In
geometry, an infinite
skew polygon or skew
apeirogon
In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes.
In some literature, the term "apeirogon" may refer only to t ...
is an infinite 2-
polytope with vertices that are not all
colinear. Infinite zig-zag skew polygons are 2-dimensional infinite skew polygons with vertices alternating between two parallel lines. Infinite helical polygons are 3-dimensional infinite skew polygons with vertices on the surface of a
cylinder.
Regular infinite skew polygons exist in the
Petrie polygons of the affine and hyperbolic
Coxeter groups. They are constructed a single operator as the composite of all the reflections of the Coxeter group.
Regular zig-zag skew apeirogons in two dimensions
A regular zig-zag skew apeirogon has (2*∞), D
∞d Frieze group
In mathematics, a frieze or frieze pattern is a two-dimensional design that repeats in one direction. Such patterns occur frequently in architecture and decorative art. Frieze patterns can be classified into seven types according to their symmetrie ...
symmetry.
Regular zig-zag skew apeirogons exist as
Petrie polygons of the three regular tilings of the plane: , , and . These regular zig-zag skew apeirogons have
internal angles of 90°, 120°, and 60° respectively, from the regular polygons within the tilings:
Isotoxal skew apeirogons in two dimensions
An
isotoxal apeirogon has one edge type, between two alternating vertex types. There's a degree of freedom in the
internal angle, α. is the
dual polygon
In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other.
Properties
Regular polygons are self-dual.
The dual of an isogonal (vertex-transitive) polygon is an isotoxal (edge- ...
of an isogonal skew apeirogon.
Isogonal skew apeirogons in two dimensions
Isogonal zig-zag skew apeirogons in two dimensions
An
isogonal skew apeirogon alternates two types of edges with various
Frieze group
In mathematics, a frieze or frieze pattern is a two-dimensional design that repeats in one direction. Such patterns occur frequently in architecture and decorative art. Frieze patterns can be classified into seven types according to their symmetrie ...
symmetries. Distorted regular zig-zag skew apeirogons produce isogonal zig-zag skew apeirogons with translational symmetry:
Isogonal elongated skew apeirogons in two dimensions
Other isogonal skew apeirogons have alternate edges parallel to the Frieze direction. These isogonal elongated skew apeirogons have vertical mirror symmetry in the midpoints of the edges parallel to the Frieze direction:
Quasiregular elongated skew apeirogons in two dimensions
An isogonal elongated skew apeirogon has two different edge types; if both of its edge types have the same length: it can't be called regular because its two edge types are still different ("trans-edge" and "cis-edge"), but it can be called quasiregular.
Example quasiregular elongated skew apeirogons can be seen as truncated Petrie polygons in truncated regular tilings of the Euclidean plane:
Hyperbolic skew apeirogons
Infinite regular skew polygons are similarly found in the Euclidean plane and in the
hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ' ...
.
Hyperbolic infinite regular skew polygons also exist as
Petrie polygons zig-zagging edge paths on all
regular tilings of the hyperbolic plane. And again like in the Euclidean plane, hyperbolic infinite quasiregular skew polygons can be constructed as truncated Petrie polygons within the edges of all truncated regular tilings of the hyperbolic plane.
Infinite helical polygons in three dimensions
An infinite
helical
Helical may refer to:
* Helix
A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is for ...
(skew) polygon can exist in three dimensions, where the vertices can be seen as limited to the surface of a
cylinder. The sketch on the right is a 3D perspective view of such an infinite regular helical polygon.
This infinite helical polygon can be mostly seen as constructed from the vertices in an infinite stack of
uniform ''n''-gonal
prism
Prism usually refers to:
* Prism (optics), a transparent optical component with flat surfaces that refract light
* Prism (geometry), a kind of polyhedron
Prism may also refer to:
Science and mathematics
* Prism (geology), a type of sedimentar ...
s or
antiprism
In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation .
Antiprisms are a subclass o ...
s, although in general the twist angle is not limited to an integer divisor of 180°. An infinite helical (skew) polygon has
screw axis
A screw axis (helical axis or twist axis) is a line that is simultaneously the axis of rotation and the line along which translation of a body occurs. Chasles' theorem shows that each Euclidean displacement in three-dimensional space has a scr ...
symmetry.
An infinite stack of
prisms
Prism usually refers to:
* Prism (optics), a transparent optical component with flat surfaces that refract light
* Prism (geometry), a kind of polyhedron
Prism may also refer to:
Science and mathematics
* Prism (geology), a type of sedimentar ...
, for example cubes, contain an infinite helical polygon across the diagonals of the square faces, with a twist angle of 90° and with a Schläfli symbol # .
An infinite stack of antiprisms, for example
octahedra
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
, makes infinite helical polygons, 3 here highlighted in red, green, and blue, each with a twist angle of 60° and with a Schläfli symbol # .
A sequence of edges of a
Boerdijk–Coxeter helix
The Boerdijk–Coxeter helix, named after H. S. M. Coxeter and A. H. Boerdijk, is a linear stacking of regular tetrahedra, arranged so that the edges of the complex that belong to only one tetrahedron form three intertwined helices. There are t ...
can represent infinite regular helical polygons with an irrational twist angle:
Infinite isogonal helical polygons in three dimensions
A stack of right
prisms
Prism usually refers to:
* Prism (optics), a transparent optical component with flat surfaces that refract light
* Prism (geometry), a kind of polyhedron
Prism may also refer to:
Science and mathematics
* Prism (geology), a type of sedimentar ...
can generate isogonal helical apeirogons alternating edges around axis, and along axis; for example a stack of cubes can generate this isogonal helical apeirogon alternating red and blue edges:
Similarly an alternating stack of prisms and antiprisms can produce an infinite isogonal helical polygon; for example, a triangular stack of prisms and antiprisms with an infinite isogonal helical polygon:
An infinite isogonal helical polygon with an irrational twist angle can also be constructed from
truncated tetrahedra
In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedron ...
stacked like a
Boerdijk–Coxeter helix
The Boerdijk–Coxeter helix, named after H. S. M. Coxeter and A. H. Boerdijk, is a linear stacking of regular tetrahedra, arranged so that the edges of the complex that belong to only one tetrahedron form three intertwined helices. There are t ...
, alternating two types of edges, between pairs of hexagonal faces and pairs of triangular faces:
References
*
Coxeter, H.S.M.; ''Regular complex polytopes'' (1974). Chapter 1. ''Regular polygons'', 1.5. Regular polygons in n dimensions, 1.7. ''Zigzag and antiprismatic polygons'', 1.8. ''Helical polygons''. 4.3. ''Flags and Orthoschemes'', 11.3. ''Petrie polygons''
{{polygons
Types of polygons