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Octagram

An octagram is a regular star polygon consisting of eight sides and eight vertices, denoted by the Schläfli symbol {8/3}, formed by connecting every third point among eight equally spaced points on a circle, resulting in a self-intersecting, non-convex figure.^[1]^[2] It possesses octagonal dihedral symmetry (D₈), a central density of 3, and an interior angle of 45°, with its area given by 2(√2 - 1) for a side length of 1.^[2] The octagram can be constructed as the second stellation of a regular octagon or as a uniform quasitruncation of a square, and its mirror image is the enantiomorphic form {8/5}.^[2] Unlike compound eight-pointed stars such as the Star of Lakshmi ({8/2}, formed by two overlapping squares), the octagram is a single, simple polygram.^[3] In higher dimensions, it relates to uniform polytopes, appearing as faces in certain 5D and higher figures.^[2] Historically, the octagram has held significant symbolic meaning across cultures, often representing celestial bodies, deities, and cosmic order due to its geometric elegance and constructibility with compass and straightedge.^[4] In ancient Mesopotamia, it symbolized the Sumerian goddess Inanna (later Ishtar in Semitic traditions), associated with love, war, and the planet Venus, appearing in iconography from the third millennium BCE.^[4] This motif influenced later artistic traditions, including medieval Islamic tessellations in architecture—such as those in the Alcazar of Seville—where it contributed to intricate geometric patterns emphasizing harmony and infinity, evolving from simpler forms in the 9th century to more complex star designs by the 16th century.^[4]

Definition and Basic Properties

Geometric Definition

An octagram is an eight-vertex star polygon {8/3} formed by connecting every third point on the vertices of a regular octagon. This self-intersecting figure exhibits the rotational and reflectional symmetries of the underlying octagon while creating a star-like shape through its overlapping sides.^[5] The construction begins with a regular octagon inscribed in a circle of circumradius R, where the eight vertices are positioned at equal angular intervals of $45^\circ. To form the octagram, connect successive vertices by skipping two intervening points (connecting every third point overall), yielding line segments that each subtend a central angle of $135^\circ. In this case, the side length s of each edge is the chord corresponding to this central angle, calculated as
s = 2 R \sin\left( \frac{135^\circ}{2} \right) = 2 R \sin(67.5^\circ).
At each vertex, the angle between adjacent sides, known as the vertex angle, measures $45^\circ, derived from the generalized interior angle sum for star polygons divided equally among the vertices.^[5]^[6] The octagram possesses isogonal symmetry under the dihedral group D_8, which includes 16 transformations preserving the figure's rotational order of eight and reflections across eight axes passing through opposite vertices or midpoints of opposite sides. From the center, eight radial rays extend to the vertices, dividing the circumcircle into equal sectors. The sides intersect at additional points, forming a smaller regular octagon at the core bounded by these intersection segments.^[5]^[7] The density of the octagram, also termed the winding number, quantifies its topological complexity and can be illustrated as the number of edges a ray from the interior must cross to reach the exterior boundary when traversing the polygon's outline. This measure reflects how the path winds around the center multiple times, enclosing regions with varying enclosure levels.^[5]

Schläfli Symbol and Density

The Schläfli symbol {n/k} provides a concise notation for regular star polygons, where n denotes the number of vertices and edges, and k represents the step or density parameter indicating how many vertices are skipped when connecting successive points on a circumscribed circle.^[5] For the octagram, n=8 and k=3 (coprime), yielding {8/3} for the simple form. The enantiomorphic form is {8/5}, the mirror image of {8/3}. When gcd(n, k) > 1, the figure is a compound rather than a single simple polygon; for example, {8/2} is a compound of two squares (gcd=2), while {8/4} is a compound of four digons (gcd=4), consisting of four diameters intersecting at the center.^[5] The density d of a star polygon, which measures the winding number or the number of times its edges encircle the center, is given by d = k for simple star polygons {n/k} where gcd(n, k) = 1 and k < n/2.^[5] In the case of {8/3}, the density is 3, meaning the edges wind around the center three times, creating a more interlaced and centrally filled appearance compared to lower-density stars like the pentagram {5/2} with d=2.^[5] For compounds like {8/4}, the density is 4, reflecting the multiple components.^[5] Higher density values in octagrams enhance the "filled" visual effect by increasing internal intersections and reducing the prominence of the central void, as the path revisits interior regions more frequently before closing.^[5] Octagrams relate to regular polygons as stellations of the octagon, where the {8/3} form arises as the second stellation by extending the octagon's sides until they intersect to form the star.^[1]

Variations and Types

Simple Octagram {8/3}

The simple octagram, denoted by the Schläfli symbol {8/3}, is a unicursal star polygon formed by connecting every third vertex of a regular octagon, resulting in a single continuous line that winds around the center with a density of 3, completing three full turns before closing after eight edges.^[5] This configuration distinguishes it as the primary non-compound regular octagram, exhibiting self-intersections that create a star-shaped figure with eight equilateral sides.^[1] The symmetry group of the {8/3} octagram is the full dihedral group D_8, which consists of 16 elements: eight rotations (by multiples of $45^\circ) and eight reflections across axes passing through opposite vertices or midpoints of opposite sides.^[5] The vertex angle at each point is $45^\circ.^[2] In the complex plane, assuming a unit circumradius, the vertices of the {8/3} octagram can be represented as e^{2\pi i \cdot 3k / 8} for k = 0, 1, \dots, 7, which traces the points in the order of connection.^[1] The side length s for this unit circumradius is given by s = 2 \sin(3\pi / 8).^[5] For plotting the star parametrically, the coordinates are

x(t) = \cos\left(\frac{6\pi t}{8}\right), \quad y(t) = \sin\left(\frac{6\pi t}{8}\right),

where t increments in steps of 1 from 0 to 7 (discretized) or continuously for the envelope curve.^[5] The {8/5} octagram is equivalent to {8/3}, as the connection step of 5 is congruent to -3 modulo 8, producing the same figure but traversed in the reverse direction.^[1]

Compound Octagram {8/2}

The compound octagram denoted by the Schläfli symbol {8/2}, equivalently 2{4}, is a regular star polygon compound consisting of two interlocked squares rotated relative to each other by 45 degrees.^[8] This figure arises as the first stellation of a regular octagon and represents a multicursal structure where the two square components share the same eight vertices on a common circumcircle.^[8] Unlike simple star polygons, which form a single connected path, the {8/2} is a uniform compound composed of distinct polygonal elements.^[8] Key properties include a central density of 2, reflecting the twofold overlap in the interior regions covered by the squares, with the density derived from the Schläfli notation indicating the number of component polygons.^[8] The eight vertices are shared among the components, and the edges intersect to form a regular octagon at the center where the squares overlap.^[8] Topologically, it is not a single connected polygon but a discrete compound maintaining uniformity through its regular construction.^[8] This compound can be constructed by overlapping two regular squares inscribed in the unit circle, with one square having vertices at (1,0), (0,1), (-1,0), and (0,-1), and the other rotated by 45 degrees with vertices at \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right), \left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right), \left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right), and \left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right).^[8] The resulting figure exhibits D₈ (dihedral group of order 16) symmetry, preserving rotational and reflectional invariance across both components.^[8]

Constructions and Mathematical Representations

Relation to the Square

The octagram relates to the square through fundamental geometric constructions that highlight shared symmetry and transformation properties. Connecting every second vertex of a regular octagon produces the compound star polygon denoted {8/2} or 2{4}, which consists of two regular squares rotated by 45 degrees relative to one another and interlocked to form an eight-pointed figure.^[8] This compound demonstrates how the square's vertices, when doubled and offset by rotation, generate a star configuration with eight edges and eight vertices. The simple octagram {8/3}, in contrast, arises from connecting every third vertex of the same octagon, extending the square's underlying structure into a non-compound star with higher density.^[5] Geometric transformations further illustrate this connection. A regular octagram possesses eight-fold rotational symmetry, allowing it to align with a square's axes upon a 45-degree rotation, matching the angular spacing of the square's diagonals and sides.^[5] Inscribed squares within an octagram can be scaled relative to the circumscribed circle; for a unit circle, the side length of the largest inscribed square connecting alternate vertices is \sqrt{2}. The vertices of a regular octagram with edge length 1 are given by all even permutations of \left( \pm \frac{\sqrt{2} - 1}{2}, \pm \frac{1}{2} \right).^[9]^[2] In terms of duality under polar reciprocity, the vertices of a regular octagram correspond to the poles of the lines forming a dual square's edges, preserving the eight-fold symmetry in projective geometry and linking the star's intersecting structure to the square's convex boundary.^[10] (Note: While primarily discussed for polyhedra, the principle extends to plane figures via polar duality.) Historically, early geometric constructions linking squares to star polygons, including octagrams, emerged in medieval extensions of Euclidean methods. Thomas Bradwardine (c. 1290–1349) analyzed star polygons' angles, describing an octagram as composed of two quadrangles (squares) with right angles, while Jan Brożek (1585–1652) explicitly constructed the star octagon as two interlocked squares using isoperimetric techniques derived from classical polygon constructions.^[11] These developments built on Euclid's foundational work in Elements for regular polygons, adapting compass-and-straightedge methods to star forms by the Renaissance.

As a Quasitruncated Square

The regular octagram arises as the quasitruncated form of the square in geometric constructions. Quasitruncation is a specialized truncation operation applied to regular polygons or tilings, involving cutting off vertices to a depth that inverts the original edges and produces star-shaped figures, extending beyond standard rectification where edges reduce to points. This process alternates truncation of vertices to points with a retrogradation step that expands the structure to form intersecting star edges, as conceptualized in extensions of uniform polytope theory to two dimensions. When applied to the square polygon {4}, quasitruncation yields the regular octagram {8/3}, denoted by the Schläfli symbol t{4/3}. The operation truncates the square's vertices until the original sides vanish, with the new edges connecting in a density-3 star configuration, creating eight points and edges from the four original vertices. The edge length of the resulting octagram relates to the original square's side length through trigonometric factors, specifically involving tan(π/8) to determine the intersection points and arm extensions. In the context of tilings, quasitruncation of the regular square tiling {4,4} produces the quasitruncated square tiling, a nonconvex uniform star tiling featuring regular octagram faces alongside square faces. This Archimedean-like tiling has the vertex configuration (4.8/3.8/3), where one square and two octagrams meet at each vertex, and the vertex figure is an isosceles triangle reflecting the equal edge lengths across faces. The tiling belongs to the family of uniform hyperbolic and Euclidean star tilings, analogous to the uniform polyhedra described by Coxeter, with notation such as the Coxeter diagram x4/3x4o capturing its symmetry.

Star Polygon Compounds

Star polygon compounds involving octagrams extend the geometry by combining one or more octagrams with other polygons or their variants, resulting in uniform figures with increased complexity and density. A prominent example is the dioctagram, formed by the superposition of a regular octagram {8/3} and its enantiomorphic retrograde {8/5}.^[12] This compound consists of two interlocked octagrams sharing the same 8 vertices on a circle, creating a multicursal figure with 16 edges.^[13] The construction of such compounds relies on density addition, where the individual density of 3 for each octagram sums to a total density of 6 for the dioctagram, reflecting the number of edge windings around the center.^[2] This additive property arises from the disjoint cycle permutations in the vertex connections, ensuring the components do not overlap in their paths but interweave uniformly.^[13] The symmetry of the compound is preserved as the dihedral group D_8 (order 16), matching that of the base octagon and providing rotational and reflectional invariance.^[2] Among known uniform compounds featuring octagrams, the dioctagram represents a planar example with octagonal symmetry. In three dimensions, the octagrammic crossed antiprism serves as an extension, incorporating two parallel octagrams as caps connected by 16 equilateral triangular sides, forming a nonconvex uniform polyhedron with D_{8d} symmetry (order 32).^[14] This structure maintains the star polygon's density characteristics while achieving higher polyhedral uniformity through antiprismatic twisting.

Symbolic and Cultural Uses

In Religion and Heraldry

The octagram, particularly in its form as an eight-pointed star, has appeared in religious symbolism across ancient and modern traditions, often denoting renewal, divine order, and spiritual guidance. Earliest depictions trace back to ancient Mesopotamian art, where it served as the Star of Ishtar, the goddess of love, war, and fertility, symbolizing cosmic authority in temple carvings and seals.^[15] In Christianity, the eight-pointed star embodies regeneration and baptism, with its octagonal form echoing the traditional shape of baptismal fonts and signifying the renewal of the soul through the Holy Spirit, as referenced in biblical accounts like Noah's ark saving eight persons as a prefiguration of baptism.^[16] It also holds specific significance for saints such as Dominic, whose order adopted it to symbolize the illumination he brought to the faith, depicted as a star appearing at his baptism and repeated in sacred art like the mosaics of the Basilica of the National Shrine of the Immaculate Conception.^[17] In Islam, the octagram manifests as the Rub el Hizb, an eight-pointed emblem formed by overlapping squares that marks quarter divisions in Quranic manuscripts to aid recitation, drawing from Surah Al-Haqqah to evoke the eight angels bearing God's Throne and symbolizing stability, protection, and cosmic balance.^[18] In Buddhism, the octagram symbolizes the Noble Eightfold Path, representing the eight practices leading to the cessation of suffering.^[19] Within the Baha'i Faith, while the nine-pointed star is the primary emblem of perfection and unity among religions, eight-pointed stars appear in architectural designs around shrines, chosen pragmatically by Shoghi Effendi to represent balanced spiritual progression without overemphasizing numerology.^[20] Heraldic applications of the octagram emerged in various historical contexts, often denoting nobility or imperial authority. In the Ottoman Empire, an eight-pointed star accompanied the crescent on naval flags from the late 18th century, evolving from earlier designs and symbolizing divine favor, as seen in banners post-1793 before the shift to a five-pointed star in 1844.^[21] The symbol's protective and harmonious connotations influenced its varied interpretations: the {8/4} form, resembling two overlapping squares, frequently evokes unity and balance in Islamic and Mesopotamian contexts, while the {8/3} interlaced variant has been associated with warding off evil across cardinal directions in esoteric traditions.^[22]

In Modern Design and Logos

In contemporary graphic design, octagrams are valued for their symmetrical geometry, often incorporated into logos and icons to evoke balance, harmony, and infinity. Vector resources and design templates frequently feature eight-pointed star motifs, such as elegant octagonal stars in gold or minimalist silhouettes, suitable for branding in sectors like technology and wellness where visual unity is emphasized.^[23]^[24] The compound octagram {8/4}, resembling two overlapping squares, appears in icon packs for digital interfaces, symbolizing interconnected stability in user experience design.^[25] The Star of Lakshmi, a compound {8/2} octagram formed by two rotated squares, has influenced modern branding by representing prosperity and directional completeness, appearing in vector symbols for commercial and cultural motifs. In architecture and contemporary art, octagrams inspire geometric installations and prints, such as hand-drawn modern fine art pieces that abstract the form into monochromatic patterns for gallery displays.^[26] Within popular culture, octagrams manifest as runes or sigils in fantasy role-playing games, where eight-pointed stars denote chaos, magic, or cosmic forces in tabletop systems like those involving radial arrow patterns. They also gain traction in body art, with octagram tattoos embodying infinity through their endless interlocking lines, alongside themes of regeneration and elemental convergence, appealing to those seeking symbols of eternal cycles.^[19] In the 2020s, digital artists have integrated octagram patterns into generative works, leveraging their symmetry for algorithmic explorations in virtual exhibitions, though specific NFT applications remain niche and experimental.^[27]

References

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