Ten-Fold Star

About Ten-Fold Star

Mark a center and draw a circle. Divide the circumference into ten equal arcs of thirty-six degrees each — ten points on the circle, the vertices of a regular decagon. From those ten points draw the ten chords that connect each vertex to the third vertex onward, and the lines cross at the heart of the figure into a sharp ten-pointed star. The angles inside this star track the golden ratio: the long diagonal of a regular pentagon is φ ≈ 1.618 times its side, and the same ratio surfaces again wherever five-fold or ten-fold symmetry meets a flat surface. The ten-fold star is the figure that brings the Persian girih tradition into contact with a piece of mathematics that classical crystallography said could not exist — five-fold symmetry that fills a plane without ever repeating itself.

Mathematical Properties

The ten-fold star is built on the dihedral group D₁₀ — ten rotations by multiples of 36° and ten axes of reflection. The simplest closed-form construction is the regular {10/3} star polygon: inscribe a regular decagon in a circle (ten vertices at 36° intervals), then draw a chord from each vertex to the vertex three positions onward. The ten chords cross at the center into a sharp ten-pointed star whose internal point-angle is 36° (180° × (10−2×3) / 10 = 36°). An alternative construction overlays two regular pentagons in a single circumscribing circle, rotated 36° relative to each other; the union of the two pentagons forms a ten-pointed star with broader internal angles than the {10/3} form. Golden-ratio relationships pervade the figure. In a regular pentagon the ratio of the diagonal to the side is φ = (1 + √5) / 2 ≈ 1.618. The ratio of the outer-circle radius (to a ten-fold-star point) to the inner-decagon radius is 1 / cos(18°) ≈ 1.051. The golden-gnomon triangle (apex 36°, base angles 72°) and the golden-acute triangle (apex 108°, base angles 36°) are the two triangles that compose the Penrose kite-and-dart and the Penrose rhomb tilings, and both triangles appear directly inside the girih-tile set built on ten-fold symmetry. The constraint that a regular pentagon does not tile the plane — five copies of a 108° angle sum to 540°, not 360° — extends to the regular decagon and to the ten-fold star. Ten-fold rotational symmetry is therefore *forbidden* under the rules of classical periodic crystallography (the crystallographic restriction theorem permits only two-, three-, four-, and six-fold rotational symmetries in periodic tilings of the plane). The girih-tile method circumvents this by tessellating with a small set of compatible polygons (decagon, pentagon, elongated hexagon, bowtie, rhombus) whose angles do all sum correctly at every vertex. The resulting surfaces show local ten-fold symmetry without being globally periodic. When the tile-set is combined with a self-similar subdivision rule — each large girih tile decomposes into a specific arrangement of smaller girih tiles at a scale ratio of φ — and the rule is iterated, the surface approaches a quasiperiodic tiling in the strict sense formalized by Penrose and de Bruijn. Lu and Steinhardt argued in 2007 that the Darb-i Imam spandrel implements exactly this construction.

Architectural Use

The ten-fold star is one of the dominant star figures of Persian, Anatolian, and Central Asian Islamic architecture from the late twelfth century onward, and is rarer in Mamluk Egyptian and Maghrebi traditions, which lean more on eight- and twelve-fold symmetries. The earliest large-scale surviving example is the Gunbad-i Qabud (Blue Tomb) in Maragha, Iran (1196–1197, Seljuk period), a decagonal tomb tower whose exterior is wrapped in glazed and unglazed brickwork in a complex ten-fold-symmetric girih-tile pattern. Emil Makovicky's 1992 analysis showed that the Maragha tomb's pattern uses the girih tile-set and includes elements that anticipate aperiodic-tiling behavior; Lu and Steinhardt cited it in their 2007 paper as an important precursor to the more developed Darb-i Imam example. The most-cited single architectural example is the Darb-i Imam shrine in Isfahan, conventionally dated to 1453, Timurid. The spandrel above each of the shrine's portals carries a ten-fold-symmetric girih pattern at two scales: a large-scale outer pattern of decagons, pentagons, and bowties visible from across the courtyard, and a small-scale subdivision inside each large tile visible up close. The two scales are related by a self-similar inflation rule that approaches a quasiperiodic regime. Peter Cromwell and Owen Beach argued in a 2018 *Nexus Network Journal* paper titled "Darb-e Imam Tessellations: A Mistake of 250 Years" that the spandrel in question is part of the 1715–1717 Safavid extension (1129 AH foundation inscription), not the original 1453 construction. If their dating holds, the chronological gap from Islamic geometric practice to Penrose's 1974 work narrows from approximately five centuries to about two and a half. The girih method itself remains documented in the twelfth century at Gunbad-i Qabud, and the geometric analysis of the pattern is unaffected by its construction date — but the headline claim depends on the dating dispute. Ten-fold-star patterns also appear on the Friday Mosque of Isfahan (multiple periods), the Bibi Khanum mosque in Samarqand (1399–1404, Timurid), the Gur-i Amir mausoleum in Samarqand (1404, Timurid), the Imam Mosque in Isfahan (1611–1638, Safavid), and on numerous smaller mosques, madrasas, and tomb-towers across the Iranian plateau. Materials include glazed brick, cuerda-seca tilework, ceramic mosaic (*kashi-kari*), and seven-color (*haft-rangi*) tile.

Construction Method

Stretch a cord from a fixed center and mark a circle. Divide the circumference into ten equal arcs — the standard artisan method uses a compass set to the circle's radius and steps off six points (since the radius chords the circle into six 60° arcs), then subdivides each pair of adjacent points to reach the ten 36° arcs by additional compass-and-straightedge construction. The ten points marked on the circumference are the vertices of a regular decagon and provide the framework for everything that follows. For the closed-form {10/3} star: from each of the ten points, draw a straight chord to the point three positions clockwise (1→4, 2→5, 3→6, and so on). The ten chords trace a sharp ten-pointed star with an internal regular decagon at its core. For the two-overlapping-pentagons construction: connect points 1-3-5-7-9 to form one pentagon, then connect points 2-4-6-8-10 to form a second pentagon rotated 36°; the union is a softer ten-pointed star with broader internal angles. Either figure becomes the central rosette of a larger girih-tile composition. The artisan extends strapwork outward from each of the ten star points into the surrounding field, choosing strapwork angles (commonly 18° or 36° from each star edge) so the strapwork connects into adjacent star figures and tessellates the wall. At the Darb-i Imam spandrel the construction operates at two scales. The artisan first laid out a large-scale girih tessellation across the spandrel — large decagons, large pentagons, large bowties, large rhombi, each carrying its strapwork lines. Then each large tile was subdivided into smaller girih tiles according to a fixed substitution rule, with the small tiles rotated and arranged so the small-scale strapwork joins smoothly. The Topkapı Scroll preserves working drawings of ten-fold girih compositions at multiple scales; some of the patterns in the scroll closely match surviving wall surfaces in Iran. Realization in material: most ten-fold-star surfaces use either glazed brick (alternating glazed and unglazed bricks set so the glazed faces form the strapwork bands), *kashi-kari* tile mosaic (individual ceramic pieces cut from larger glazed sheets and set in mortar), or *haft-rangi* seven-color tile (larger tiles with the multi-color pattern fired onto each tile before laying).

Spiritual Meaning

The ten-fold star, like the rest of the Islamic geometric vocabulary, is theologically read inside the doctrine of tawhid — the unity of God that exceeds depiction. The figure does not symbolize the divine; it occupies a surface that declines to depict, opening an attention that moves through structure rather than fixing on representation. Within Persian Islamic culture, where the ten-fold star is most concentrated, the figure carries no canonical doctrinal meaning. There is no Qurʾanic text or hadith that fixes a reading on the number ten or on the ten-pointed star. Some classical and modern Sufi-influenced commentators have proposed symbolic readings — ten as the number of completeness in numerical mysticism, ten as the count of the Ismaʿili imams in some traditions, ten as the sum of the five pillars of Islam doubled — but these are commentator readings overlaid on a geometric form whose architectural use long predates and runs broader than any of them. The ten-fold star's spiritual register is the wider Islamic geometric tradition's: the patterns are non-representational ornaments that gesture toward divine unity by declining to image it, and that organize the architectural surface around the spaces where the worshipper's body returns — the mihrab, the dome above the prayer hall, the spandrels above the entry portal. Where the ten-fold star becomes most concentrated theologically is in shrine architecture rather than mosque architecture. The Darb-i Imam shrine in Isfahan is a Shia memorial site associated with two descendants of the Imams, and the ten-fold geometric programs at such shrines may carry resonances specific to Twelver Shia devotional culture — though the patterns themselves are not Shia symbols and appear on Sunni buildings as well. The most precise theological statement that can be made is the same one that applies to girih generally: these are not secret teachings encoded for initiates. They are public craft, recorded in the Topkapı Scroll and the Tashkent Scrolls as workshop reference, transmitted master to apprentice as construction technique, and visible to anyone who walks into the building. The depth lies in what they are, not in what is hidden behind them.

Significance

Set a compass on a center point and step ten equal arcs around its circumference, 36° at a time. The ten points that fall on the circle are the vertices of a regular decagon; the star figure that arises when those points are connected by chords every third position is the {10/3} star polygon, ten sharp points around an inner decagon. The internal angle at each star point is 36° — half the 72° interior angle of a regular pentagon, the same 36° that appears in the long-side-to-short-side ratio of a golden gnomon triangle. Where this figure becomes mathematically distinctive, in contrast to the eight-fold or twelve-fold star, is its relation to the plane. A regular hexagon tiles the plane without gaps; a square tiles the plane; a regular triangle tiles the plane. A regular pentagon does not — pentagons cannot meet edge-to-edge without leaving wedge-shaped holes, because 360° does not divide cleanly by 108°. The same constraint propagates to the decagon and to the ten-fold star: five-fold and ten-fold symmetry are *forbidden symmetries* under the rules of classical periodic crystallography. Islamic geometric design met this constraint by inventing the girih-tile method. With five equilateral polygons — decagon, pentagon, elongated hexagon, bowtie, rhombus — an artisan can tessellate a surface in which ten-fold stars appear at every decagon center and pentagonal star-fragments at every pentagon center, with the strapwork joining smoothly across all tile edges. The patterns tessellate locally; they do not need to repeat globally to do their architectural work. The deepest claim about these patterns, made by Peter Lu and Paul Steinhardt in their 2007 *Science* article 'Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture' (315: 1106–1110), concerns the Darb-i Imam shrine in Isfahan, conventionally dated to 1453. Lu and Steinhardt argued that the shrine's spandrel uses ten-fold-symmetry girih tiles at two scales — a large-scale pattern visible from the courtyard and a small-scale subdivision visible up close — with the small tiles dividing the large ones according to a self-similar subdivision rule. Iterated, this rule generates a near-quasiperiodic tiling: a pattern that exhibits long-range order with ten-fold rotational symmetry but never repeats itself exactly. The same kind of pattern was formalized by Roger Penrose in 1974 in his published aperiodic tilings, and the underlying mathematics was further developed in the 1980s in connection with Daniel Shechtman's discovery of physical quasicrystals (Nobel Prize in Chemistry, 2011). On the conventional 1453 dating, the Darb-i Imam spandrel predates Penrose's 1974 publication by approximately five centuries. Cromwell and Beach's 2018 redating of the relevant panel to the 1715–1717 Safavid extension would narrow this gap to approximately two and a half centuries. The girih method as such — the tile-set, the substitution rule, the placement-based generation of star-and-polygon patterns — is independently attested in twelfth-century Maragha and in the late-fifteenth-century Topkapı Scroll, neither of which is in dispute. The careful framing of this finding, which Lu and Steinhardt make and which subsequent commentary (Cromwell 2009, Makovicky 1992, the 2018 *Nexus Network Journal* re-examination by Cromwell and Beach) has sharpened, is *near-quasiperiodic* rather than perfectly quasiperiodic. The medieval artisans worked at the scale of a wall, not at the scale of a mathematical proof; the spandrel is finite, contains small inconsistencies, and shows the self-similar subdivision rule applied a limited number of times rather than infinitely. The artisans were not anticipating Penrose's theorems; they were constructing a specific wall with a tile-set that, used this way, happens to enter the quasiperiodic regime. Both findings stand: the medieval Islamic geometric tradition reached the quasiperiodic structure by craft means before its formal mathematical theory, and the formal theory of aperiodic tilings remains the work of Penrose, de Bruijn, and others in the 1970s and 1980s. This is the geometric impossibility-becoming-possibility moment of the ten-fold star — a forbidden symmetry under classical crystallography, reached on real walls in real buildings, by tile-placement rules that did not need to know the mathematics they were sitting inside.

Connections

The ten-fold star sits inside the girih-tile tradition and shares all of its links: to the Topkapı Scroll (Topkapı Palace Library MS H. 1956), to the Tashkent Scrolls of Bukhara, to Persian and Central Asian architecture from the Seljuk period onward, and — via the Lu and Steinhardt 2007 *Science* paper — to Penrose tilings (1974) and to physical quasicrystals (Shechtman 1982, Nobel Prize 2011). Within Islamic geometric design the ten-fold star pairs naturally with the five-fold star and with the eight-fold and twelve-fold stars built on different rotational symmetries. The ten-fold star is geometrically and culturally distinct from the ten-fold rosettes and dasara forms in South Asian mandalas, which use different construction methods on different theological grounds. The Bahaʾi nine-pointed star, the Christian and folk five-pointed star, and the eight-pointed-star Seal of Solomon — these are all neighboring figures in the broader visual vocabulary of religious geometry, but they do not derive from the Islamic ten-fold tradition or vice versa. The golden ratio φ, the irrational number 1.618…, runs through the ten-fold star at every level — in the diagonal-to-side ratio of the regular pentagon, in the inflation ratio between the large and small girih tiles at Darb-i Imam, in the dimensions of the Penrose kite-and-dart tiles. This is not a transmission across traditions; it is the same mathematical regime being reached from different directions.

Further Reading

• Lu, Peter J., and Paul J. Steinhardt. "Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture." *Science* 315, no. 5815 (Feb 23, 2007): 1106–1110.
• Cromwell, Peter R., and Owen Beach. "Darb-e Imam Tessellations: A Mistake of 250 Years." *Nexus Network Journal* 20, no. 3 (2018): 567–582. DOI: 10.1007/s00004-018-0391-y.
• Cromwell, Peter R. "The Search for Quasi-Periodicity in Islamic 5-fold Ornament." *The Mathematical Intelligencer* 31, no. 1 (2009): 36–56.
• Makovicky, Emil. "800-Year-Old Pentagonal Tiling from Marāgha, Iran, and the New Varieties of Aperiodic Tiling It Inspired." In *Fivefold Symmetry*, edited by István Hargittai, 67–86. Singapore: World Scientific, 1992.
• Necipoğlu, Gülru. *The Topkapı Scroll: Geometry and Ornament in Islamic Architecture* (Topkapı Palace Museum Library MS H. 1956). Santa Monica: Getty Center for the History of Art and the Humanities, 1995.
• Bonner, Jay. *Islamic Geometric Patterns: Their Historical Development and Traditional Methods of Construction.* New York: Springer, 2017.
• Penrose, Roger. "The Role of Aesthetics in Pure and Applied Mathematical Research." *Bulletin of the Institute of Mathematics and its Applications* 10 (1974): 266–271.
• Hattstein, Markus, and Peter Delius, eds. *Islam: Art and Architecture.* Cologne: Könemann, 2000. (For the Gunbad-i Qabud and Maragha context.)