Girih Tile

About Girih Tile

Set down a regular decagon. To its sides fit an elongated hexagon, then a bowtie, then a rhombus, then a regular pentagon — five tile shapes total, every edge the same length, every interior angle a multiple of thirty-six degrees. Each tile carries on its face a fixed run of strapwork lines. When an artisan composes a wall by butting the tiles edge to edge, the strapwork crosses each seam without breaking, and a star-and-polygon pattern emerges from a tessellation rather than from a draftsman's compass tracing. This is the girih-tile method documented in Persian and Timurid design scrolls — a tile-set that lets a craftsman generate large, complex geometric surfaces by placement decisions instead of by drawing every line from a single center.

Mathematical Properties

Five tiles, one edge length, every interior angle a multiple of 36°. The tiles are: a regular decagon (interior angles 144°), a regular pentagon (108°), an elongated hexagon with angles 72°-144°-144°-72°-144°-144°, a bowtie or 'butterfly' with four vertices and interior angles 72°-72°-72°-216° (the 216° vertex is reflex, making the tile non-convex), and a thin rhombus with angles 72° and 108°. Every tile is equilateral; the common edge length lets any tile be placed against any other without gaps. Each tile carries a fixed decoration — straight strapwork lines that meet each edge at the same angle on every tile in the set, so that when two tiles share an edge their decorations join into continuous lines. The resulting star-and-polygon pattern is a property of the tessellation rather than of any single tile. Local five-fold and ten-fold rotational symmetry appears at the decagon and pentagon vertices; the strapwork generates ten-pointed stars at decagon centers and five-pointed star fragments at pentagon centers. Globally the patterns can be either periodic (most surviving examples) or aperiodic. The vertex-condition rules are what constrain the tessellation. At every vertex where multiple tiles meet, the interior angles must sum to exactly 360°. Because each tile's interior angles are all integer multiples of 36°, the allowed vertex configurations form a finite, enumerable set — most modern catalogues count between fifteen and twenty primary vertex-types — and the artisan's placement decisions are constrained to choices among these configurations. The strapwork-continuity condition is a second, independent constraint: at each shared edge, the strapwork lines on the two adjoining tiles must enter and exit at the same angle (in the standard girih set, 72° from the edge), so the decoration continues smoothly across the seam. The two constraints together — vertex-angle closure and strapwork continuity — give the system its rule-driven character. The Darb-i Imam spandrel uses a self-similar subdivision rule — each large girih tile decomposes into a specific arrangement of smaller girih tiles at a scale ratio of roughly the golden ratio φ ≈ 1.618 — which, iterated, generates a near-quasiperiodic pattern. The substitution rule is reversible (each small tile can be aggregated back into the large tile it belongs to) and self-consistent across the full tile-set. This is the same kind of inflation-deflation rule that drives modern aperiodic tilings: a recursive substitution that breaks periodicity while preserving long-range order. The pattern shares its symmetry class (decagonal, ten-fold) with Penrose's 1974 kite-and-dart tilings, and the inflation rule is mathematically analogous to Penrose's. The Penrose tilings can be derived from the girih substitution rules and vice versa — they are different presentations of the same underlying decagonal quasiperiodic structure. Lu and Steinhardt argued (2007) that the Darb-i Imam example shows the medieval Islamic geometers had reached the inflation-rule construction by craft means; the careful framing is *near-quasiperiodic*, not provably perfect, because the wall is finite and contains small placement inconsistencies. The girih tile-set is not the only path to these patterns — direct compass-and-straightedge construction can generate the same surface — but the tile-set method is what enables the patterns at architectural scale without redrafting. The wallpaper group of a periodic girih pattern (when one is in use) is typically *p6m* or *pmm*, depending on the underlying repeat-cell. The quasiperiodic patterns at Darb-i Imam do not have a wallpaper group at all — they have no translational symmetry, which is precisely the property that placed them outside classical crystallography until Penrose and de Bruijn formalized quasiperiodic order in the 1970s and 1980s, and until Daniel Shechtman's 1982 discovery of physical quasicrystals showed that nature also reaches this regime.

Architectural Use

Girih tile patterns cover surfaces across the Persian, Anatolian, and Central Asian Islamic architectural traditions from roughly the late twelfth century onward. The earliest surviving large-scale girih work is on the Gunbad-i Qabud (Blue Tomb) in Maragha, Iran, dated 1196–1197 — a Seljuk-period decagonal tomb tower whose exterior is wrapped in glazed and unglazed brick tile in a pattern Emil Makovicky and later Lu and Steinhardt analyzed as an early girih-tile composition. Maragha in the late twelfth century was an intellectual center; the observatory founded there a century later under Nasir al-Din al-Tusi (1259) made it a hub of mathematical and astronomical work, and the geometric sophistication of the Gunbad-i Qabud is consistent with the city's broader scholarly milieu. The Friday Mosque of Isfahan (Masjid-i Jameʿ) preserves girih-pattern brickwork from its Seljuk renovations of the eleventh and twelfth centuries, including patterns on its two great brick domes (the south dome built 1086–1087 under Nizam al-Mulk, the north dome 1088–1089 under Taj al-Mulk). The most-discussed single example is the Darb-i Imam shrine in Isfahan, conventionally dated to 1453, where the spandrels above the side portals carry a girih-tile pattern at two scales — a large-scale outer 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 rule. Darb-i Imam is the wall Lu and Steinhardt argued in 2007 reaches a near-quasiperiodic regime. Peter Cromwell and Owen Beach, in a 2018 *Nexus Network Journal* paper titled "Darb-e Imam Tessellations: A Mistake of 250 Years," have since argued that the spandrel in question is part of the 1715–1717 Safavid extension to the shrine (datable from a 1129 AH inscription on the building) rather than from the 1453 construction. If the later dating holds, the chronological gap to Penrose's 1974 work narrows from approximately five centuries to about two and a half. The girih method itself remains attested in the twelfth century (Gunbad-i Qabud, 1196–1197), and the geometric properties of the spandrel are independent of when it was laid — but the dating dispute affects the chronological headline directly, not merely "some panels." Girih patterns also appear on the Karatay Madrasa in Konya (1251, Anatolian Seljuk), whose dome interior is one of the most-celebrated geometric programs in the Anatolian tradition; on the Bibi Khanum mosque in Samarqand (1399–1404, Timurid); on the Gur-i Amir mausoleum in Samarqand (1404, Timurid), Timur's own tomb; and on the Madrasa-i Mader-i Shah and the Imam Mosque in Isfahan from the Safavid period (seventeenth century). The Registan complex in Samarqand — the Ulugh Beg, Sher-Dor, and Tilya-Kori madrasas, built between 1417 and 1660 — carries girih programs on multiple facades. The patterns are applied in several material registers depending on region and period. In Seljuk and Timurid Iran the typical medium is cut-glazed-brick (turquoise glaze on terracotta, set so the glazed faces trace the strapwork) and *kashi-kari* mosaic tile (small ceramic pieces cut from larger glazed sheets and laid in mortar). In Safavid Iran the cuerda-seca and seven-color *haft-rangi* techniques allow multi-color patterns to be fired onto larger tiles before laying, which reduces labor at some cost in line-sharpness. In Anatolian Seljuk work the medium is more often carved stone and brick. In Mughal India girih-derived patterns appear in red sandstone, in white marble inlay (*parchin kari*), and in carved screen-jali. The Topkapı Scroll preserves 114 such patterns as ink drawings — a working pattern-book that an architect or master-craftsman could consult to choose a design and then realize it in tile or brick.

Construction Method

Begin by selecting the tile-set. The Persian tradition codified five shapes — regular decagon, regular pentagon, elongated hexagon, bowtie, and thin rhombus — every shape equilateral with the same edge length and every interior angle a multiple of 36°. The artisan first decides where the largest tile, the decagon, will sit. On a major wall this is usually a centered placement on a vertical or horizontal axis, often aligned with the qibla direction or with the architectural axis of the building. From the decagon's ten edges the tile-set radiates outward. A common opening sequence places a pentagon at each of two adjacent decagon edges, a bowtie between them along the third edge, an elongated hexagon to bridge to the next decagon position, and so on across the surface. The placements are not free — at any junction where multiple tile-edges meet, the strapwork lines carried on each tile must continue smoothly across all the joining edges, which constrains which tiles can sit at that junction. Working out a coherent surface is a placement puzzle, not a drawing problem. The artisan's working method, reconstructed from the Topkapı Scroll and from surviving wall surfaces, proceeds in stages. First, a small-scale design study is drafted on paper, working out the tile placements for the full surface in compressed form. Second, the design is transferred to a full-scale cartoon — drawn on paper or scribed directly into a layer of plaster — using compass-and-rule construction for the underlying polygon framework. Third, the strapwork lines are added on top of the polygon framework, drawn through the tiles' interiors to connect the edge-crossings. Fourth, the polygon framework is erased or covered, leaving the strapwork visible as the apparent pattern even though the polygons were the underlying construction logic. Once the tessellation is laid out, the strapwork lines drawn on each tile are realized in the chosen material. For brick: the artisan cuts glazed and unglazed bricks to fit the polygon shapes and lays them so the glazed faces form the strapwork bands; this is the dominant Seljuk and early Timurid method, visible at Gunbad-i Qabud and at the Friday Mosque of Isfahan. For *kashi-kari* (tile mosaic): individual ceramic pieces are cut from larger glazed sheets and set into mortar following the polygon outlines, with the colors arranged so the strapwork reads cleanly against the field. For *haft-rangi* (seven-color cuerda-seca tile, dominant in Safavid work): larger square or rectangular tiles carry the multi-color pattern fired onto each tile, with colored glazes separated by manganese-oxide line-blocks; the tiles are then laid in mortar to compose the surface. For stucco: the polygons are scribed into wet plaster and the strapwork channels carved as raised lines. For wood and inlay: the polygon framework is cut from a planar substrate (often walnut or beech) and the strapwork inlaid in contrasting wood, bone, or mother-of-pearl. At Darb-i Imam the artisans worked at two scales — laying out a large-scale tile pattern across the spandrel, then subdividing each large tile into smaller girih tiles according to a self-similar rule. The two-scale method requires that the subdivision rule be fixed in advance: every instance of a large decagon decomposes into the same arrangement of small tiles, every instance of a large bowtie into the same arrangement of small tiles, and so on. This consistency across the spandrel is what allowed Lu and Steinhardt to identify the construction as a deliberate iteration of an inflation rule rather than as an ad-hoc decorative choice. The Topkapı Scroll (MS H. 1956) preserves working drawings of this kind: 114 patterns in ink, several of which match surviving wall surfaces in Iran and Central Asia. The scroll is a craftsman's reference, not a presentation manuscript — it shows the construction lines, the polygon outlines, and the strapwork together, exactly as a workshop would need them. The Tashkent Scrolls (Bukhara, c. sixteenth–seventeenth century, held in the Institute of Oriental Studies, Tashkent) preserve a related artisan tradition with similar working-drawing conventions.

Spiritual Meaning

The theological frame for girih, as for Islamic geometric design generally, is tawhid — the doctrine of divine unity that holds God to be one in a way that exceeds depiction. Aniconism in Islamic art is not a prohibition on art; it is a recognition that the divine cannot be represented and so should not be approximated by image. The geometric, the vegetal (arabesque), and the calligraphic are the three large fields that open in that recognition. Geometry is one way of working with surfaces that decline to depict. The girih patterns, like the wider Islamic geometric tradition, have been read by scholars such as Titus Burckhardt (*Art of Islam*, 1976) and Keith Critchlow (*Islamic Patterns*, 1976) as visual gestures toward a unity whose internal articulation can be entered indefinitely. The decagon at the center of a girih pattern is a finite shape; the strapwork that radiates from it can in principle continue without bound, with each star inviting the eye to its center and then outward to the next. The viewer's attention does not arrive at a depicted object; it moves through a structure that does not resolve into representation. This is the architectural-theological move that Islamic geometric ornament makes possible: a surface that organizes itself rigorously without ever offering an image to fix on. The worshipper looks at the wall and does not see God in it, does not see a saint in it, does not see a narrative in it; what is there is articulated order, and the worshipper's recitation, posture, and orientation toward Mecca do the work that figurative imagery does in other religious traditions. This reading is widely held but should not be over-extended. There is no documented esoteric girih lore — no secret transmission of mystical meaning from master to apprentice that the public is not allowed to know. The girih method is craft knowledge, preserved in working pattern-books (the Topkapı Scroll, the Tashkent Scrolls) and transmitted as construction technique. Sufi orders certainly used buildings ornamented with girih and developed reflections on those spaces, but the girih method itself is not a Sufi secret. The patterns belong to the *maʿmar* (master builder) tradition first and to the wider Muslim architectural culture second. Where individual Sufi commentators have offered geometric readings — for example in some Naqshbandi or Mevlevi-adjacent texts that draw on the rotational symmetries as a figure of the divine name's repetition in *dhikr* — these are interpretive applications of a publicly-available form, not the unveiling of a hidden teaching that the form itself encodes. Some buildings carry Qurʾanic inscriptions interwoven with girih bands, integrating the geometric and calligraphic registers; the patterns there frame the recitation rather than substitute for it. Where the meaning becomes most concentrated is in the patterning of mosque interiors at the points the worshipper's body returns to — the mihrab niche, the dome above the prayer hall, the entry portal — where the geometric surface marks the place of prayer without depicting either the worshipper or what is being prayed to. Shia shrine architecture, including Darb-i Imam itself, intensifies this organization around the tomb of an imam or imamzada (a descendant of the Imams); the geometric program orders the architectural approach to the place of intercession without imaging either the deceased or the relationship the pilgrim has come to enact. The depth of the form lies in this restraint. Girih organizes a space for prayer or pilgrimage without making any image-claim about what prayer reaches.

Significance

Lay the regular decagon first and the rest of the tile-set falls into place around it — pentagon in the corners, bowtie in the throat between two decagons, rhombus where the angles narrow, elongated hexagon as a connective tissue. The significance of girih is that this five-shape kit is enough. A craftsman who knows the set can compose surfaces of any size without re-drafting; the design has been delegated to the tile, not held in the master's drawing. That delegation is what makes girih historically distinctive within Islamic geometric design. Before about 1200 CE the dominant construction was compass-and-straightedge — radial grids drawn from a center, the strapwork traced along the resulting intersections. That method works at the scale of a panel but becomes unwieldy at the scale of a façade. The tile-set method, documented by Peter Lu and Paul Steinhardt in 2007, replaces drafting with placement: the wall is laid out as a tessellation of equilateral polygons whose edges all have the same length, and the underlying star-and-polygon pattern emerges from how the tiles fit together. The shift is not cosmetic. It is a shift in where the design intelligence lives. Compass-and-straightedge construction holds the whole figure in a single radial logic — every line referenced back to the center and the chosen base angle. A girih composition holds the figure in the joint conditions of the tile-set itself, locally at every edge. Two craftsmen working at opposite ends of a long wall can produce a coherent surface without either of them needing to see the whole, provided both know the tile-set and the strapwork-junction rules. This is the geometric equivalent of moving from a centrally-drafted blueprint to a modular construction system, and it is what made architectural-scale geometric ornament reproducible at Timurid scale. Girih tilework carries this method into a theological frame that does not require it but accepts it readily. The Qurʾanic move away from figurative imagery, grounded in tawhid (the unity of God that exceeds depiction), opened a wide architectural field for non-figurative ornament — calligraphic, vegetal, and geometric. Girih became one of the languages of that field. The patterns are not symbols of God; they are surfaces that decline to depict, that gesture instead toward a unity whose internal articulation can be entered but never circumscribed. A worshipper standing before a mihrab framed in girih sees a surface that opens repeatedly into smaller orders without resolving into representation — the eye finds a ten-pointed star, follows the strapwork outward to a pentagon, finds another star, and the surface neither depicts nor terminates. Titus Burckhardt and Keith Critchlow, in their 1976 studies of Islamic ornament, both read this organization as a visual analogue of the metaphysics it sits within: a unity whose articulation does not exhaust it. Their reading is widely held but should not be made esoteric. The girih method is craft knowledge, transmitted master-to-apprentice through working drawings, and the theological reading is a public Islamic interpretation of a public artisan tradition. The 1453 Darb-i Imam shrine in Isfahan is the most-cited single example. Lu and Steinhardt argued that the shrine's spandrel uses girih tiles at two scales — a large-scale pattern visible from across the courtyard and a small-scale subdivision visible up close — with the small tiles dividing the large ones according to a self-similar rule. Run that rule enough times and the result approaches a quasiperiodic tiling: a pattern that locally looks regular but never repeats itself exactly. Mathematically this is equivalent to the tilings Roger Penrose published in 1974. If the spandrel is 1453, the medieval Islamic pattern predates Penrose by about five centuries; if Peter Cromwell and Owen Beach's 2018 redating to the 1715–1717 Safavid extension is correct (see architectural_use below), the gap narrows to about two and a half centuries. The careful claim here, which Lu and Steinhardt make and which later commentary has sharpened, is *near-quasiperiodic* — the medieval pattern shows the self-similar subdivision rule and the local quasiperiodic order, but the spandrel is a finite wall and includes small inconsistencies in tile placement. The artisans were not proving theorems about aperiodic tilings; they were constructing a wall with tools that, used this way, happen to produce a structure that turned out to be deep. The distinction preserves both findings. The medieval Islamic geometric tradition reached the quasiperiodic regime by craft methods well before the formal theory, and the formal mathematical theory of aperiodic tilings is still Penrose's achievement. Both can be true. A further point worth holding: the discovery direction has changed since Lu and Steinhardt's paper. Mathematicians and historians have begun looking at other Islamic geometric surfaces — buildings in Maragha, Bukhara, Konya, Cairo — for further examples of quasiperiodic or near-quasiperiodic structure, and the question is now whether the Darb-i Imam spandrel is unique or whether it sits inside a broader, less-formalized tradition. Peter Cromwell's 2009 review in *The Mathematical Intelligencer* argued for caution: not every visually-complex Islamic geometric pattern is quasiperiodic, and the specific claim requires the self-similar inflation rule to be in evidence. What remains undisputed is that the girih-tile method, as such, is documented in surviving design scrolls, is geometrically equivalent to a powerful piece of modern mathematics, and was developed and transmitted as public craft within working artisan traditions across the Persian and Central Asian Islamic world.

Connections

Girih sits at the center of an unusually rich set of cross-tradition links. The most direct is to Penrose tiling — the Lu and Steinhardt 2007 *Science* paper made the connection explicit, and the two systems share decagonal symmetry, golden-ratio relationships, and aperiodic order. The connection is mathematical-structural, not influence-based: there is no transmission line from medieval Isfahan to Oxford in 1974. Within Islamic tradition girih is closely tied to the ten-fold star and eight-fold star as the star-shapes the strapwork generates, and to muqarnas, which uses the same five tile shapes projected into three dimensions to build honeycomb vaults. Girih is distinct from zellige, the Maghrebi tradition of cut-glazed-terracotta tilework that uses different polygon shapes and different construction logic, and from sebka, the Andalusian lozenge-grid surface treatment. The Topkapı Scroll (Topkapı Palace Library MS H. 1956), the late-fifteenth-century Timurid pattern-book studied in depth by Gülru Necipoğlu, is the surviving design source — 114 patterns drawn in ink, including the girih tile-set used at Darb-i Imam. The Tashkent Scrolls (Bukhara, c. sixteenth–seventeenth century) preserve a related artisan tradition. The girih method is publicly documented craft knowledge, not esoteric transmission.

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.
• Necipoğlu, Gülru. *The Topkapı Scroll: Geometry and Ornament in Islamic Architecture* (Topkapı Palace Museum Library MS H. 1956). With an essay on the geometry of the muqarnas by Mohammad al-Asad. 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.
• Broug, Eric. *Islamic Geometric Patterns.* London: Thames & Hudson, 2008.
• Critchlow, Keith. *Islamic Patterns: An Analytical and Cosmological Approach.* London: Thames & Hudson, 1976.
• 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*, ed. István Hargittai, 67–86. Singapore: World Scientific, 1992.