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Monogon
Monogon
Monogon
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Monogon
On a circle, a monogon is a tessellation with a single vertex, and one 360-degree arc edge.
TypeRegular polygon
Edges and vertices1
Schläfli symbol{1} or h{2}
Coxeter–Dynkin diagrams or
Symmetry group[ ], Cs
Dual polygonSelf-dual

In geometry, a monogon, also known as a henagon, is a curve, considered by some as a polygon with one edge and one vertex. It has Schläfli symbol {1}.[1]

In Euclidean geometry

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In Euclidean geometry a monogon is a degenerate polygon because its endpoints must coincide, unlike any Euclidean line segment. Most definitions of a polygon in Euclidean geometry do not admit the monogon.

In spherical geometry

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In spherical geometry, a monogon can be constructed as a vertex on a great circle (equator). This forms a dihedron, {1,2}, with two hemispherical monogonal faces which share one 360° edge and one vertex. Its dual, a hosohedron, {2,1} has two antipodal vertices at the poles, one 360° lune face, and one edge (meridian) between the two vertices.[1]


Monogonal dihedron, {1,2}
Monogonal hosohedron, {2,1}

See also

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Look up monogon in Wiktionary, the free dictionary.

References

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Monogon

View on Grokipedia
from Grokipedia
A monogon, also known as a henagon, is a degenerate consisting of a single edge connected to a single vertex, forming a closed loop without area in . It is not generally recognized as a valid polygon in standard Euclidean contexts due to its failure to enclose a positive area, though the term is employed in certain mathematical disciplines such as and to describe 1-gons or embedded loops in diagrams. In non-Euclidean geometries, particularly , the monogon assumes a non-degenerate form as a hemispherical bounded by a single arc that loops back to its starting point, with a marked vertex on the boundary. This realization is unique up to among spherical surfaces of zero with one boundary component and no interior conical singularities, as duplicating and gluing two such monogons along their boundaries yields a standard with a single marked point. Such constructions highlight the monogon's role in studying curves and irreducible surfaces in curved spaces. The concept of the monogon extends to broader theory and tessellations, where it serves as a foundational element in abstract classifications of regular figures, often denoted despite its impracticality in physical constructions. In applications like and , monogons appear as basic building blocks for analyzing intersections and laminar structures on surfaces.

Definition and Symbolism

Definition

A monogon is defined as a polygon consisting of exactly one edge and one vertex, with the edge connecting back to the same vertex to form a closed loop. This structure represents the simplest possible polygonal figure, though it deviates from conventional that enclose an interior region. In standard mathematical definitions, a monogon is regarded as a degenerate case of a , as are typically required to have at least three sides to form a non-degenerate closed with positive area. The term arises in contexts like theory or , where such degenerate forms are useful for generalization, but it is not recognized as a true in due to the coincidence of its endpoints. The name "monogon" originates from the Greek prefix "mono-" (μονός, meaning "one" or "single") combined with "-gon," derived from γωνία (gōnía, meaning "angle" or "corner"), following the etymological pattern of names. It is alternatively called a henagon, using the Greek prefix "hena-" (ἕν, also meaning "one"). In abstract visualization, a monogon resembles a simple loop or a point with a self-intersecting edge that has effectively zero length, emphasizing its role as a boundary without interior. It is sometimes denoted by the {1} as a notational shorthand in studies.

Schläfli Symbol

The Schläfli symbol is a compact notation for classifying regular polytopes and tessellations, where the symbol {n} specifically denotes a regular n-gon in two dimensions. For the monogon, the Schläfli symbol is {1}, representing a degenerate regular polygon consisting of one edge. This symbol extends the notation to theoretical cases beyond standard polygons, and it appears in higher-dimensional contexts, such as extensions involving apeirogons {∞} or star polygons like {5/2}, where {1} contributes to the systematic framework for degenerate or infinite structures. The derivation follows directly from the general form {n} for regular polygons: substituting n=1 yields a figure with a single side, differing from the multi-sided configurations for n≥3 that form closed shapes in the plane. In usage, {1} completes the sequence of regular polygon symbols, paralleling {3} for the and {4} for the square, thereby ensuring theoretical consistency in the classification of all positive integer-sided regular figures.

Historical Development

Early Concepts

The foundations of polygon theory were established in ancient Greece, where the concept of a polygon emerged as a plane figure bounded by straight lines forming angles. The suffix "-gon" in polygon nomenclature originates from the Greek word gōnia, meaning "angle" or "corner," highlighting the emphasis on angular structures in geometric figures. In Euclid's Elements (c. 300 BCE), polygons are defined in Book I, Definition 19 as rectilinear figures contained by straight lines, with classifications beginning at the trilateral figure (bounded by three straight lines, or a triangle) and extending to quadrilaterals (four lines) and multilaterals (more than four lines). This framework implicitly sets a minimum of three sides for non-degenerate plane polygons, with no explicit discussion of figures possessing one or two sides, as such constructions would violate the requirement for distinct bounding lines not lying in the same straight line. During the Renaissance and early modern period, mathematicians began exploring geometric figures in curved spaces, laying groundwork for degenerate polygons like the digon (a two-sided figure). In the 19th century, abstract approaches to polygon classification by side count gained prominence, with Augustin-Louis Cauchy contributing to the study of polygonal chains and their rigidity. The transition to non-Euclidean geometries in the 1820s–1830s, pioneered independently by Nikolai Lobachevsky and János Bolyai, introduced hyperbolic spaces where polygon angle sums are less than in Euclidean geometry, indirectly accommodating degenerate cases by relaxing constraints on side counts and allowing theoretical exploration of minimal polygons without explicit dismissal. Lobachevsky's Imaginary Geometry (1829–1830) and Bolyai's Appendix (1832) focused on triangles and higher polygons but provided a framework where lower-sided figures could be conceptualized as limits in curved metrics.

Modern Formalization

The modern formalization of the monogon emerged in the mid-20th century as part of broader efforts to classify regular figures beyond traditional Euclidean constraints, with H.S.M. Coxeter playing a pivotal role. The term "monogon" (from Greek monos, "single," and gōnia, "angle") was coined by analogy to higher polygons. In his foundational text Regular Polytopes (first published 1948, third edition 1973), Coxeter incorporated the monogon into the complete spectrum of regular polygons and polytopes, treating it as a degenerate case with {1} that fits within the systematic enumeration of symmetry groups and tessellations. This inclusion extended the theory to encompass one-sided figures, allowing for a unified framework that bridges finite and infinite regular structures. Coxeter elaborated on the monogon in Regular Polytopes, describing it as a valid theoretical object akin to a point graph or a single-vertex, single-edge , emphasizing its role in projective and combinatorial contexts. These works established the monogon as a legitimate element in theory, using extended Schläfli symbols to classify degenerate and infinite polytopes alongside classical examples like the {2}. During the and , the monogon's acceptance grew within the developing field of theory, which abstracted geometric realizations to combinatorial posets independent of embedding spaces. This shift, building on Coxeter's groundwork and contributions from figures like Jacques Tits, considered the monogon in some combinatorial and geometric contexts, though it is not a standard abstract polytope of rank 1 due to issues with its face lattice structure (e.g., failing the diamond condition for a single vertex and edge). Key publications, such as Coxeter's extensions of Schläfli notation, facilitated the classification of such degenerate forms, enabling their integration into broader studies of reflection groups and symmetry. This formalization influenced subsequent research by the late 20th century, paving the way for applications in and where abstract polytopes model complex structures without physical realizability constraints. Similarly, in abstract theory, the monogon's consideration as a base case informed enumerations of higher-dimensional symmetries and their realizations.

Representations in Geometry

In Euclidean Geometry

In , the monogon manifests as a highly degenerate figure, consisting of a single edge whose endpoints coincide, effectively collapsing into a single point with zero and no enclosed area. This representation fails to form a bounded region, as the edge overlaps itself completely, resembling a self-intersecting loop at a point rather than a proper curve. Such a structure violates the foundational axioms of polygons in the plane, which require at least three non-collinear sides connected end-to-end to form a simple closed Jordan curve that separates the plane into an interior and exterior. With only one side, the monogon cannot satisfy this condition, rendering it invalid as a standard and instead classifying it among degenerate cases where all vertices lie on a straight line—or, in this extreme, coincide entirely. Mathematically, this degeneracy implies that the interior angle at the single vertex is undefined, interpretable as either 360° (a full around ) or 0° (no angular deviation), with no meaningful perimeter distinct from the point itself. Visually, it appears as a self-overlapping of extent, underscoring its non-simple status and inability to bound any positive-area domain in flat space.

In

In , the monogon emerges as a valid polygonal figure due to the intrinsic positive of , which permits closure of a single-sided without the degeneracy observed in . It is constructed along a , featuring a single vertex on the great circle and a single edge that loops fully around , spanning 360 degrees and thereby dividing the surface into two equal hemispherical regions. This configuration realizes the monogon as part of a monogonal , denoted by the {1,2}, where two monogonal faces—each a hemisphere—meet along the shared 360-degree edge, with the single vertex on the shared edge. Visually, the monogonal resembles a spherical lens, formed by the two bulging hemispherical surfaces joined seamlessly at their circular boundary, encapsulating the entire in a symmetric, doubled structure. The to the monogonal is the monogonal , with {2,1}, consisting of two antipodal vertices at the sphere's poles connected by a single edge along a meridian, and bounded by one lune-shaped face that spans the full 360 degrees around . This appears as a spherical dual, where the expansive lune face wraps the entire globe, linking the polar vertices in a manner that highlights the monogon’s role in dual spherical tilings.

Properties

Combinatorial Properties

The monogon possesses a minimal combinatorial structure as an abstract , consisting of exactly one vertex and one edge, where the edge forms a loop incident to the single vertex at both ends. This configuration defines it as the degenerate case of an n-gon for , serving as the foundational element in the sequence of regular polygons labeled by the {1}. Abstractly, the monogon bounds a single face, yielding an of χ = 1 computed via V - E + F = 1 - 1 + 1 = 1, consistent with the topology of a polygonal disk. Topologically, its boundary is homeomorphic to the circle S¹, while as a one-dimensional , it corresponds to a loop graph with a single looped edge. As the n=1 member of the n-gon family, the monogon establishes it as a limiting case where higher polygonal structures converge to this looped form without introducing additional vertices or edges.

Symmetry and Duality

The monogon possesses minimal symmetry among regular polygons, with its group being the trivial C1C_1, consisting solely of the identity transformation corresponding to a 360° . Including reflections, the full is the CsC_s of order 2, generated by a single reflection through the vertex. The monogon is self-dual, meaning it is isomorphic to its dual ; this follows from its {1}, which is self-reciprocal under the reversal operation that defines duality for regular polytopes, and from the combinatorial correspondence where the single vertex maps to the single face in the dual. In higher-dimensional regular polytopes and compounds, the monogon extends as a 1-faceted 2-dimensional cell, inheriting no non-trivial rotational symmetries beyond the identity, which underscores its role in limiting cases of polytope constructions. In contrast to the {2}, whose symmetry group is the D2D_2 of order 4 (featuring a 180° and two reflections), the monogon's CsC_s represents the endpoint of minimal in the sequence of regular polygons.
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