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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
Infinite-order apeirogonal tiling Poincaré disk model of the hyperbolic plane: Type: Hyperbolic regular tiling: Vertex configuration: ∞ ∞: Schläfli symbol {∞,∞} Wythoff symbol
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.
In geometry, an order-2 apeirogonal tiling, apeirogonal dihedron, or infinite dihedron [1] is a tessellation (gap-free filling with repeated shapes) of the plane consisting of two apeirogons. It may be considered an improper regular tiling of the Euclidean plane, with Schläfli symbol {∞, 2}.
In geometry, an apeirogonal tiling is a tessellation of the Euclidean plane, hyperbolic plane, or some other two-dimensional space by apeirogons. Tilings of this type include: Order-2 apeirogonal tiling, Euclidean tiling of two half-spaces; Order-3 apeirogonal tiling, hyperbolic tiling with 3 apeirogons around a vertex