represents the number of alternate reflections in the two mirrors that become equal to its opposite, ie for q=4, ABAB = BABA. When q is odd, then p1 = p2, e.g. 3{5}3 means AAA = BBB = 1; ABABA = BABAB.
The example octagon is represented as 4{4}2, which belongs to symmetry group AAAA = BB = 1, ABAB = BABA.
Real conjugates
In the ordinary, or real plane, we can construct a visible figure as the real conjugate of some complex polygon. Likewise in ordinary space, we can construct a visible figure as the real conjugate of some complex polyhedron.To obtain the real conjugate, we discard the imaginary part of any coordinate. For example the complex point (a + ib) has real conjugate a.
The real conjugate of a complex edge is a line with the vertex points distributed along it (not generally evenly spaced). The second of the two octagon projections above shows the real conjugates of the sides.