Search results
Results from the WOW.Com Content Network
In 1926 John Flinders Petrie took the concept of a regular skew polygons, polygons whose vertices are not all in the same plane, and extended it to polyhedra.While apeirohedra are typically required to tile the 2-dimensional plane, Petrie considered cases where the faces were still convex but were not required to lie flat in the plane, they could have a skew polygon vertex figure.
In geometry, a skew apeirohedron is an infinite skew polyhedron consisting of nonplanar faces or nonplanar vertex figures, allowing the figure to extend indefinitely without folding round to form a closed surface. Skew apeirohedra have also been called polyhedral sponges.
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.
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.
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}. These regular zig-zag skew apeirogons have internal ...
A regular skew polygon is a faithful symmetric realization of a polygon in dimension greater than 2. In 3 dimensions a regular skew polygon has vertices alternating between two parallel planes. A regular skew n-gon can be given a Schläfli symbol {p}#{} as a blend of a regular polygon p and an orthogonal line segment { }. [3]
Regular skew polyhedra can also be constructed in dimensions higher than 4 as embeddings into regular polytopes or honeycombs. For example, the regular icosahedron can be embedded into the vertices of the 6-demicube; this was named the regular skew icosahedron by H. S. M. Coxeter. The dodecahedron can be similarly embedded into the 10-demicube. [4]