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Apeirogon
Apeirogon
Apeirogon
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Apeirogon (regular)
Edges and vertices∞
Schläfli symbol{∞}
Coxeter–Dynkin diagrams
Internal angle (degrees)180°
Dual polygonSelf-dual
A partition of the Euclidean line into infinitely many equal-length segments can be understood as a regular apeirogon.

In geometry, an apeirogon (from Ancient Greek ἄπειροv apeiron 'infinite, boundless' and γωνία gonia 'angle') or infinite polygon is a polygon with an infinite number of sides. Apeirogons are the rank 2 case of infinite polytopes. In some literature, the term "apeirogon" may refer only to the regular apeirogon, with an infinite dihedral group of symmetries.[1]

Definitions

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Geometric apeirogon

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Given a point A0 in a Euclidean space and a translation S, define the point Ai to be the point obtained from i applications of the translation S to A0, so Ai = Si(A0). The set of vertices Ai 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.[1]

A regular apeirogon can be defined as a partition of the Euclidean line E1 into infinitely many equal-length segments. It generalizes the regular n-gon, which may be defined as a partition of the circle S1 into finitely many equal-length segments.[2]

Hyperbolic pseudogon

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The regular pseudogon is a partition of the hyperbolic line H1 (instead of the Euclidean line) into segments of length 2λ, as an analogue of the regular apeirogon.[2]

Abstract apeirogon

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An abstract polytope is a partially ordered set P (whose elements are called faces) with properties modeling those of the inclusions of faces of convex polytopes. The rank (or dimension) of an abstract polytope is determined by the length of the maximal ordered chains of its faces, and an abstract polytope of rank n is called an abstract n-polytope.[3]: 22–25 

For abstract polytopes of rank 2, this means that: A) the elements of the partially ordered set are sets of vertices with either zero vertex (the empty set), one vertex, two vertices (an edge), or the entire vertex set (a two-dimensional face), ordered by inclusion of sets; B) each vertex belongs to exactly two edges; C) the undirected graph formed by the vertices and edges is connected.[3]: 22–25 [4]: 224 

An abstract polytope is called an abstract apeirotope if it has infinitely many elements; an abstract 2-apeirotope is called an abstract apeirogon.[3]: 25 

A realization of an abstract polytope is a mapping of its vertices to points a geometric space (typically a Euclidean space).[3]: 121  A faithful realization is a realization such that the vertex mapping is injective.[3]: 122 [note 1] Every geometric apeirogon is a realization of the abstract apeirogon.

Symmetries

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The order-3 apeirogonal tiling, {∞,3}, fills the hyperbolic plane with apeirogons whose vertices lie along horocyclic paths.

The infinite dihedral group G of symmetries of a regular geometric apeirogon is generated by two reflections, the product of which translates each vertex of P to the next.[3]: 140–141 [4]: 231  The product of the two reflections can be decomposed as a product of a non-zero translation, finitely many rotations, and a possibly trivial reflection.[3]: 141 [4]: 231 

In an abstract polytope, a flag is a collection of one face of each dimension, all incident to each other (that is, comparable in the partial order); an abstract polytope is called regular if it has symmetries (structure-preserving permutations of its elements) that take any flag to any other flag. In the case of a two-dimensional abstract polytope, this is automatically true; the symmetries of the apeirogon form the infinite dihedral group.[3]: 31 

A symmetric realization of an abstract apeirogon is defined as a mapping from its vertices to a finite-dimensional geometric space (typically a Euclidean space) such that every symmetry of the abstract apeirogon corresponds to an isometry of the images of the mapping.[3]: 121 [4]: 225 

Moduli space

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Generally, the moduli space of a faithful realization of an abstract polytope is a convex cone of infinite dimension.[3]: 127 [4]: 229–230  The realization cone of the abstract apeirogon has uncountably infinite algebraic dimension and cannot be closed in the Euclidean topology.[3]: 141 [4]: 232 

Classification of Euclidean apeirogons

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The symmetric realization of any regular polygon in Euclidean space of dimension greater than 2 is reducible, meaning it can be made as a blend of two lower-dimensional polygons.[3] This characterization of the regular polygons naturally characterizes the regular apeirogons as well. The discrete apeirogons are the results of blending the 1-dimensional apeirogon with other polygons.[4]: 231  Since every polygon is a quotient of the apeirogon, the blend of any polygon with an apeirogon produces another apeirogon.[3]

In two dimensions the discrete regular apeirogons are the infinite zigzag polygons,[5] resulting from the blend of the 1-dimensional apeirogon with the digon, represented with the Schläfli symbol {∞}#{2}, {∞}#{}, or .[3]

In three dimensions the discrete regular apeirogons are the infinite helical polygons,[5] with vertices spaced evenly along a helix. These are the result of blending the 1-dimensional apeirogon with a 2-dimensional polygon, {∞}#{p/q} or .[3]

Generalizations

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Higher rank

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Main articles: Apeirotope and Apeirohedron

Apeirohedra are the rank 3 analogues of apeirogons, and are the infinite analogues of polyhedra.[6] More generally, n-apeirotopes or infinite n-polytopes are the n-dimensional analogues of apeirogons, and are the infinite analogues of n-polytopes.[3]: 22–25 

See also

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Notes

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Apeirogon

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An apeirogon is a polygon with an infinite number of sides and vertices, serving as the limiting case of a regular n-gon as the number of sides n approaches infinity. The term originates from the Ancient Greek words ἄπειρος (apeiros), meaning "infinite" or "boundless," and γωνία (gōnía), meaning "angle." In its regular form, denoted by the Schläfli symbol {∞}, an apeirogon generalizes the structure of finite regular polygons and represents a fundamental infinite polytope of rank 2. A regular apeirogon in Euclidean geometry can be realized as a partition of the one-dimensional Euclidean line into infinitely many equal-length segments, with vertices at integer coordinates when the edge length is 1. This configuration forms the only regular tiling of 1-dimensional space, consisting of an infinite sequence of dyads (line segments) and exhibiting apeirogonal symmetry, denoted as the Coxeter group W₂ of order 2∞. In this linear embedding, the internal angles are π radians, and it serves as a degenerate tiling {∞, 2} on the Euclidean plane. Unlike finite polygons, the apeirogon has no bounded area in the plane but extends infinitely, with its vertex figure being a dyad of length 2. In non-Euclidean geometries, apeirogons take on more varied forms. In , a regular apeirogon can be inscribed in a when the condition tanh(s/2) sec(θ/2) = 1 holds, where s is the edge length and θ is the internal angle; if this value exceeds 1, it inscribes in a hypercycle or equidistant curve, sometimes termed a pseudogon. These hyperbolic apeirogons enable regular tilings of the hyperbolic plane, such as the apeirogonal tiling {∞, k}, and contribute to infinite like apeirohedra. No regular apeirogons exist on due to its positive . The concept of the apeirogon emerged in the study of regular polytopes and Coxeter groups, prominently featured in H. S. M. Coxeter's foundational work Regular Polytopes (first published ), where it is described as an infinite analogue of finite polygons and used to construct higher-dimensional apeirotopes through operations like blending with segments. Apeirogons play a key role in abstract regular polytopes for arbitrary Coxeter groups, appearing as facets, vertex figures, or 2-faces in classifications of infinite polytopes, and have applications in understanding symmetries and tilings in low-dimensional spaces.

Introduction and Definitions

Basic Definition

An apeirogon is a generalized consisting of a countably infinite number of sides and vertices, extending the definition of a finite to the limiting case where the number of sides tends to infinity. The term "apeirogon" derives from the ἄπειρος (apeiros), meaning "infinite" or "boundless," and γωνία (gōnía), meaning "." Unlike finite polygons, which possess finite perimeters and enclose finite areas, an apeirogon features an infinite perimeter due to its unbounded sequence of edges and does not enclose a bounded area but extends infinitely. The regular apeirogon, characterized by equal side lengths and equal vertex angles in the limit, is represented by the Schläfli symbol {∞}. This figure serves as a degenerate form of a , where the interior angles approach 180 degrees as the number of sides becomes infinite.

Historical Context

The concept of the apeirogon traces its origins to 19th-century advancements in projective and higher-dimensional geometry, where mathematicians like explored infinite configurations and points at infinity in projective spaces, laying groundwork for unbounded polygonal structures. Ludwig Schläfli advanced these ideas in his seminal 1852 work on polytopes, developing the notation {p,q,...} for regular figures, which was later extended to infinite-sided cases like the apeirogon denoted by {∞} as part of a systematic classification of higher-dimensional geometries. A pivotal milestone came in 1948 with H.S.M. Coxeter's book Regular Polytopes, which formalized the apeirogon as a regular infinite within the framework of Euclidean and non-Euclidean geometries, integrating it into the broader theory of regular polytopes and their symmetries. Coxeter's treatment highlighted the apeirogon's role in infinite regular honeycombs and tilings, building on earlier symbolic notations but providing the first comprehensive geometric realization. The abstract perspective on apeirogons was rigorously developed by Peter McMullen and Egon Schulte in their 2002 book Abstract Regular Polytopes, where they defined abstract versions as combinatorial structures with infinite facets, emphasizing their group-theoretic properties independent of embedding spaces. In , the study of apeirogons, including pseudogons as analogs inscribed in hypercycles, continued to evolve, as discussed in works like Norman W. Johnson's 2018 book Geometries and Transformations, which explores their partitioning of hyperbolic lines into infinite regular segments and transformations under hyperbolic isometries. Pre-1948 literature reveals gaps in formal treatment, with informal mentions of infinite polygons appearing in studies of tilings and Fuchsian groups, such as Poincaré's analysis of hyperbolic surfaces, where unbounded polygonal tilings were discussed without explicit regularization.

Types and Realizations

Geometric Apeirogon

In the , a geometric apeirogon is constructed as an infinite of equal-length edges connecting a countably infinite of distinct vertices. The vertices are generated by starting with an initial point A0A_0 and applying successive translations by a fixed vector v\mathbf{v} of length equal to the edge length, yielding positions An=A0+nvA_n = A_0 + n \mathbf{v} for each nn. This results in a straight infinite aligned along the direction of v\mathbf{v}, partitioning a line into equal segments. Such a realization forms the regular apeirogon with {∞}\{\infty\}, where all edges are congruent and successive edges meet at 180-degree internal angles, making it degenerate as a flat figure. An alternative coplanar embedding produces an infinite chain, with vertices alternating between two while maintaining equal edge lengths and consistent turning angles, though not collinear overall. The geometric apeirogon arises as the limiting case of a regular nn-gon with fixed edge length as n→∞n \to \infty, where the increasing number of sides and approaching 180-degree vertex angles flatten the figure into an infinite straight chain of equal edges. Finite-density realizations, such as the straight chain or , consist of discrete vertices extending unbounded across the plane without filling any bounded region densely. In contrast, infinite-density versions, approached via the limit of an nn-gon with fixed circumradius, place vertices densely along a , while unbounded variants like spirals maintain discrete placement but expand outward indefinitely.

Hyperbolic Pseudogon

The hyperbolic pseudogon is a regular infinite-sided realized in the hyperbolic plane when the tanh(ss/2) sec(θ\theta/2) > 1, where ss is the edge length and θ\theta is the internal angle; in this case, it is inscribed in a hypercycle or equidistant curve. It consists of an infinite sequence of vertices lying on a hypercycle, connected by edges each of fixed hyperbolic length. This construction partitions the hypercycle into countably infinite segments of equal length, generating a figure whose involves reflections across the edges. The vertices accumulate asymptotically toward two ideal endpoints on the boundary at infinity. In the , the hypercycle appears as a (not orthogonal to the boundary). The vertices are positioned along this arc such that the hyperbolic distance between consecutive ones is constant, causing the points to accumulate toward the two ideal endpoints. These ideal points function as limiting "vertices at ," distinguishing the pseudogon from finite polygons, as the figure never closes but "encloses" an infinite region on one side, reflecting hyperbolic space's properties. The hyperbolic distance formula in this model, d=\arccosh(1+2∣z1−z2∣2(1−∣z1∣2)(1−∣z2∣2))d = \arccosh\left(1 + \frac{2|z_1 - z_2|^2}{(1 - |z_1|^2)(1 - |z_2|^2)}\right), ensures constant edge lengths despite Euclidean contraction near the boundary. The gon's edges relate to , but unlike the horocycle-inscribed apeirogon, the configuration uses the hypercycle for vertices. In some apeirogonal tilings, pseudogons appear when the angle condition leads to hypercycle inscription. The "" designation highlights the ideal endpoints at , preventing closure and yielding infinite area. For comparison, the regular hyperbolic apeirogon (when tanh(ss/2) sec(θ\theta/2) = 1) has vertices on a horocycle, approaching a single ideal point, and edges tangent to another horocycle; this form is central to regular tilings like {\infty, k}. A degenerate linear realization on a exists but is analogous to the Euclidean case with 180° angles.

Abstract Apeirogon

An abstract apeirogon is defined combinatorially as a rank-2 infinite , realized as a (poset) whose elements consist of the empty face at rank -1, infinitely many vertices at rank 0, infinitely many edges at rank 1, and the entire figure at rank 2, ordered by the incidence relation among these faces. In this poset, the covering relations ensure that each edge is incident to exactly two vertices, and each vertex is incident to exactly two edges, forming an infinite chain of alternating vertices and edges that extends bidirectionally without bound. This structure encodes the essential combinatorial properties of an infinite , independent of any metric or . The abstract apeirogon features two infinite flags, reflecting its unbounded nature, and its acts transitively on the edges, preserving the incidence relations while permuting the infinite set of edges freely. The Coxeter diagram for the regular abstract apeirogon, denoted by the {∞}, is represented as a single node with an ∞ label, signifying infinite valence in terms of the unbounded number of sides. A key of the abstract apeirogon is that its facet lattice—the poset induced by the vertices and edges—is isomorphic to the order complex of the infinite dihedral group, which serves as the defining for this structure. This property positions the abstract apeirogon as the universal object among all such rank-2 polytopes of type {∞}. Up to , there exists a unique abstract apeirogon fulfilling these conditions, known as the universal {∞}, which provides the foundational combinatorial template for all regular apeirogons.

Symmetries and Structure

Symmetry Groups

The primary symmetry group associated with the apeirogon is the infinite dihedral group Dih(∞)\mathrm{Dih}(\infty), which captures the isometries preserving its infinite structure. This group arises as the denoted by [∞][\infty], an affine acting on the real line or its embedding in the . Dih(∞)\mathrm{Dih}(\infty) is generated by two reflections σ1\sigma_1 and σ2\sigma_2, satisfying the relation (σ1σ2)∞=id(\sigma_1 \sigma_2)^\infty = \mathrm{id}, meaning the product σ1σ2\sigma_1 \sigma_2 has infinite order and corresponds to a along the apeirogon's axis. The group's actions include these translations, which shift the infinite of edges and vertices by multiples of the fundamental distance (twice the separation between the reflection hyperplanes), and the individual reflections, which reverse orientation across specific points such as vertices or edge midpoints. Additionally, 180° rotations around midpoints of edges emerge as compositions within the group, equivalent to point reflections in the one-dimensional realization, preserving the apeirogon's regularity. A standard presentation of Dih(∞)\mathrm{Dih}(\infty) in terms of a rotation and a reflection is ⟨r,s∣s2=r2=(rs)∞=1⟩\langle r, s \mid s^2 = r^2 = (rs)^\infty = 1 \rangle, where rr generates the 180° rotations around edge midpoints and ss denotes a reflection, with the infinite-order relation reflecting the unbounded nature of the figure. For the regular apeirogon {∞}\{\infty\}, the full symmetry group is Dih(∞)\mathrm{Dih}(\infty). This structure aligns with the affine [∞][\infty], emphasizing the apeirogon's role as a one-dimensional regular polytope.

Moduli Space

The of apeirogons consists of all possible realizations of the abstract apeirogon up to similarity transformations, including translations, rotations, and scalings, thereby classifying distinct geometric configurations by their intrinsic shapes. Generally, the of a faithful realization of the abstract apeirogon is a of infinite . The realization cone has uncountably infinite algebraic and cannot be closed in the .

Euclidean Apeirogons

Classification

The linear apeirogon, denoted {∞}, is the basic regular Euclidean apeirogon, realized as vertices at points on a straight line with edge length 1. Euclidean apeirogons are classified combinatorially and topologically into types including the and more general equilateral and helical forms, with distinctions based on self-intersection and parameters. The equilateral apeirogon is a primary skew type, denoted by the symbol {∞}#{2}, which features a of edges that alternate direction in a plane, effectively "closing" the turn after two steps to form a non-self-intersecting infinite path generated by glide reflections. This form is regular when the turns are symmetrical and even, lying coplanar but not collinear, and serves as a foundational simple apeirogon in with d=1. Helical generalizations extend this structure to {∞}#{p/q}, where p and q are coprime positive integers, describing an infinite polygon that winds q times around an axis over p steps, producing a skew path in three-dimensional Euclidean space that projects to a zigzag in the plane perpendicular to the axis. These forms maintain equilateral edges and equal turning angles but introduce a helical twist, with the turning ratio p/q determining the winding behavior; for instance, {∞}#{1/3} spirals with a single wind over three steps. Topologically, Euclidean apeirogons are distinguished as simple (non-self-intersecting) or types, where simple apeirogons trace a single infinite path without crossings, while apeirogons exhibit self-intersections forming multiple intertwined paths with greater than 1. The parameter d = |p/q| quantifies this for helical and variants, with d = 1 for the simple {∞}#{2} as a special case, and higher densities yielding configurations that compound multiple simple paths. Enumeration of these apeirogons yields infinite families parameterized by the rational density d = |p/q| in lowest terms, encompassing all coprime pairs (p, q) and capturing both planar zigzags and spatial helices as discrete types under combinatorial equivalence. criteria for these families, up to affine transformations of , rely on the rational turning ratios p/q, where two apeirogons are equivalent if their ratios match after reduction, preserving the topological and invariants. The turning angle θ, related to these ratios, parameterizes continuous variations within each discrete type but is secondary to the combinatorial .

Examples

One prominent example of a Euclidean apeirogon is the simple zigzag form, denoted by the Schläfli symbol {∞}#{2}. This structure lies in a plane with vertices alternating between two , forming an infinite chain of equal-length edges that indefinitely without closing or intersecting itself beyond adjacent edges. The exact realization traces an infinite straight path with regular alternations along the direction perpendicular to the parallel lines. A helical example is the apeirogon {∞}#{3/2}, which extends non-planarly in three-dimensional Euclidean space. This configuration winds twice around every three edges, producing a structure with density parameter d = 3/2 that is periodic along the axis and spirals along a helical path. Vertices follow a parametric curve such as (a \cos \beta t, a \sin \beta t, b t) for integer t, where a and b determine the radius and pitch, and 0 < \beta < \pi ensures equal edge lengths without self-intersection in the infinite limit. The star apeirogon {∞/5}#{2} represents a more complex Euclidean realization, extending the {5/2} infinitely with 5/2. It forms a planar or slightly skewed infinite figure where edges intersect in a star-like pattern that repeats without bound, analogous to a perpetual pentagrammic lace. For computational visualization of these apeirogons, finite approximations can be plotted using parametric equations that accumulate positions via fixed turning angles \phi_k. Specifically, the vertex coordinates are given by xn=∑k=1ncos⁡ϕk,yn=∑k=1nsin⁡ϕk,x_n = \sum_{k=1}^n \cos \phi_k, \quad y_n = \sum_{k=1}^n \sin \phi_k, with \phi_k constant for regular turning (e.g., \phi = \pi for basic zigzag alternations, or fractional multiples for helical/star variants), allowing truncation at large n to approximate the infinite form. In three dimensions, a z-component can be added as bz_n for helical paths. These methods reveal the apeirogon's asymptotic behavior, such as linear extension for zigzags or cylindrical winding for helices. Euclidean apeirogons also arise as limits in uniform polyhedra or Euclidean honeycombs, such as infinite prismatic structures, where increasing the side count of polygons approaches infinite-sided faces.

Generalizations

Higher Rank Polytopes

The concept of the apeirogon extends naturally to higher-rank abstract polytopes through the notion of apeirotopes, which are infinite regular polytopes of rank greater than 2. In particular, an apeirohedron is a rank-3 abstract polytope denoted by the Schläfli symbol {∞,3}, featuring infinitely many triangular faces meeting three at each edge, with the overall structure manifesting as an infinite-sided . A canonical geometric realization is the infinite skew polyhedron, where faces wind helically around a central axis in Euclidean 3-space. More generally, an n-apeirotope is a rank-n with a single infinite , characterized by the {∞, p_1, \dots, p_{n-2}}, where the ∞ indicates infinitely many facets, and the subsequent p_i denote the finite branching factors for lower-dimensional elements. These structures maintain the combinatorial regularity of finite polytopes but incorporate at one end of the incidence , leading to unbounded vertex figures while keeping facets and ridges finite in local configuration. The groups of n-apeirotopes correspond to infinite Coxeter groups, represented by linear Coxeter forming a chain of nodes with an ∞-labeled branch at one terminal node, signifying parallel hyperplanes that do not intersect. These groups are universal in the sense that the apeirotope is generated by reflections across the hyperplanes defined by the , ensuring flag-transitivity and the axioms. Representative examples include the rank-4 apeirotope {∞,3,3}, which arises as a limiting case of the cubic {4,3,4} under affine deformation, yielding an infinite structure with cubic vertex figures and infinite prismatic facets. Similarly, {∞,3,4} describes an apeirogonal with order-3 apeirogonal tiling cells, three meeting at each edge and four at each vertex, octahedral vertex figures, and apeirogonal bases, realized as a skew apeirotope in 3-dimensional .

Infinite Tilings and Honeycombs

Apeirogons serve as fundamental cells in certain uniform tilings of , particularly in degenerate or infinite configurations. For instance, the order-2 apeirogonal tiling, denoted by the {∞,2}, represents an infinite that divides the into two half-planes along a straight line, with each apeirogon consisting of infinitely many collinear edges of equal length. This tiling arises as a limiting case of regular polygonal tilings and is characterized by internal angles of π radians, effectively forming a single infinite edge shared by two apeirogonal faces. In higher dimensions, apeirogonal prisms contribute to Euclidean , such as apeirogonal prismatic , where infinite strips of squares are extruded along an infinite direction to fill space periodically. In hyperbolic geometry, apeirogons play a prominent role in the construction of regular honeycombs, often appearing as cells or vertex figures. The {∞,3,3} honeycomb, for example, features cells that are order-3 apeirogonal tilings {∞,3}, with three meeting at each edge and four at each vertex, and the vertex figure is a tetrahedron {3,3}. This structure fills hyperbolic 3-space compactly, with vertices lying on horocycles and edges of finite length determined by hyperbolic metrics, as detailed in Coxeter's enumeration of infinite regular honeycombs incorporating the ∞ symbol in their Schläfli notation. Similarly, the {4,4,∞} honeycomb uses infinite-order square tiling vertex figures to tile hyperbolic space with square tiling cells, extending the concept of infinite-sided polygons to three-dimensional packings. These honeycombs are paracompact, meaning their cells are unbounded but the overall structure is locally finite. Coxeter also describes {3,∞,3} (with infinite-order triangular tiling cells and order-3 apeirogonal vertex figures) and {3,3,∞} (with tetrahedral cells and infinite-order triangular tiling vertex figures) as additional paracompact examples. Apeirogons also feature in infinite compounds derived from kaleidoscopic constructions, where reflection groups generate stellated or density-compounded arrangements of infinite polygons. Coxeter's work on regular star-polytopes and describes such compounds, including those with apeirogonal facets emerging from infinite dihedral symmetries in hyperbolic kaleidoscopes. These constructions produce uniform compounds with infinite , useful for understanding in non-compact spaces. In practical applications, apeirogonal structures inform the design of infinite patterns in , such as repetitive hyperbolic motifs in tiling-based facades that simulate endless geometric progression, drawing from Escher-inspired hyperbolic tilings. In , apeirogons model periodic structures in simulations of lattices or infinite graphs, facilitating algorithms for and visualization in unbounded domains. Coxeter's infinite regular , including examples like {5,3,∞} with apeirogonal vertex figures, provide seminal benchmarks for these models, emphasizing their role in scaling finite polyhedral approximations to infinite geometries.
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