Apeirogon
Encyclopedia
An apeirogon is a degenerate
polygon
with a countably
infinite number of sides. It is the limit of a sequence of polygons with more and more sides.
Like any polygon, it is a sequence of line segments (edges) and angles (corners). But whereas an ordinary polygon has no ends because it is a closed circuit, an apeirogon can also have no ends because you can never make the infinite number of steps needed to get to the end in either direction. Closed apeirogons also exist. They occur when the corners form sequences (one in each direction, starting from any point) whose limits converge on the same point. Such a point is called an accumulation point, and any closed apeirogon must have at least one of them.
. Its Schläfli symbol is {∞}.
If the corner angles are 180 degrees, the overall form of the apeirogon resembles a straight line
:
...
...
This line may be considered as a circle
of infinite radius, by analogy with regular polygons with great number of edges, which resemble a circle.
For some time, people thought this was the only regular example. Then Branko Grünbaum
discovered two more.
If the corner angles alternate either side of the figure, the apeirogon resembles a zig-zag, and has 2*∞ Frieze group
symmetry.
...
...
If each corner angle is displaced out of the plane of the previous angle, the apeirogon resembles a three-dimensional helix
. A polygon such as this which does not lie in a plane, is said to be skew
. The sketch on the right is a 3D perspective view of such a regular skew apeirogon.
This polygon can be constructed from a sequential subset of edges within an infinite stack of uniform
n-gonal antiprism
s, although unlike the antiprisms, the twist angle is not limited to an integer divisor of 180°. This polygon has screw axis
.
Degeneracy (mathematics)
In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class....
polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
with a countably
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
infinite number of sides. It is the limit of a sequence of polygons with more and more sides.
Like any polygon, it is a sequence of line segments (edges) and angles (corners). But whereas an ordinary polygon has no ends because it is a closed circuit, an apeirogon can also have no ends because you can never make the infinite number of steps needed to get to the end in either direction. Closed apeirogons also exist. They occur when the corners form sequences (one in each direction, starting from any point) whose limits converge on the same point. Such a point is called an accumulation point, and any closed apeirogon must have at least one of them.
Regular apeirogons
A regular apeirogon has equal-length sides and equal corner angles, just like any regular polygonRegular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...
. Its Schläfli symbol is {∞}.
If the corner angles are 180 degrees, the overall form of the apeirogon resembles a straight line
Line (mathematics)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
:
...
...This line may be considered as a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
of infinite radius, by analogy with regular polygons with great number of edges, which resemble a circle.
Skew forms
Branko Grünbaum
Branko Grünbaum is a Croatian-born mathematician and a professor emeritus at the University of Washington in Seattle. He received his Ph.D. in 1957 from Hebrew University of Jerusalem in Israel....
discovered two more.
If the corner angles alternate either side of the figure, the apeirogon resembles a zig-zag, and has 2*∞ Frieze group
Frieze group
A frieze group is a mathematical concept to classify designs on two-dimensional surfaces which are repetitive in one direction, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art...
symmetry.
...
...If each corner angle is displaced out of the plane of the previous angle, the apeirogon resembles a three-dimensional helix
Helix
A helix is a type of smooth space curve, i.e. a curve in three-dimensional space. It has the property that the tangent line at any point makes a constant angle with a fixed line called the axis. Examples of helixes are coil springs and the handrails of spiral staircases. A "filled-in" helix – for...
. A polygon such as this which does not lie in a plane, is said to be skew
Skew polygon
In geometry, a skew polygon is a polygon whose vertices do not lie in a plane. Skew polygons must have at least 4 vertices.A regular skew polygon is a skew polygon with equal edge lengths and which is vertex-transitive....
. The sketch on the right is a 3D perspective view of such a regular skew apeirogon.
This polygon can be constructed from a sequential subset of edges within an infinite stack of uniform
Prismatic uniform polyhedron
In geometry, a prismatic uniform polyhedron is a uniform polyhedron with dihedral symmetry. They exist in two infinite families, the uniform prisms and the uniform antiprisms...
n-gonal antiprism
Antiprism
In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles...
s, although unlike the antiprisms, the twist angle is not limited to an integer divisor of 180°. This polygon has screw axis
Screw axis
The screw axis of an object is a line that is simultaneously the axis of rotation and the line along which a translation occurs...
.
See also
- Apeirogonal prismApeirogonal prismIn geometry, an apeirogonal prism or infinite prism is the arithmetic limit of the family of prisms; it can be considered an infinite polyhedron or a tiling of the plane....
- Apeirogonal antiprismApeirogonal antiprismIn geometry, an apeirogonal antiprism or infinite antiprism is the arithmetic limit of the family of antiprisms; it can be considered an infinite polyhedron or a tiling of the plane.If the sides are equilateral triangles, it is a uniform tiling...
- ApeirohedronApeirohedronAn apeirohedron is a polyhedron having infinitely many faces. Like an ordinary polyhedron it forms a surface with no border. But where an ordinary polyhedral surface has no border because it folds round to close back on itself, an apeirohedron has no border because its surface is unbounded.Two main...
- CircleCircleA circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
- Closed, convex round figure that can count as a regular apeirogon.