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Heptagram

A heptagram, also known as a septagram, is a regular seven-pointed star polygon formed by connecting the vertices of a regular heptagon in a non-adjacent sequence, resulting in one of two distinct configurations: {7/2}, which connects every second vertex, and {7/3}, which connects every third vertex.^[1] These star polygons are characterized by their Schläfli symbol {p/q}, where p = 7 denotes the number of vertices and q represents the step size in connecting them, with q and p being coprime to produce a simple, non-compound figure.^[2] The {7/2} heptagram has a density of 2, meaning its edges wind around the center twice before closing, while the {7/3} has a density of 3, creating a more intricate interlacing pattern; both exhibit rotational symmetry of order 7 and equal side lengths when regular.^[2] The circumradius R for a unit edge length is given by R = sin((7 - 2q)π / 14) / sin(2qπ / 7), yielding specific geometric proportions that have fascinated mathematicians.^[2] The study of heptagrams traces back to the 14th century, when English scholar Thomas Bradwardine first explored star polygons systematically in his geometric writings.^[2] Later formalized in modern geometry by H.S.M. Coxeter, these figures are part of the broader class of uniform polyhedra and tilings, with applications in understanding symmetry and tessellations.^[2] Beyond mathematics, heptagrams hold symbolic importance in various cultural and esoteric traditions, often invoked for protection akin to the pentagram, such as warding off evil when applied to structures.^[3] Though their primary notability remains in geometric theory.^[1]

Mathematical Foundations

Definition and Schläfli Symbols

A heptagram is a seven-pointed star polygon formed by connecting every q-th point out of seven equally spaced points on a circle, where q = 2 or 3, resulting in self-intersecting edges that create the star shape, in contrast to the non-intersecting edges of the convex heptagon (q = 1).^[2] The regular heptagrams are denoted using Schläfli symbols as {7/2} and {7/3}, where the numerator 7 indicates the number of vertices and the denominator q represents the density, or the number of times the figure winds around its center before closing.^[2] The {7/2} heptagram, with density 2, winds twice around the center, while the {7/3} heptagram, with density 3, winds three times.^[2] Note that {7/1} and {7/6} both represent the convex heptagon, and {7/4} is equivalent to {7/3} since 4 ≡ -3 mod 7, yielding the same figure when traversed in the opposite direction.^[2] The Schläfli symbol notation, which systematically classifies regular polygons and star polygons like the heptagram, was introduced by Swiss mathematician Ludwig Schläfli in his 1852 work Theorie der vielfachen Kontinuität.^[4]

Geometric Properties and Construction

A regular heptagram is a seven-pointed star polygon denoted by the Schläfli symbols {7/2} and {7/3}, formed by connecting every second or third vertex, respectively, of seven equally spaced points on a circle. The vertices lie on the circumference of the circle and can be given by the coordinates (\cos(2\pi k / 7), \sin(2\pi k / 7)) for k = 0, 1, \dots, 6 when inscribed in the unit circle. These coordinates derive from the standard placement of a regular heptagon's vertices, with the star edges creating self-intersections that increase the figure's complexity; for {7/2}, the density is 2, meaning the boundary winds twice around the center before closing, resulting in 7 vertices and internal intersections that divide the 7 edges into 21 visible line segments. The side length of the edges in a unit circumradius is $2 \sin(\pi k / 7) for the {7/k} variant, where k=2 or 3.^[2] The vertex angles, or apex angles at the points of the star, differ between the two forms due to their connection steps. For the {7/2} heptagram, each vertex angle measures approximately 77.14°, calculated via the formula \frac{(n - 2k) \times 180^\circ}{n} with n=7 and k=2. In contrast, the {7/3} heptagram has sharper vertex angles of approximately 25.71°, using the same formula with k=3. These angles represent the turning angle at each tip and highlight the relative "pointiness" of the stars; the {7/2} and {7/3} are isogonal conjugates, such that the vertex figure of one corresponds to the edge connections of the other, reflecting their duality in the complete heptagonal star compound. Area formulas for heptagrams are more involved than for simple polygons due to overlaps, but for a {7/k} inscribed in a unit circle, the area can be determined using the shoelace formula applied to the ordered vertex sequence (starting at a vertex and stepping by k modulo 7), yielding the net signed area.^[5]^[2] Construction of a regular heptagram begins with approximating a regular heptagon, as exact compass-and-straightedge construction of the heptagon is impossible (requiring angle trisection). One approximate method involves drawing a circle and marking arcs of $360^\circ / 7 \approx 51.428^\circ successively from an initial point using a protractor or iterative approximation with ruler and compass (e.g., via neusis construction for higher precision). Label the seven points sequentially as 0 through 6 around the circle. For the {7/2} heptagram, connect point 0 to 2, 2 to 4, 4 to 6, 6 to 1, 1 to 3, 3 to 5, and 5 back to 0. For {7/3}, connect every third point: 0 to 3, 3 to 6, 6 to 2, 2 to 5, 5 to 1, 1 to 4, and 4 back to 0. These connections form the star within the circumscribed circle shared with the parent heptagon.^[6]^[2] Heptagrams relate closely to the regular heptagon {7/1}, sharing the same vertices and circumradius, but forming star shapes by selecting longer chords as edges (skipping 1 or 2 vertices instead of adjacent ones). In a regular heptagon of side length 1, the {7/2} edge corresponds to the shorter diagonal (skipping one vertex), with length approximately 1.801 times the heptagon's side, while the {7/3} edge is the longer diagonal (skipping two vertices), approximately 2.247 times the side; these ratios arise from the chord lengths \frac{\sin(m \pi / 7)}{\sin(\pi / 7)} for step size m=2 or 3, and approximate certain algebraic numbers without exact closed-form expressions beyond cubic solutions. This inscription allows heptagrams to be viewed as stellations of the heptagon, enhancing its geometric density without altering the bounding circle.^[7]

Historical and Cultural Context

Origins in Ancient and Medieval Traditions

The heptagram, a seven-pointed star polygon, finds its earliest associations in ancient Mesopotamian astronomy, where the number seven symbolized the seven known celestial bodies—Sun, Moon, Mercury, Venus, Mars, Jupiter, and Saturn—central to Babylonian cosmological models from the second millennium BCE.^[8] These planetary concepts influenced later symbolic representations, though direct depictions of the connected heptagram form remain elusive in surviving cuneiform records or artifacts. The reverence for seven as a sacred number extended to related cultures, reflecting harmony in the cosmos, but archaeological evidence for the geometric figure itself is limited until the Hellenistic era. In Islamic geometric traditions, the heptagram appeared in patterns symbolizing the seven heavens or divine order, as seen in medieval girih tile designs.^[9] In the Hellenistic period, the heptagram emerged more distinctly as a religious emblem in Ptolemaic Egypt, symbolizing the syncretic deity Serapis, a fusion of Greek Zeus and Egyptian Osiris-Apis created around 300 BCE to unify multicultural worship. Priests of Serapis wore diadems adorned with a seven-pointed star, representing divine protection and celestial authority, as seen in Roman-era portrait busts from the 2nd–3rd centuries CE.^[10]^[11] This usage bridged Greek mathematical interest in polygons with Egyptian mysticism, though the figure's construction as a {7/3} or {7/2} star polygon aligns more with later Euclidean geometry than explicit ancient treatises. During the medieval period, the heptagram appeared in European esoteric manuscripts, particularly in Kabbalistic and grimoire traditions, where it evoked the seven lower sephirot or planetary intelligences. In 13th-century Jewish mysticism, such as the Zohar, seven symbolized divine emanations, with the heptagram later illustrating Netzach (victory and Venus) in diagrammatic texts.^[12] By the 15th century, it featured in Christian and Hermetic grimoires like the Liber Juratus Honorii and British Library Sloane MS 313, inscribed within sigils such as the Sigillum Dei Aemeth to invoke creator names and ward against spirits.^[13] These manuscripts, often blending astrology and theology, used the heptagram for talismanic protection, marking a shift from astronomical symbolism to practical occult application up to the Renaissance. A similar astrological heptagram appears in the 15th-century University of Pennsylvania LJS 226, combining it with pentagrams and heptagons for planetary invocation.

Evolution in Modern Symbolism

In the 19th century, the heptagram underwent a notable revival within Western esoteric traditions, particularly in Freemasonry, where the seven-pointed star represents the seven virtues—faith, hope, charity, fortitude, justice, temperance, and prudence—reflecting the figure's association with intellectual and spiritual enlightenment and the number seven's longstanding significance in Masonic ritual as a representation of completeness and divine order.^[14] This adoption draws on the seven liberal arts and sciences as a core element of the Fellowcraft degree, though the star's symbolism extends to broader themes of perfection.^[15] Éliphas Lévi's Dogme et Rituel de la Haute Magie (1856) played a pivotal role in popularizing star polygons, including the heptagram, in modern occultism, framing it as an emblem of life, perfection, and the harmonious integration of spiritual forces with the material world. Lévi's synthesis of Kabbalistic, Hermetic, and alchemical ideas elevated the heptagram from medieval obscurity to a key icon in Western esotericism, influencing subsequent generations of mystics who viewed its seven points as denoting cosmic balance and eternal cycles.^[16]^[17] The 20th century saw further expansion of the heptagram's symbolism in emerging spiritual movements. The Theosophical Society, founded in 1875, incorporated the number seven into its teachings on the chakras as seven energy centers, indirectly associating the heptagram with esoteric anatomy and universal harmony, though its primary emblem remained a composite seal. Similarly, in the Bahá'í Faith, the concept of seven—evident in Bahá'u'lláh's The Seven Valleys (c. 1856), which outlines stages of spiritual love and unity—underscored themes of progression and divine manifestation, even as the faith's official symbols favored nine-pointed stars.^[18]^[19] Standardization efforts in the late 20th century facilitated the heptagram's broader digital adoption. Unicode version 1.1 (1993) introduced star symbols in the Dingbats block, with subsequent versions adding various approximations, enabling its use in computing and design by the 2000s. This technical recognition supported the symbol's integration into post-2000 digital media, including gaming—such as protective sigils in Supernatural (2005–2020)—and non-fungible tokens (NFTs), where it appears in esoteric-themed digital art collections symbolizing rarity and mystical value.^[20] Cross-cultural shifts marked the heptagram's transition from secretive occult use to public emblem. In the 2010s, environmental activism repurposed the symbol in icons denoting the seven continents, emphasizing planetary interconnectedness amid climate challenges, as seen in eco-art and awareness campaigns linking ecological stewardship to ancient geometric wisdom. This evolution highlights the heptagram's adaptability, moving from esoteric exclusivity to accessible icons of perfection and collective aspiration across diverse fields.^[21]

Symbolic Applications

In Religion and Occult Practices

In occult traditions, the heptagram serves as a key symbol in planetary magic, representing the seven classical planets—Saturn, Jupiter, Mars, Sun, Venus, Mercury, and Moon—arranged in order of their apparent orbital speeds from slowest to fastest. This configuration aligns with the sephiroth on the Tree of Life and the seven days of the week, facilitating rituals for invoking or banishing planetary influences. In Thelema, founded by Aleister Crowley in 1904, the {7/4} heptagram appears in the Seal of the A∴A∴ (Astrum Argentum), enclosing the name BABALON in its points to invoke the goddess as a force of universal manifestation and liberation. Crowley's Enochian workings, building on John Dee's system, incorporate heptagrams to channel the seven planetary hierarchies, emphasizing their role in traversing the Aethyrs for spiritual ascent.^[22]^[23] Within Hermeticism, the heptagram ritual involves tracing the figure within a magical circle, with the top point oriented eastward. Invocations draw the heptagram clockwise from the planet's assigned point using a lotus wand, while banishings proceed counterclockwise with a sword, accompanied by divine names such as IHVH ALHIM for Saturn. This practice extends to summoning the Heptarchic Royalty or the 49 Good Angels, underscoring the heptagram's utility in aligning personal energy with cosmic forces. In neopagan and Wiccan contexts, the heptagram functions as a protective talisman, drawn during spells to ward off negative energies by harmonizing the seven directions (north, south, east, west, above, below, center). It also aids in chakra work, linking the seven energy centers to planetary correspondences—for instance, the crown chakra to the Sun (violet, associated with thought and divine connection) and the root to Saturn (red, grounding material existence)—often visualized in rainbow hues to reflect the spectrum of spiritual evolution.^[22]^[24] Anthroposophy, developed by Rudolf Steiner around 1910, integrates the heptagram into meditative exercises on the seven planetary stages of human evolution, as seen in the planetary seals where dual seven-pointed stars overlay to symbolize etheric metabolic processes and cosmic rhythms. These seals, drawn with seven ticks on a circle connected to form intersecting stars, evoke the seven rays of spiritual influence—often depicted in rainbow colors to represent the prismatic revelation of divine wisdom through human development. Such visualizations prohibit literal idolization, permitting artistic use to foster inner transformation without dogmatic attachment.^[25]

In Heraldry, Flags, and Emblems

The heptagram, particularly in its form as a seven-pointed star, finds limited but significant application in national flags and emblems, where it often symbolizes unity, federation, or spiritual completeness. In the flag of Australia, adopted in 1903 and formalized in subsequent proclamations, the Commonwealth Star—a prominent seven-pointed white star positioned below the Union Jack—represents the federation of the six original states plus one point for all territories past, present, and future. This design was refined in 1908 to include the seventh point, aligning with the crest of the Commonwealth Coat of Arms and emphasizing national sovereignty. Additionally, the four larger stars depicting the Southern Cross constellation on the flag's fly are seven-pointed, chosen to evoke the southern sky and distinguish the design from British naval ensigns.^[26] The flag of Jordan provides another example of the seven-pointed star in official symbolism, incorporated in 1928 into the red triangle adjacent to the hoist. This white heptagram-like star signifies the seven verses of Al-Fatiha, the opening surah of the Quran, underscoring Islamic faith and Arab unity within the Hashemite Kingdom. While not an interlaced polygon, its seven points evoke protective and harmonious qualities in vexillological contexts. In heraldry, seven-pointed stars appear sparingly, often in modern arms to denote rarity or specific virtues like divine perfection, though the fully interlaced {7/3} heptagram remains uncommon outside esoteric traditions. For instance, the coat of arms granted to David Grabovac, Honorary Consul, features an azure field with a seven-pointed star above a fess chequy, blending heraldic convention with symbolic elevation. Scottish heraldry, while rich in stellar charges like mullets, rarely employs the seven-pointed form in clan badges, such as those of the MacDonalds, which favor crosses and hands over stars. Organizational emblems occasionally adopt stylized versions; the three-diamond mark of Mitsubishi, registered in 1914, is rooted in family crests representing integrity, longevity, and maritime prowess.^[27]^[28] Design considerations for heptagrams in these contexts prioritize clarity and symbolism under vexillological principles. Upright orientation, with one point vertical, conveys stability and sovereignty, as seen in the Australian Commonwealth Star, while stylizations range from solid-filled for bold visibility at distance to outlined for intricate detail in emblems. In Armenia's cultural emblems, such as church ornamentation, the seven-pointed star symbolizes the ancient god Vahagn and protective forces, appearing in medieval stonework but absent from the 1992 national coat of arms centered on Mount Ararat.

In Law Enforcement and Organizational Insignia

The heptagram, or seven-pointed star, appears in various law enforcement badges across the United States, particularly among sheriff's departments, where it serves as an outline to denote authority and justice. For instance, the Santa Barbara County Sheriff's Office in California employs a seven-pointed star badge, with each point signifying one of the following values: Service, Honor, Ethics, Respect, Integrity, Fairness, and Fidelity.^[29] Similarly, the municipal seal of Florissant, Missouri, incorporates a seven-pointed star to symbolize county sheriff's departments, highlighting their role in maintaining public order alongside municipal police.^[30] These designs often feature metallic finishes, such as gold or silver plating, to emphasize institutional authority and visibility in the field.^[31] In Australia, the Australian Federal Police (AFP), established in 1979, adopted a seven-pointed star as the central element of its badge, drawing from the Commonwealth Star to represent the nation's six states and combined territories.^[32] This configuration underscores the AFP's federal mandate to operate across jurisdictional boundaries, with the star encircled by a laurel wreath to signify vigilance and protection.^[33] The badge's design has remained consistent since the agency's formation under the Australian Federal Police Act 1979, prohibiting unauthorized use to safeguard its emblematic integrity.^[32] The symbolic rationale for the heptagram in these contexts emphasizes completeness and moral fortitude, often alluding to seven virtues or principles essential to law enforcement, such as courage, justice, and fidelity, akin to the seven days of the week denoting a full cycle of duty.^[34] This choice distinguishes enforcement insignia from simpler five-pointed stars used elsewhere, reinforcing a sense of comprehensive guardianship through its intricate, interlocking form.^[31]

In Art, Literature, and Popular Culture

In literature, the heptagram symbolizes divine perfection and religious devotion. In George R. R. Martin's A Song of Ice and Fire series, the Faith of the Seven—the dominant religion in the fictional continent of Westeros—employs the seven-pointed star as its primary emblem, representing the seven aspects of God: the Father, Mother, Warrior, Maiden, Smith, Crone, and Stranger. This symbol underscores themes of faith, power, and societal structure throughout the narrative. In visual art, the heptagram features prominently in early abstract works exploring spirituality and geometry. Swedish artist Hilma af Klint created the series The Seven-Pointed Star in 1908, part of her Paintings for the Temple collection, where the shape serves as a central motif to depict cosmic harmony and the evolution of the soul. These oil paintings, produced under spiritual guidance, use the heptagram's intersecting lines to convey interconnected realms of existence, marking af Klint as a pioneer of non-representational art. In popular media, the heptagram appears across music, television, and branding to evoke mysticism and unity. The progressive rock band Tool integrates the seven-pointed star into live performances, album aesthetics, and merchandise since the 1990s, interpreting it as a sigil for personal growth, planetary energies, and collective consciousness, as explained by guitarist Adam Jones. In television, the symbol is central to the Game of Thrones adaptation of Martin's series (2011–2019), where it adorns septs, clerical robes, and artifacts of the Faith of the Seven, reinforcing motifs of piety amid political intrigue. Additionally, the logo of DC Shoes, a prominent skateboarding brand, incorporates a {7/3} heptagram within its "C" since the early 2000s, blending geometric symbolism with urban youth culture.^[35]^[36]

Simple Heptagrams

Simple heptagrams are the two regular, non-compound star polygons denoted by the Schläfli symbols {7/2} and {7/3}, formed by connecting every second or third vertex of a regular heptagon, respectively. These figures exhibit the full symmetry of the dihedral group D₇, including rotational symmetry of order 7, where the figure maps onto itself after rotations by multiples of 360°/7 ≈ 51.43°.^[2]^[37] The {7/2} heptagram features obtuse vertex angles of approximately 77.14° and a stubbier profile with 7 intersection points, enclosing a smaller heptagon at its core that provides a compact, balanced structure suitable for symbolic designs such as jewelry pendants. Its lower density results in fewer overlaps, creating a relatively open interior with the intersections outlining the inner heptagon rather than additional stars.^[1]^[38]^[39] In contrast, the {7/3} heptagram has acute vertex angles of approximately 25.71° at its points, yielding a more slender and pointy appearance with 14 intersection points due to its higher density, which produces greater overlaps. This variant is equivalent to {7/4} via reflection, as the connection step of 4 is the supplement of 3 modulo 7, resulting in a mirror-image form with identical geometric properties. The increased intersections enhance its intricate, layered aesthetic, making it prevalent in architectural motifs where visual complexity is desired.^[2]^[38]^[40]

Compound and Stellar Heptagrams

Compound heptagrams represent more intricate configurations achieved by superimposing multiple simple heptagrams in a symmetric arrangement. A prominent example is the {14/4} tetradecagram (or 2{7/2}), formed by two {7/2} heptagrams rotated relative to each other by an angle of \pi/7, resulting in a figure with 14 vertices and a density of 4, where the winding number indicates the number of edge crossings. This compound exhibits heptagonal dihedral symmetry and can be visualized as a denser star enclosing the individual components.^[41] Similarly, the {14/6} great tetradecagram (or 2{7/3}) arises from the superposition of two {7/3} great heptagrams, also rotated by \pi/7, yielding 14 vertices and a higher density of 6. This variant has found application in geometric tilings, particularly in exploring aperiodic or hyperbolic arrangements where its intersecting structure contributes to complex periodic patterns.^[42] Stellar heptagrams, such as the great heptagram {7/3}, are obtained as stellations of the regular heptagon. Although Kepler–Poinsot polyhedra, such as the great stellated dodecahedron, incorporate stellated faces like pentagrams in their regular star configurations, analogous non-regular polyhedral extensions may feature heptagram faces, albeit without achieving the purity of icosahedral symmetry.^[43] Other variants include retrograde forms, such as the {7/5} heptagram, which is equivalent to the {7/2} heptagram but traversed in the opposite direction, effectively serving as its mirror image. Construction of these compounds typically involves aligning multiple heptagrams via rotations that are rational multiples of $2\pi/7, with the total number of distinct vertices determined by the least common multiple of the component densities scaled by the number of elements; for instance, two-component compounds yield 14 points, while three-component ones produce 21 points.^[42]

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Tetradecagon - Wikipedia
There are two regular star polygons: {14/3} and {14/5}, using the same vertices, but connecting every third or fifth points.Missing: heptagram | Show results with:heptagram
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Tetradecagram - Polytope Wiki
It is one of two regular 14-sided star polygons in 2D, with the other being the great tetradecagram. It is the uniform truncation of the great heptagram.Missing: compound | Show results with:compound
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Uniform Polygons - polytope.net
Feb 29, 2020 · A uniform polygon is considered to be a regular polygon. There are an infinity of them, they can be either convex or star polygons.
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Great heptagram - Polytope Wiki
The great heptagram is a polygon with 7 sides. It is created by taking the second stellation of a heptagram. A regular great heptagram has equal sides and equal ...Missing: mathematical | Show results with:mathematical
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Heptagram - Wikipedia
There are two regular heptagrams, labeled as {7/2} and {7/3}, with the second number representing the vertex interval step from a regular heptagon, {7/1}.