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Given a point A 0 in a Euclidean space and a translation S, define the point A i to be the point obtained from i applications of the translation S to A 0, so A i = S i (A 0).The set of vertices A i with i any integer, together with edges connecting adjacent vertices, is a sequence of equal-length segments of a line, and is called the regular apeirogon as defined by H. S. M. Coxeter.
The angled edges of an apeirogonal antiprism represent a regular zig-zag skew apeirogon.. A regular zig-zag skew apeirogon has (2*∞), D ∞d Frieze group symmetry.. Regular zig-zag skew apeirogons exist as Petrie polygons of the three regular tilings of the plane: {4,4}, {6,3}, and {3,6}.
In geometry, a polygon is traditionally a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain. These segments are called its edges or sides , and the points where two of the edges meet are the polygon's vertices (singular: vertex) or corners .
Heptagram – star polygon with 7 sides; Octagram – star polygon with 8 sides Star of Lakshmi (example) Enneagram - star polygon with 9 sides; Decagram - star polygon with 10 sides; Hendecagram - star polygon with 11 sides; Dodecagram - star polygon with 12 sides; Apeirogon - generalized polygon with countably infinite set of sides
A skew apeirogon in two dimensions forms a zig-zag line in the plane. If the zig-zag is even and symmetrical, then the apeirogon is regular. Skew apeirogons can be constructed in any number of dimensions. In three dimensions, a regular skew apeirogon traces out a helical spiral and may be either left- or right-handed.
Skew polygons must have at least four vertices. The interior surface and corresponding area measure of such a polygon is not uniquely defined. Skew infinite polygons (apeirogons) have vertices which are not all colinear. A zig-zag skew polygon or antiprismatic polygon [2] has vertices which alternate on two parallel planes, and thus must be ...
Special polygons in hyperbolic geometry are the regular apeirogon and pseudogon uniform polygons with an infinite number of sides. In Euclidean geometry , the only way to construct such a polygon is to make the side lengths tend to zero and the apeirogon is indistinguishable from a circle, or make the interior angles tend to 180° and the ...
Infinite-order apeirogonal tiling Poincaré disk model of the hyperbolic plane: Type: Hyperbolic regular tiling: Vertex configuration: ∞ ∞: Schläfli symbol {∞,∞} Wythoff symbol