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Star polygon

A star polygon is a non-convex polygon formed by connecting every q-th point out of p equally spaced points lying on the circumference of a circle, where p and q are coprime positive integers and q < p/2, and denoted using the Schläfli symbol {p/q}.^[1] The density of such a polygon is q, representing the number of times the figure winds around its center before closing.^[1] Notable examples include the pentagram {5/2}, a five-pointed star, and the hexagram {6/2}, known as the Star of David.^[2]^[3] For a star polygon {p/q} with unit edge lengths and (p,q)=1, the circumradius R is given by the formula R = \frac{\sin\left(\frac{p-2q}{2p}\pi\right)}{\sin\left(\frac{2q}{p}\pi\right)}.^[1] When p and q are not coprime, the resulting figure is a star polygon compound rather than a single polygon.^[1] Regular star polygons were first systematically studied in the 14th century by the English mathematician Thomas Bradwardine in his work Geometria speculativa.^[4]

Definitions and Basics

Definition

A star polygon is a non-convex polygon formed by connecting vertices of a regular polygon in a non-adjacent order, resulting in self-intersecting edges.^[1] Unlike convex polygons, which have all interior angles less than 180 degrees and no intersecting sides, star polygons feature edges that cross each other, producing a star-like appearance.^[1] Key attributes of star polygons include their self-intersections, which create intricate internal structures, and their density, which measures how many times the boundary winds around the center, distinguishing it from simple polygons.^[5] This density measures the complexity of the figure's winding around its center, distinguishing it from simple polygons.^[5] Regular star polygons were first systematically studied in the 14th century by Thomas Bradwardine, with Johannes Kepler providing further descriptions in 1619 in his work Harmonices Mundi, using the term "stellatio" to refer to star figures formed by extending edges of polygons.^[6] Kepler's exploration laid foundational insights into these non-convex forms, integrating them into broader geometric and harmonic studies.^[7] A basic example is the pentagram, denoted by the Schläfli symbol {5/2}, which connects every second vertex of a regular pentagon to form a five-pointed star.^[2] This figure exemplifies the self-intersecting nature of star polygons and serves as the simplest such construction.^[2]

Terminology and Notation

The standard notation for regular star polygons is the Schläfli symbol \{n/d\}, introduced by Ludwig Schläfli and systematized by H.S.M. Coxeter, where n represents the number of vertices (or sides) and d denotes the density or step size, indicating that every d-th vertex is connected in sequence around a circle. This symbol applies to non-convex polygons with $1 < d < n/2, ensuring the figure is neither convex nor retrograde, and for simple (primitive) star polygons, \gcd(n,d)=1 to guarantee a single connected component without decomposition into multiples.^[1] (Coxeter, Regular Polytopes, 3rd ed., 1973) Variations in notation exist, with density sometimes denoted by k instead of d, particularly in earlier literature or computational contexts, though \{n/d\} remains the conventional form for describing the topological and geometric properties of these figures. For compound star polygons, where \gcd(n,d)=g > 1, the symbol \{n/d\} represents a regular compound consisting of g intertwined copies of the primitive star polygon \{n/g, d/g\}, often notated explicitly as g\{n/g, d/g\} to emphasize the multiplicity. A star polygon is termed primitive (or simple) if \gcd(n,d)=1, resulting in a single, indivisible component; otherwise, it decomposes into a compound.^[1] Key terminology distinguishes simple star polygons, which form a single connected path, from compound ones that interlace multiple such paths. Regular star polygons are isogonal, possessing equal interior angles at each vertex due to their rotational symmetry. These two-dimensional constructs differ from star polyhedra, which extend the concept to three-dimensional regular compounds with star polygonal faces. As an illustrative example, the pentagram corresponds to \{5/2\}.^[1]^[8]

Types of Star Polygons

Regular Star Polygons

Regular star polygons represent the most symmetric class of star polygons, characterized by being equilateral—all sides of equal length—and equiangular—all vertex angles equal—while exhibiting self-intersections and possessing rotational symmetry of order [n](/page/N+), where [n](/page/N+) is the number of vertices. These figures maintain the same high degree of symmetry as convex regular n-gons but achieve their star-like appearance through a non-adjacent connection of vertices. The Schläfli symbol \{n/k\} briefly denotes such a polygon, with k indicating the density or step size in vertex connections, ensuring \gcd(n,k)=1 and $1 < k < n/2.^[1]^[9] Classic examples illustrate the diversity within this category. The pentagram \{5/2\}, or five-pointed star, arises from connecting every second vertex of a regular pentagon, resulting in a figure with five intersecting sides that enclose a pentagonal core. For seven vertices, the heptagrams \{7/2\} and \{7/3\} provide two non-isomorphic forms, each with seven sides and distinct intersection patterns; these are mirror images, reflecting the chiral nature possible in odd-n cases. The octagram \{8/3\} similarly connects every third vertex of a regular octagon, yielding an eight-pointed star with a more intricate overlapping structure. These examples highlight how varying k alters the visual density while preserving overall regularity.^[1]^[2]^[9] The symmetry group governing a regular star polygon \{n/k\} is the dihedral group D_n, which includes n rotational symmetries around the center and n reflection symmetries across axes passing through vertices or midpoints of sides; this group acts transitively on the vertices, independent of the specific k value. This full dihedral symmetry ensures that the figure looks identical after any of its $2n transformations, underscoring its regularity despite self-intersections. Regarding the vertex configuration, the n vertices lie precisely at the positions of a regular n-gon inscribed in a circle, providing the foundational equispaced arrangement. Furthermore, the intersection points generated by the edges often form additional regular polygons; notably, in the pentagram \{5/2\}, the five primary inner intersections delineate a smaller regular pentagon.^[1]^[9]^[10]

Compound Star Polygons

Compound star polygons are formed by the superposition of multiple regular polygons or star polygons that share a common center and vertex set, resulting in a regular compound figure. These arise specifically when constructing a star polygon denoted by the Schläfli symbol {n/d}, where the greatest common divisor g = \gcd(n, d) > 1. In such cases, the figure decomposes into g identical primitive components, each being a regular { (n/g)/(d/g) } polygon, with the components rotated by multiples of 360^\circ / n relative to one another around the center. This decomposition ensures the overall symmetry remains regular, as described in foundational geometric analyses.^[1] A classic example is {9/3}, where g = 3, yielding a compound of three equilateral triangles denoted as 3{3/1}. Similarly, {10/4} with g = 2 produces two interlocked pentagrams, or 2{5/2}. Another simple case is {6/2} = 2{3/1}, consisting of two triangles. In three dimensions, Kepler's stella octangula serves as a polyhedral analog, comprising two interpenetrating regular tetrahedra, highlighting the extension of compound principles beyond the plane.^[1]^[11] The enumeration of compound star polygons for a fixed n counts the distinct {n/d} where 1 < d < n/2 and \gcd(n, d) > 1, accounting for the equivalence {n/d} = {n/(n-d)}. For n = 8, there are two such compounds: {8/2} = 2{4/1} (two squares) and {8/4} = 4{2/1} (four digons). This systematic counting reveals the limited but structured variety of regular compounds for each n.^[1]

Construction Methods

Vertex Connection Approach

The vertex connection approach constructs a star polygon by linking vertices of a regular convex n-gon in a specific skipping pattern, using the Schläfli symbol {n/d} to denote the figure where d indicates the step size.^[1] To form {n/d}, begin at one vertex of the n-gon and connect it to the vertex d positions away in a consistent direction, repeating this process until returning to the starting point; if n and d are coprime (gcd(n,d)=1), the path is primitive and closes after exactly n steps, producing a single connected component.^[1] The parameter d measures the winding of the connection around the center, determining how many times the edges overlap before completing the figure.^[1] The symbols {n/d} and {n/(n-d)} represent the same star polygon, though one is the mirror image of the other due to opposite traversal directions.^[1] Degenerate cases occur when the skipping pattern fails to produce a proper star, such as d=1, which yields the convex regular n-gon {n/1} without intersections.^[1] For even n, d=n/2 results in a digonal degenerate like {4/2}, consisting of n/2 overlapping diameters rather than a closed polygonal path.^[12] For odd n, d=(n-1)/2 produces a non-degenerate star, but cases where gcd(n,d)>1 generally form compounds rather than single stars.^[1] A representative illustration is the pentagram {5/2}, constructed from a regular pentagon with vertices labeled sequentially as A, B, C, D, E. Start at A and connect to C (skipping B), then from C to E (skipping D), E to B (skipping A), B to D (skipping C), and D back to A (skipping E); this sequence generates five line segments that intersect at five interior points, forming the characteristic five-pointed star.^[2] The intersections occur at points that divide each segment in the golden ratio, creating a smaller regular pentagon at the center.^[2]

Stellation Process

The stellation process for generating star polygons involves extending the sides of a regular convex polygon outward until they intersect, forming a star-shaped envelope that encloses the original figure. This method, pioneered by Johannes Kepler in his 1619 work Harmonices Mundi, applies to any regular n-gon where n ≥ 5, producing a non-convex polygon with intersecting edges.^[13] Kepler described stellate polygons, particularly pentagonal variants, as extensions where the sides meet to create pointed forms, emphasizing their geometric harmony.^[13] In practice, starting from a regular convex n-gon, each side is prolonged infinitely in both directions symmetrically around its midpoint, and the outermost intersection points define the vertices of the resulting star polygon. This continuous extension contrasts with discrete vertex connections and yields a single envelope per polygon, though multiple intersection layers may appear internally. For instance, the stellation of a regular pentagon produces the pentagram, denoted as the star polygon {5/2}, where the extended sides intersect to form a five-pointed star with density 2.^[13]^[2] The two-dimensional stellation process serves as an analog to three-dimensional stellation of polyhedra, where faces are extended to form star polyhedra; notably, the faces of Kepler's small stellated dodecahedron consist of such pentagrams obtained via this method.^[14] Variations of the stellation process include full stellation, which extends all sides completely to form the complete star envelope, and partial stellation, where only select sides or segments are extended, yielding intermediate or compound figures. Degenerate cases, where extensions align without forming a closed star, tie briefly to vertex-skipping constructions but are not central to the stellation method.^[13]

Properties and Characteristics

Geometric Properties

Star polygons, particularly the regular ones denoted by the Schläfli symbol {n/d} where n is the number of vertices and d is the density (with gcd(n, d) = 1 and 1 < d < n/2), exhibit specific geometric attributes that distinguish them from convex polygons. The interior angle at each vertex, which forms the sharp points of the star, is calculated using the formula \frac{(n - 2d) \times 180^\circ}{n}. This expression derives from the sum of interior angles being (n - 2d) × 180°, adjusted for the intersecting nature and density of the figure, yielding an average angle per vertex that decreases with higher density. For example, the pentagram {5/2} has interior angles of 36° each. The side lengths of a regular star polygon are equal, a property that underscores their equilateral nature. When inscribed in a unit circle (radius 1), each side spans a central angle of $2\pi d / n radians, resulting in a side length of $2 \sin(\pi d / n). This chord length formula arises directly from the geometry of the circle and the step size d in connecting the vertices.^[1] Such uniformity in side lengths contributes to the overall symmetry of the figure, which belongs to the dihedral group D_n with 2n elements, including n rotational and n reflection symmetries.^[1] More precise computations of the area often involve decomposing the star into triangular sectors. Regular star polygons are inherently isotoxal, meaning all edges are of equal length and the symmetry group acts transitively on the edges, ensuring no distinction among them under rotation or reflection. This edge-transitivity is a defining feature, as explored in analyses of polygonal metamorphoses, and holds for all {n/d} with coprime n and d.^[8]

Density and Interiors

The density d of a regular star polygon denoted by the Schläfli symbol \{n/d\}, where n and d are coprime positive integers with $1 < d < n/2, represents the number of times the polygon's boundary winds around its center as the figure is traversed once. This parameter, also called the step density, quantifies the "tightness" of the connections between vertices on the circumscribed circle, with d = 1 yielding a convex regular n-gon and higher d producing more interlaced structures.^[1]^[15] In topological terms, the density corresponds to the winding number of the boundary curve around the center point, a formal measure of enclosure defined as the integer w = \frac{1}{2\pi} \int_C d\theta, where C is the closed polygonal path and \theta is the angle subtended at the point. For points in the plane not on the boundary, the winding number is constant within each connected region and changes by \pm 1 when crossing an edge, reflecting the oriented enclosure by the curve. The center lies in a region with winding number d, while the exterior unbounded region has winding number 0; intermediate regions have winding numbers between 0 and d. This assignment distinguishes the interiors topologically, with higher-winding regions being more deeply enclosed by the boundary.^[16]^[17] The self-intersecting edges of the star polygon form a planar graph that divides the plane into bounded (interior faces) and one unbounded region. To count the total number of regions, apply Euler's formula for connected planar graphs: V - E + F = 2, where V is the number of vertices, E the number of edges, and F the number of faces (regions, including the unbounded one). The vertices consist of the original n points plus the interior intersection points; the number of intersections is I = n(d-1), so V = n + I = nd. Each of the n original edges is crossed $2(d-1) times, dividing it into $2d - 1 segments, yielding E = n(2d - 1). Substituting into Euler's formula gives F = E - V + 2 = n(2d - 1) - nd + 2 = n(d - 1) + 2. Thus, a simple regular star polygon \{n/d\} divides the plane into n(d - 1) + 2 regions in total. For example, the pentagram \{5/2\} has $5(2 - 1) + 2 = 7 regions: 6 bounded and 1 unbounded.^[18]^[19]

Applications and Contexts

In Tilings and Tessellations

Star polygons play a significant role in both non-periodic and periodic tilings, extending beyond convex polygons to create complex, symmetric patterns in Euclidean, hyperbolic, and quasiperiodic contexts. In non-periodic tilings with five-fold rotational symmetry, pentagrams denoted by the Schläfli symbol \{5/2\} are incorporated without translational periodicity, as seen in hierarchical substitutions that produce infinite, non-repeating arrangements.^[20] Similarly, snub tilings in the hyperbolic plane utilize octagons \{8\} as faces in uniform configurations, where four triangles and one octagon meet at each vertex, yielding chiral semiregular patterns with high symmetry.^[21] Isotoxal tilings, characterized by equal edge lengths and transitive symmetries across edges, incorporate star polygons to form uniform arrangements; for instance, the 3.12.12 tiling features triangles alternating with stellated dodecagons, ensuring all edges are congruent while maintaining vertex uniformity.^[22] These constructions leverage the edge-transitive properties of star polygons, allowing seamless integration in semi-regular tessellations like those derived from rectified or expanded forms.^[22] In hyperbolic geometry, infinite families of regular star polygons \{n/d\} with n > 2d tile the plane when the density exceeds that of Euclidean space, satisfying the condition \frac{2d}{n} + \frac{1}{k} > 1 for vertex figure k, resulting in edge-to-edge coverings with negative curvature.^[23] Examples include \{7/3\} or higher-order stars surrounding vertices in uniform hyperbolic tilings, enabling boundless expansions not possible in flat space.^[23] Historically, Islamic architecture employed girih tiles—sets of decagons, pentagons, rhombuses, bowties, and hexagons—to construct quasiperiodic patterns incorporating \{5/2\} pentagrams and \{7/3\} heptagrams, achieving near-aperiodic decagonal symmetry as early as the 15th century in structures like the Darb-i Imam shrine.^[24] These tiles facilitated self-similar hierarchies with ratios approximating the golden number, producing intricate, non-repeating motifs that prefigure modern quasicrystal models.^[24]

In Art and Culture

Star polygons have held profound symbolic significance across various cultures, often representing mystical, protective, or divine concepts. In ancient Babylonian mysticism, the octagram {8/3}, an eight-pointed star, symbolized the goddess Ishtar, associated with love, war, and the planet Venus, appearing in artifacts and iconography as early as the 3rd millennium BCE.^[25] Similarly, the Pythagoreans in the 6th century BCE revered the pentagram {5/2}, known as the pentalpha, as a emblem of perfection, health, and the harmony of the five elements, using it as a secret sign of recognition among followers; this symbol later evolved into a key motif in Western occult traditions, denoting magical protection and the human microcosm. In Judaism, the hexagram {6/2}, or Star of David (Magen David), emerged as a prominent emblem by the medieval period, symbolizing divine protection and the union of opposites, with roots tracing back to its decorative use in ancient synagogues such as the 2nd-3rd century CE synagogue at Capernaum alongside other geometric motifs.^[26] In art history, star polygons enriched visual expressions during the Renaissance and in Islamic decorative traditions. Albrecht Dürer's 1514 engraving Melencolia I incorporates geometric elements interpretable as a hexagram {6/2}, evoking the Seal of Solomon and themes of intellectual melancholy intertwined with alchemical symbolism.^[27] In Islamic art, star polygons like heptagrams {7/3} featured prominently in geometric ornamentation, symbolizing cosmic order and infinity; for instance, intricate star motifs, including seven-pointed forms, adorned architectural elements in structures such as the Alhambra in Granada, reflecting the mathematical sophistication of Nasrid-era design from the 14th century.^[28] In modern culture, star polygons continue to influence national symbols and visual media. The pentagram {5/2} appears as a green emblem on Morocco's flag, adopted in 1915 and retained post-independence, representing the five pillars of Islam and cultural heritage.^[29] Stellated and star polygonal shapes also permeate contemporary design, appearing in corporate logos—such as abstract star forms in branding for companies like Paramount Pictures—blending ancient symbolism with modern aesthetics.^[30] The cultural evolution of star polygons illustrates a transition from ancient mystical talismans, like the Babylonian octagram tied to deity worship, to enduring symbols in religious iconography and, ultimately, to versatile elements in global contemporary design, underscoring their timeless appeal in conveying harmony, protection, and the infinite.^[31]

References

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A star polygon is formed by connecting every qth point out of p points on a circle, where p and q are positive integers. If q=1, it is a regular polygon.
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The pentagram, also called the five-point star, pentacle, pentalpha, or pentangle, is the star polygon {5/2}. It is a pagan religious symbol.
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### Definition of Density for Star Polygons
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**Summary:**
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