Regular Complex Polytopes
The only complex polytopes to have been systematically studied are the regular 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.
Read more about this topic: Complex Polytope
Famous quotes containing the words regular and/or complex:
“He hung out of the window a long while looking up and down the street. The world’s second metropolis. In the brick houses and the dingy lamplight and the voices of a group of boys kidding and quarreling on the steps of a house opposite, in the regular firm tread of a policeman, he felt a marching like soldiers, like a sidewheeler going up the Hudson under the Palisades, like an election parade, through long streets towards something tall white full of colonnades and stately. Metropolis.”
—John Dos Passos (1896–1970)
“We must open our eyes and see that modern civilization has become so complex and the lives of civilized men so interwoven with the lives of other men in other countries as to make it impossible to be in this world and out of it.”
—Franklin D. Roosevelt (1882–1945)