Octagram Star

About Octagram Star

Take a square. Lay a second square on top of it, rotated 45° around the same center. The figure you have made is an octagram — an eight-pointed star with a regular octagon at its core and dihedral symmetry of order 16 (D8). The construction is two compass-marks and four straight cuts. The result has shown up under different names in at least four distinct traditions: as the Khātim Sulaymān (Seal of Solomon) and *Rub al-Ḥizb* in Islam, as the Star of Lakshmi (*Aṣṭa-Lakṣmī yantra*) in Hinduism, as the Seal of Solomon in Jewish and Western magical lore, and as a recurring octagram across Babylonian and Mesopotamian iconography much earlier. This page covers the octagram as the two-overlapping-squares construction — the {8/2} star polygon — distinct from the radiating-rayed [eight-fold star](/sacred-geometry/eight-fold-star/) used as a girih rosette in Islamic architecture.

Mathematical Properties

The {8/2} octagram carries the dihedral symmetry group D8, of order 16 — eight rotational positions (each a multiple of 45°) and eight reflection axes (four through opposite points, four through opposite edges of the central octagon). The figure has eight points, sixteen edges (eight from each of the two squares), and a regular octagonal core where the two squares intersect.

The Schläfli symbol {8/2} indicates a star polygon connecting every second vertex of a regular octagon — but because gcd(8, 2) = 2, the {8/2} polygon is *degenerate* in the strict sense: it does not trace as a single connected line but as two distinct squares offset by 45°. This is sometimes written 2{4} (two squares) to make the compound nature explicit. The non-degenerate eight-pointed star polygons are {8/3} (a single connected line with sharper points, used as the basis for many Islamic eight-fold rosettes) and the regular octagon {8/1}.

The two squares of the {8/2} octagram inscribed in a circle of radius R have side length R√2. The points of the octagram lie on the circle of radius R. The internal angle at each point of the star is 45°. The central octagonal core has internal angles of 135° (the standard regular-octagon internal angle). For the unit-circumradius case (R=1), the figure can be computed numerically: each square has side √2 and area 2, and the central octagon (the intersection of the two squares) has area 2(√2 − 1) ≈ 0.828, giving a total octagram area of approximately 3.172.

The construction by compass and ruler is among the simplest in the regular-star-polygon family: inscribe a square in a given circle (the four cardinal points of the circle define one square); inscribe a second square at 45° to the first (the four points midway between the cardinal points define the second). The eight points of the octagram are the eight vertices of the two combined squares.

As a tiling motif, the {8/2} octagram does not periodically tile the plane on its own — eight-fold rotational symmetry is forbidden by the crystallographic restriction theorem, which permits only orders 2, 3, 4, and 6 for periodic tilings. In Islamic architectural use the octagram is typically deployed as an isolated motif (a section-divider, a flag-element, a single tile) rather than as the building block of a periodic tessellation. When eight-fold patterns are tessellated across a wall surface, the underlying lattice is square (4-fold, permitted), and the eight-fold rosettes carry their octagonal symmetry locally at the lattice nodes while the global pattern repeats with four-fold symmetry. The eight-fold Penrose-like quasi-periodic tilings (the octagonal Ammann-Beenker tiling, discovered by Robert Ammann in the 1970s) are a separate mathematical object, not used in classical Islamic architectural practice as far as the documentary record shows.

Architectural Use

The octagram appears in Islamic architecture in three distinct functional registers, each with its own typical execution.

*As the Rub al-Ḥizb (Quranic textual marker).* The octagram ۞ marks the end of each quarter-section of the 60-part *ḥizb* division of the Qur'an in the margins of many manuscripts and printed editions, from the Mamluk period (13th-16th centuries) onward. Mamluk Qur'ans illuminated in Cairo carry some of the most elaborate examples — the marginal octagram is often inscribed with gold and lapis lazuli, set against a square or circular ornamental field. Ottoman Qur'ans from the 16th-19th centuries continue the tradition. The function is liturgical and structural: a reciter following the monthly recitation cycle uses the octagrams to locate the daily station of reading.

*As the Khātim Sulaymān (Seal of Solomon) on flags, emblems, and ornamental panels.* The octagram appears on the historical flags of several Islamic dynasties — the Karamanid emirate of Anatolia, several Seljuk-successor states, and on Ottoman regimental standards and Janissary corps insignia. Modern Islamic state symbols carrying eight-pointed stars include the seal of Algeria and several emirate flags. Architectural deployment of the Khātim Sulaymān as an isolated decorative element is widespread across Anatolian Seljuk, Mamluk, and Maghrebi work; the Seljuk Star at the entrance to many 13th-century Anatolian mosques and madrasas is one common form.

*As an architectural element in zellige and tile work.* In Maghrebi zellige (cut-tile mosaic) tradition the octagram-from-two-squares is one of the foundational figures cut by the apprentice in early training, used both as a standalone medallion and as the building block of larger compositions. The Ben Yousef Madrasa in Marrakech (16th century) and the Bou Inania Madrasa in Fez (14th century) both carry octagram zellige panels. In Persian and Central Asian tile work the octagram appears in the spandrels and dado panels of mosques and madrasas; the Friday Mosque of Isfahan, the Shah Mosque of Isfahan (Safavid 1611-1629), and the Timurid mosques of Samarkand all carry examples.

Outside the architectural register proper, the octagram appears in textile and carpet design — many Anatolian and Persian carpets carry an octagram medallion at center, often called a *Gül* or 'rose' in Turkmen carpet terminology — and in metalwork, woodwork, and book binding.

A note on confusion with the rosette eight-fold star: the architectural panels in classical Islamic geometric work that look like elaborate eight-pointed rosettes (radiating, with internal sub-figures, integrated into girih-tile compositions) are typically *not* the {8/2} two-squares octagram but the {8/3} rosette eight-fold star — a different geometric object handled on the [eight-fold star](/sacred-geometry/eight-fold-star/) page. The Rub al-Ḥizb marginal symbol and the standalone Khātim Sulaymān are the dominant uses of the {8/2} octagram proper.

Construction Method

The octagram is among the simplest figures in the regular-polygon repertoire to construct by compass and ruler.

*Method 1 — two squares inscribed in a circle.* Draw a circle of any radius. Mark the four cardinal points where horizontal and vertical diameters meet the circle. Connect these four points with straight lines to form the first inscribed square. Bisect each quadrant by drawing diagonals or by compass-construction of the angle bisector, generating four additional points midway between the cardinals. Connect these four with straight lines to form the second inscribed square, rotated 45° to the first. The eight points of the octagram are the eight vertices of the two combined squares.

*Method 2 — by direct construction of the octagon.* Inscribe a regular octagon in a circle by the standard compass-and-ruler method (eight points equally spaced around the circumference, each 45° apart). Connect every second vertex with straight lines, producing the {8/2} two-squares figure as the chord pattern.

*Method 3 — by the girih-tile system.* Within the modular girih system codified by ~1200 CE (Lu and Steinhardt 2007), the octagram appears as one of the standard equilateral polygons in the artisan's library — though, as noted, the dominant Islamic eight-fold architectural use is the rosette {8/3} rather than the {8/2} octagram.

For the *Rub al-Ḥizb* manuscript marker specifically, the construction is typically executed at small scale (1-2 cm across) in colored ink with gold leaf. The Mamluk scribal tradition prepared the figure as part of the manuscript-illumination program: the parchment was ruled, the octagram positioned in the outer margin at the precise point where the quarter-section ended, and the figure was drawn first in ink, then filled with mineral pigments (lapis ultramarine, malachite green, vermilion) and gold leaf bound with gum arabic.

For zellige tile execution, the octagram is one of the first figures cut by an apprentice in the Maghrebi tradition. Two squares are cut from a single firing of glazed terracotta tile, scored on the surface with a steel point along guide lines, and chipped with a sharp hammer (the *menqach*) — first roughly along the guide line, then refined to the final octagram shape. The two squares are then laid adjacent on the prepared mortar bed, overlapping at 45° to form the octagram. The skilled *maâlem* can cut and lay an octagram in roughly fifteen minutes.

For architectural-stone execution, the figure is laid out at full scale on the wall surface with snapped chalk-string axes, scored into the stone with a steel point, and carved with chisel and mallet to the depth and profile required. Mamluk minbar work and Anatolian Seljuk portal carving both preserve high-quality examples.

The Unicode codepoint U+06DE ARABIC START OF RUB EL HIZB encodes the figure for digital text. The character was originally classified as a combining mark in early Unicode, with subsequent revisions reclassifying it; the canonical reference is the *Unicode Standard* and the Unicode Consortium's Proposal Review PR-171 (Changing the properties of U+06DE from a combining mark) documents the technical history.

Spiritual Meaning

The octagram's meaning depends on which tradition is reading it, and the honest move is to keep the readings distinct rather than collapse them into a universal eight-symbolism.

*In Islam* the octagram carries two distinct meanings tied to two of its names. As the *Rub al-Ḥizb*, it has a primarily functional and liturgical meaning — it marks the recitation-stations of the Qur'an across the monthly cycle, and its presence in a manuscript is the structural aid of a reader who chants the text aloud over thirty days. As the *Khātim Sulaymān*, it carries the prophetic and royal associations of Sulaymān — the prophet-king of Islamic and biblical tradition, who possessed (in the Qur'anic narrative) the seal-ring by which he commanded the jinn and ruled with wisdom. The seal-of-Sulaymān reading is older than the strictly Islamic adoption; it sits inside a wider Mesopotamian and biblical tradition of the king-prophet whose authority is sealed by a divine token. In contemporary Islamic use the eight-pointed star often carries the more general associations of paradise (the eight gates of Paradise in some hadith traditions), the eight angels who carry the throne of God on the Day of Resurrection (Qur'an 69:17), and the resurrection itself.

*In Hindu tradition* the same {8/2} figure as the Star of Lakshmi or Aṣṭa-Lakṣmī yantra represents the eight forms of Lakshmi — eight kinds of wealth or prosperity. In Vāstu-śāstra and tantric ritual practice the eight points carry the Aṣṭa-Dikpāla, the eight guardians of the directions, each with specific attributes and offerings. The figure is consecrated through *prāṇa-pratiṣṭha* (life-installation) rites in the tantric tradition; once consecrated it is treated as the actual residence of Lakshmi and her eight forms, not as a representation of them.

*In Mesopotamian iconography* (Sumerian, Akkadian, Babylonian) the eight-pointed star is the symbol of Inanna / Ishtar — goddess of love, war, sexual desire, and Venus the morning-and-evening star. The eight points connect to Venus's role as both morning and evening star and to the cycle of its observable motion through the sky.

*In Western magical and Jewish tradition* the Seal of Solomon as octagram is less commonly named than the hexagram form, and the better-documented Western-magical use is Heinrich Cornelius Agrippa's *De Occulta Philosophia* (1531-1533), whose discussion of planetary seals includes octagram figures. The octagram appears occasionally in medieval Jewish manuscript decoration as well.

What unites these readings, when anything does, is the number eight as carrying meanings of completion-beyond-seven, of resurrection (the eighth day, post-Sabbath), of multiplicity-of-blessing, and of the cardinal-and-intercardinal directions. These are independent inheritances rather than a single underlying meaning — the form is geometrically simple enough that it has been independently arrived at and independently invested with meaning by multiple cultures.

The Satyori reading does not propose a universal octagram-meaning. It proposes that the eight-pointed star is the place where one of the simplest geometric constructions (two squares, 45°) has carried distinct theological readings across at least four major traditions, that the construction is older than any of the readings, and that the readings are real where they are made and silent where they are not.

Significance

Three things are true of the octagram at once, and pulling them apart is the work of this page.

*First, it is a specific mathematical object.* The octagram in the strict sense is the {8/2} regular star polygon — the star figure produced by connecting every second vertex of a regular octagon, equivalent to overlaying two squares at 45° rotation. Its Schläfli symbol is {8/2} (sometimes written 2{4}, denoting its construction as a compound of two squares). It carries D8 dihedral symmetry, sixteen rotation-and-reflection operations, eight points, and a regular octagonal core where the two squares overlap. As a star polygon it is technically *degenerate* — strictly, {8/2} is reducible (gcd(8,2)=2) so it is not a single connected stroke, but two distinct squares; the non-degenerate eight-pointed star polygons are {8/3} and the more familiar single-stroke octagram. In ornamental practice the {8/2} two-squares figure is the dominant form and is what most traditions mean by 'eight-pointed star.'

*Second, it is a multi-tradition motif with tradition-specific names and readings.* Calling the octagram 'the Islamic eight-pointed star' or 'the Hindu eight-pointed star' or 'the Jewish Seal of Solomon' isolates one tradition's reading of a figure that is shared across several. The honest framing is that the {8/2} two-overlapping-squares figure is a geometric object that has been adopted independently or through long-running diffusion by several traditions, each of which has its own name and theological-or-iconographic reading for it.

In Islamic tradition the octagram is called the *Khātim Sulaymān* (the Seal of the Prophet Sulaymān/Solomon) and the *Rub al-Ḥizb* (literally 'a quarter of the section'). The Rub al-Ḥizb function is textual: the Qur'an is divided into 60 *ḥizb* sections, each subdivided into four quarters, and the octagram ۞ marks the end of each quarter-section in the margins of many Qur'anic manuscripts and printed editions. The Unicode character at U+06DE is named ARABIC START OF RUB EL HIZB and was originally encoded as a combining mark, with subsequent Unicode revisions reclassifying it. Beyond its textual-marker function, the octagram appears on flags, emblems, and architectural surfaces across the Islamic world, often called the Seljuk Star in Anatolian contexts.

In Hindu tradition the same {8/2} figure is the Star of Lakshmi or *Aṣṭa-Lakṣmī yantra*, representing the eight emanations of Lakshmi, goddess of prosperity — Ādi-Lakṣmī, Dhana-Lakṣmī, Dhānya-Lakṣmī, Gaja-Lakṣmī, Santāna-Lakṣmī, Vīra-Lakṣmī, Vijaya-Lakṣmī, and Vidyā-Lakṣmī. In Vāstu-śāstra (Hindu architectural canon) the same eight points carry the Aṣṭa-Dikpāla — the eight guardians of the eight directions: Indra (east), Agni (southeast), Yama (south), Nirṛti (southwest), Varuṇa (west), Vāyu (northwest), Kubera (north), and Īśāna (northeast).

In Jewish and Western magical tradition the eight-pointed star sometimes carries the name Seal of Solomon, though more frequently the Seal of Solomon is a hexagram (six-pointed) or pentagram. The octagram form appears occasionally in medieval Jewish manuscript decoration; the more solidly documented Western-magical use is Heinrich Cornelius Agrippa's *De Occulta Philosophia* (1531-1533), whose discussion of planetary seals includes octagram figures.

In Mesopotamian iconography the eight-pointed star is the symbol of Inanna / Ishtar — the Sumerian and Akkadian goddess of love, war, and Venus — visible on cylinder seals and boundary stones from the third millennium BCE, more than two thousand years before any of the later traditions name it.

*Third, the octagram is geometrically distinct from the radiating eight-fold star used in Islamic architectural rosettes.* The eight-fold rosette ([eight-fold star](/sacred-geometry/eight-fold-star/)) used in Islamic *girih* work is built from eight radiating points around a central node, generated from chord patterns and inscribed within an octagon — typically the {8/3} star polygon, which is a single connected line, or a more complex rosette with internal sub-figures. The octagram {8/2} is two distinct overlapping squares; the rosette {8/3} is one continuous strapwork line. Both appear in Islamic ornament but they are different figures with different construction methods and different uses.

The Satyori reading: the octagram is the place where a single very simple geometric construction (two squares, 45°) carries genuinely different meanings across genuinely different traditions, and the form is widely enough shared that no tradition has exclusive claim to it. The honest move is to name the construction, name each tradition's reading, and leave the meanings as distinct rather than collapsed into a universal symbol-of-eight.

Connections

The octagram sits inside a family of eight-fold geometric forms on this site. The [eight-fold star](/sacred-geometry/eight-fold-star/) is the radiating-rayed Islamic rosette form built as a {8/3} star polygon — a distinct figure from this {8/2} two-overlapping-squares octagram. The [girih tile](/sacred-geometry/girih-tile/) set is the construction system that produces the rosette eight-fold star (and the ten- and twelve-fold variants); the octagram proper is simpler and pre-dates the girih system. The [twelve-fold star](/sacred-geometry/twelve-fold-star/) and [hexagonal tessellation](/sacred-geometry/hexagonal-tessellation/) cover other major Islamic geometric registers.

The cross-tradition resonances are real and specific. The Hindu [yantra](/sacred-geometry/yantra/) tradition uses the same {8/2} two-squares figure as the Aṣṭa-Lakṣmī yantra — the cross-traditional naming is documented across Sanskrit and Persian sources from the medieval period onward. The Mesopotamian Inanna/Ishtar eight-pointed star pre-dates both the Islamic and the Hindu forms by more than two millennia and is the plausible source of long-running iconographic diffusion. The Buddhist eight-spoke [Dharma wheel](/sacred-geometry/dharma-wheel/) — the *dharmacakra* with its eight spokes representing the Noble Eightfold Path — is iconographically related at the level of the number-eight but constructed differently (a wheel-with-spokes rather than two overlapping squares).

In Western traditions, the [rose window](/sacred-geometry/rose-window/) and Christian baptismal-font architecture make heavy use of eight-sidedness — the octagonal baptismal font at the Lateran Baptistery in Rome (4th century) was the prototype, encoding the resurrection on the 'eighth day' (the day after the seven of creation). The octagonal church plan at San Vitale in Ravenna (6th century) and Charlemagne's Palatine Chapel at Aachen (early 9th century) carry the same eight-sidedness into floor-plan architecture rather than into ornament.

Further Reading

• Necipoğlu, Gülru. *The Topkapı Scroll: Geometry and Ornament in Islamic Architecture*. Santa Monica: Getty Center for the History of Art and the Humanities, 1995.
• Critchlow, Keith. *Islamic Patterns: An Analytical and Cosmological Approach*. London: Thames & Hudson, 1976.
• Burckhardt, Titus. *Art of Islam: Language and Meaning*. Trans. J. Peter Hobson. London: World of Islam Festival Publishing, 1976.
• Grünbaum, Branko, and G. C. Shephard. *Tilings and Patterns*. New York: W. H. Freeman, 1987.
• Mookerjee, Ajit. *Yantra: The Tantric Symbol of Cosmic Unity*. London: Thames & Hudson, 1981.
• Black, Jeremy, and Anthony Green. *Gods, Demons and Symbols of Ancient Mesopotamia: An Illustrated Dictionary*. Austin: University of Texas Press, 1992.
• Schimmel, Annemarie. *The Mystery of Numbers*. New York: Oxford University Press, 1993.
• Unicode Consortium. *The Unicode Standard, Version 15.0*. Mountain View, CA: Unicode Consortium, 2022. (Entry: U+06DE ARABIC START OF RUB EL HIZB.)