Hexagonal Tessellation

About Hexagonal Tessellation

Three regular polygons can tile the Euclidean plane edge-to-edge with no gaps and no overlaps: the equilateral triangle, the square, and the regular hexagon. These are the only three. Among them the hexagon is the most efficient — Thomas Hales proved in 2001 (*Discrete & Computational Geometry* 25: 1-22) that the regular hexagonal honeycomb minimizes the perimeter needed to divide the plane into regions of equal area. The same property is the reason Islamic artisans built their six-fold patterns on a hexagonal lattice, the reason bees converge on hexagonal cells in honeycomb (via surface tension on initially-cylindrical wax, per Karihaloo et al. 2013), and the reason crystallography classifies the p6 and p6m wallpaper groups among the seventeen plane symmetry groups. Three independent paths to the same form, each for its own reasons. This page is the umbrella for hexagonal tessellation in Islamic geometric practice.

Mathematical Properties

The regular hexagonal tessellation is the dual of the regular triangular tessellation: each vertex of the hexagonal tiling is the center of a triangle in the triangular tiling, and vice versa. It is one of the three regular tilings of the Euclidean plane (the others being the triangular tiling {3,6} and the square tiling {4,4}), and the only one whose Schläfli symbol is {6,3} — six-sided polygons meeting three at a vertex. Each interior angle of a regular hexagon is 120°; three of these meet at each vertex to fill 360°.

The wallpaper symmetry group of the regular hexagonal tiling is p6m — six-fold rotational symmetry combined with mirror reflections through six axes per cell, giving twelve symmetry operations per fundamental domain (the 30°-60°-90° triangle that is one-twelfth of a hexagon). This is the highest-symmetry wallpaper group, and the hexagonal lattice is the densest-packing lattice in the plane: the hexagonal packing of equal circles is the densest possible (the packing density is π/(2√3) ≈ 0.9069, a result first proved by Carl Friedrich Gauss in 1831 for lattice packings and extended to arbitrary packings in the early 20th century by Axel Thue (1890, with rigorous version 1910) and László Fejes Tóth (1940).

The honeycomb conjecture, proved by Thomas Hales in 2001 (*Discrete & Computational Geometry* 25: 1-22), establishes the regular hexagonal tiling as the perimeter-minimizing partition of the Euclidean plane into regions of unit area. The proof transports the planar tiling to a torus to take advantage of the torus's compactness, then proves that a particular functional involving the perimeter of each cell is uniquely minimized by the regular hexagon. The earlier convex-cell version of the result, proved by László Fejes Tóth in 1943, established the same minimization under the constraint that all cells be convex; Hales removed the convexity constraint and allowed disconnected cells and gaps.

The hexagonal tiling supports several derivative tessellations of architectural relevance. The triangular sub-tiling divides each hexagon into six equilateral triangles by drawing the three long diagonals — generating the {3,6} triangular tiling as a refinement. The trihexagonal tiling (semi-regular, with vertex configuration 3.6.3.6) alternates triangles and hexagons. The truncated hexagonal tiling (vertex configuration 3.12.12) is a semi-regular tiling using triangles and dodecagons. The rhombitrihexagonal tiling (vertex configuration 3.4.6.4) combines triangles, squares, and hexagons. Each of these can be derived from the hexagonal lattice by specific subdivisions, and each appears in Islamic geometric practice in different contexts.

Under the crystallographic restriction theorem, six-fold rotational symmetry is the highest periodic-tiling rotation order permitted in the Euclidean plane (the others being 2, 3, and 4). This makes the hexagonal lattice the natural substrate for any pattern requiring six-fold or twelve-fold visual symmetry — the twelve-fold work places twelve-fold rosettes at the lattice nodes, the global symmetry remains six-fold p6m, and the visual twelve-fold appears as a local feature of the rosette.

In three dimensions the hexagonal close packing (hcp) and face-centered cubic (fcc) are the two densest sphere-packings, both with density π/(3√2) ≈ 0.7405 — a result conjectured by Johannes Kepler in 1611 and proved by Thomas Hales in 1998 (the Kepler conjecture, published in *Annals of Mathematics* 162 in 2005, with formal verification completed by the Flyspeck project in 2014).

Occurrences in Nature

The hexagonal form is the destination of multiple independent processes in the natural world. Naming the mechanism for each is the work of being precise.

*Honeybee comb.* Worker bees of *Apis mellifera* and related species build wax comb in hexagonal cells. The traditional explanation, going back to Pappus of Alexandria and reiterated by Darwin in *Origin of Species* (1859), credited the bees with geometric calculation. The modern explanation, established by B. L. Karihaloo, K. Zhang, and J. Wang in 'Honeybee combs: how the circular cells transform into rounded hexagons' (*Journal of the Royal Society Interface*, 2013), is that bees build cylindrical cells initially. Surface tension at the triple junctions where three wax walls meet — operating on wax that is softened by the heat of the bees' bodies clustered around the comb — pulls the cylinders into hexagonal cross-section. The bees do not compute hexagons; the physics of surface-tension minimization at triple junctions imposes 120° wall-meeting angles, and 120° angles tessellate the plane into hexagons. A subsequent paper by Bauer and Bienefeld (*Scientific Reports*, 2016) argued that surface tension alone may be insufficient and proposed a mechanical-shaping contribution from the bees' active manipulation of the wax — the field has not fully settled on a single mechanism, but the general picture is one of physical relaxation rather than geometric construction.

*Basalt columns.* When thick basalt lava cools, thermal contraction stress generates a network of cracks at the cooling surface. The cracks propagate inward (perpendicular to the cooling surface) and meet at angles that minimize total fracture energy — under the same surface-tension-at-triple-junctions logic, that minimum is at 120°. The result is the polygonal columnar jointing visible at the Giant's Causeway in Northern Ireland, the Devil's Postpile in California, and Fingal's Cave on Staffa. The columns are typically hexagonal but include occasional pentagons and heptagons where the cooling was uneven — perfect hexagonality requires uniform conditions.

*Snowflakes.* Water molecules in ordinary ice (Ih structure, space group P63/mmc) form a hexagonal crystal lattice through tetrahedral hydrogen bonding. The macroscopic snowflake inherits the six-fold symmetry of the underlying crystallography; the dendritic branches grow outward from the central seed crystal at the angles the lattice permits. Wilson Bentley's snowflake photographs (Vermont, late 19th and early 20th centuries) document thousands of distinct snowflakes, every one carrying the same underlying six-fold symmetry with no two having identical branch detail. The variation is in the growth history; the symmetry is in the crystallography.

*Graphene and carbon nanostructures.* Graphene — a single-atom-thick sheet of carbon — is a hexagonal lattice of carbon atoms held by sp² orbital hybridization. The carbon atoms bond at 120° angles in the plane of the sheet, generating the hexagonal lattice as the equilibrium structure. The same hexagonal carbon lattice rolled into a cylinder is a carbon nanotube; wrapped into a closed surface with twelve pentagonal defects is the buckminsterfullerene molecule (C60, the buckyball), discovered in 1985 by Harry Kroto, Robert Curl, and Richard Smalley (1996 Nobel Prize in Chemistry). The same lattice in three-dimensional stack is graphite.

*Convection cells.* When a fluid layer is heated from below, it organizes into convection cells whose horizontal cross-section is typically hexagonal — Bénard cells, observed by Henri Bénard in 1900 and analyzed by Lord Rayleigh in 1916 (now Rayleigh-Bénard convection). The same triple-junction surface-tension logic applies: cells with three-way boundaries at 120° angles minimize the energy of the cellular pattern.

*Dried mud, Voronoi tessellations in cell biology, fault-crack patterns in cooling igneous and metamorphic rock.* The same mechanism — triple-junction equilibrium at 120° — recurs across radically different physical contexts. In each case the hexagon is not designed; it is the equilibrium shape that results from any process minimizing energy across a dense packing of equal-area regions in the plane.

The Islamic geometric tradition's hexagonal tessellation work does not derive from observation of nature — there is no documentary evidence that medieval Islamic artisans were modeling honeycomb or basalt. The convergence is independent: nature converges on the hexagon by surface-tension physics; the artisan converges on it by ease of compass-and-rule construction; the mathematician converges on it by perimeter-minimization optimality. The form is the destination of multiple paths, not the artifact of one of them.

Architectural Use

Hexagonal tessellation enters Islamic architecture as the lattice substrate for six-fold, three-fold, and (via overlaid grids) twelve-fold patterns, across the full geographic range of Islamic ornament from Andalusi Spain to Mughal India.

*Early Persian and Central Asian work.* The Samanid Mausoleum at Bukhara (c. 914-943 CE) is one of the early Persian instances of sophisticated geometric brickwork, with hexagonal-derived patterns appearing across its façades. The Friday Mosque (Jameh) of Isfahan — particularly the northeast dome chamber dated to the late 11th-century Seljuk additions — uses hexagonal lattice substrates for many of its tile and brick patterns. The Friday Mosque of Yazd (15th-century construction over earlier work) carries extensive hexagonal tessellations and six-fold star patterns; the tympanum above the eastern portal is one of the most-analyzed multilayered geometric patterns in the modern academic literature on Islamic two-level patterns (Lu and Steinhardt 2007 and subsequent work).

*Anatolian Seljuk work.* The Alâeddin Mosque at Konya (1220) and the Karatay Madrasa at Konya (1251-1252) carry six-fold and hexagonal-derived patterns in their tile and brick decoration. The Çifte Minareli Madrasa at Erzurum (late 13th century) extends the same tradition.

*Mamluk Egyptian work.* Mamluk minbars and Qur'an illumination from the 14th-15th centuries make heavy use of hexagonal-grid tessellations. The minbar of Sultan Qa'itbay in the Madrasa of Sultan Qa'itbay (Cairo, 1472-1474) is one of the canonical Mamluk examples.

*Andalusi and Maghrebi work.* The Alhambra in Granada (Nasrid, 13th-14th centuries) carries hexagonal-derived patterns alongside its more famous quasi-periodic decagonal work. Edith Müller's 1944 doctoral dissertation 'Gruppentheoretische und strukturanalytische Untersuchung der maurischen Ornamente aus der Alhambra in Granada' counted eleven of the seventeen plane symmetry groups represented at the Alhambra (later analyses by Branko Grünbaum and others have revised the count to thirteen). The popular claim that the Alhambra contains all seventeen wallpaper groups is overstated and not supported by the careful group-theoretic surveys. The Ben Yousef Madrasa in Marrakech (16th century, Saadian rebuild on earlier Almoravid foundations) and the Bou Inania Madrasa in Fez (14th century, Marinid) carry hexagonal zellige work.

*Mughal Indian work.* The Taj Mahal (1632-1653), the Tomb of Akbar at Sikandra (1605-1613), and the Itimad-ud-Daulah at Agra (1622-1628) use hexagonal-grid tessellations in their *jali* screens and *pietra dura* inlay. The Mughal tradition tends to subordinate strict geometric tessellation to the floral-and-calligraphic register, but hexagonal substrate work appears throughout.

*Ottoman work.* The Süleymaniye Mosque in Istanbul (1550-1557, Mimar Sinan) and the Selimiye Mosque at Edirne (1568-1574, also Sinan) carry hexagonal-derived patterns in their iznik tile revetments. Süleymaniye carpet pages — bound prayer-rugs and Qur'an covers from the same period — are noted for hexagonal grid compositions.

In most of these instances the hexagonal tessellation is not the surface pattern as such — the surface carries the elaborated rosette-and-strapwork ornament that the lattice supports. The hexagon is the underlying construction grid that the artisan snaps onto the wall with chalk-string, and from which the final visible pattern develops by inscribing rosettes at nodes and filling interstices with secondary polygons. The lattice itself is often invisible in the finished work; its presence is inferred from the pattern's six-fold symmetry.

Construction Method

Hexagonal tessellation is among the simplest exact constructions in compass-and-rule geometry. The Topkapı Scroll and the Tashkent Scrolls — the two main surviving artisan-scroll references — present the hexagon construction as one of the foundational figures in the apprentice's first training, prerequisite to the more elaborate rosette work.

*Constructing a single hexagon.* Draw a circle of any radius *r* with center *O*. Set the compass to the same radius *r*. Place the compass point on the circumference of the circle and swing an arc that intersects the circle. Move the compass point to that intersection and swing another arc. Continuing around the circle, the six step-points are reached after exactly six steps — the hexagon's vertices. Connect the six points with straight lines. The construction works because the side of a regular hexagon inscribed in a circle equals the radius of the circle: the six radii to the vertices and the six sides between them form six equilateral triangles, so each side is exactly the radius.

*Constructing the hexagonal lattice.* From the central hexagon, repeat the same compass-and-rule procedure at each of the six vertices, producing six adjacent hexagons. Each new hexagon shares two sides with its neighbor; the seven-hexagon cluster (one central plus six surrounding) is the basic unit. Repeat outward across the wall surface — at scale, the artisan snaps the central east-west and north-south reference cords first, marks the principal lattice nodes with chalked string at the calculated intervals, and then either inscribes the full hexagonal lattice in chalk or works directly with the lattice nodes as construction reference points.

*Deriving sub-patterns from the lattice.* The triangular sub-tiling is generated by drawing the three long diagonals of each hexagon, producing six equilateral triangles per hexagon. The six-pointed star (hexagram) is generated within each hexagon by drawing alternating vertex-to-vertex chords. The six-fold rosette is generated by inscribing a circle in each hexagon and constructing a star pattern within the circle. The twelve-fold rosette is generated by overlaying two hexagonal lattices offset by 30°, with twelve-fold rosettes constructed at the nodes of the combined lattice.

*Materials and execution.* The materials and the execution vary by region and period. In Timurid Samarkand and Persian-Safavid Isfahan the hexagonal lattice is laid out at full scale on the wall, the rosettes and interstitial polygons are cut from glazed tile (*moʿarraq* mosaic technique), and the cut pieces are laid into prepared mortar according to the lattice plan. In Mamluk Cairo the lattice may be carved directly into stone or laid as inlaid marble (*opus sectile*). In Anatolian Seljuk work it is laid in brick (*hazārbāf*) with glazed-tile inserts. In Maghrebi *zellige* work the cut-tile mosaic is the dominant medium — the *maâlem* cuts the hexagonal and interstitial tiles by hand at his low workshop table, then composes them on the prepared mortar bed.

The Topkapı Scroll (Topkapı Palace Library MS H. 1956, Persian, c. 15th century) preserves the most complete surviving artisan record of these construction methods, with diagrams showing the step-by-step construction from circle to lattice to rosette. Gülru Necipoğlu's 1995 study *The Topkapı Scroll* is the standard modern reference and includes facsimile reproductions of the scroll's hexagonal-construction pages.

The construction methods are not secret. They are openly preserved in the artisan-scroll tradition, taught in working ateliers in Fez, Marrakech, Tunis, Istanbul, and Isfahan today, and documented in the academic literature for any reader with access to the major monographs of Necipoğlu, Critchlow, Bonner, and Broug.

Spiritual Meaning

Islamic theological reading of hexagonal tessellation works at two levels. At the general level, the same *tawhid*-centered theology of geometric ornament applies: the underlying mathematical order of which the hexagonal lattice is one of three regular tilings points back to the One who established that order. The number three (the three regular tilings: triangle, square, hexagon) and the number six (the rotational symmetry order) carry their own associations in Islamic theology — six is the number of days of creation in the Qur'anic cosmogony (Qur'an 7:54, 11:7, 25:59 and other passages affirming that God created the heavens and earth in six days), the six directions of space (the four cardinal plus zenith and nadir), and six is the number of articles of faith (*īmān*) in standard Sunni catechetical formulation.

The Twelver Shi'i tradition has its own six- and twelve-fold associations that connect through the hexagonal substrate to twelve-fold star work — twelve being the number of imams. Sufi traditions occasionally associate six with the Six Pillars (variant lists in different orders) but the symbolism is not as load-bearing as in the Twelver context.

At the specific level, hexagonal tessellation in Islamic ornament does not encode a fixed iconography. Following Titus Burckhardt and Keith Critchlow, the consensus reading is that the lattice carries the underlying numerical-and-geometrical order of which all six-fold instances (the days of creation, the cardinal directions, the natural hexagons of honeycomb and basalt and snowflake) are particular cases. The artisan's hexagonal tessellation is a *contemplation* of that order through the multiplicity it generates, not a coded reference to any one of the six-fold associations.

A word about the convergence with nature. The Islamic tradition's hexagonal work does not derive from observation of honeycomb or basalt — there is no documentary record of medieval artisans studying bee comb or columnar lava in their geometric training. The convergence is real, but it is a convergence on a form that is mathematically optimal and naturally favored, not a transmission of natural pattern into religious art. The Satyori reading of this convergence does not claim that the artisans 'rediscovered' the hexagon of nature; it claims that any process of dividing a plane into equal cells under any pressure converges on the hexagon, and that the Islamic tradition's hexagonal work is one of the human inheritances of that mathematical-and-physical destination.

What the hexagonal tessellation does not carry, in the documentary record of Islamic ornament, is a secret-Sufi-knowledge framing — the construction is the simplest in the artisan repertoire (compass set to radius, step around the circle, six points), shown step by step in the Topkapı Scroll and the Tashkent Scrolls, taught in the artisan workshops of Cairo, Isfahan, Samarkand, Fez, and Istanbul, and laid by working *maâlems* whose apprentices pass the method forward. The depth of the form is in its public visibility and its centuries of cumulative elaboration, not in concealment.

Significance

Three independent disciplines have, at different times and for different reasons, converged on the hexagon as the optimal division of a plane into equal cells. Naming the convergence is the substance of this page.

*The mathematical case.* Among regular polygons, only the equilateral triangle, the square, and the regular hexagon tile the Euclidean plane edge-to-edge — the constraint is that the polygon's internal angle must divide 360° evenly, which for regular polygons gives 60° (triangle, 6 around a vertex), 90° (square, 4 around a vertex), and 120° (hexagon, 3 around a vertex). The hexagon is uniquely efficient among the three. Pappus of Alexandria (c. 320 CE) conjectured that hexagons give the least-perimeter division of the plane into equal areas. The conjecture stood unproven for sixteen hundred years. László Fejes Tóth proved a constrained version in 1943 (under the hypothesis that cells are convex). Thomas Hales gave the full proof in 1999, published in 2001 in *Discrete & Computational Geometry* 25, pages 1-22 — the same Hales who four years earlier had proved the Kepler conjecture on sphere-packing. The honeycomb conjecture is now a theorem.

The seventeen wallpaper groups of plane symmetry — classified in the 19th century by Evgraf Fedorov (1891) and independently rederived by George Pólya (1924) — include p6 (six-fold rotation) and p6m (six-fold rotation with mirror axes) as the two groups carrying hexagonal symmetry. These are among the symmetry classes the crystallographic restriction theorem permits (orders 2, 3, 4, 6 — six being the highest order possible in a periodic plane tiling).

*The natural case.* Honeybee comb is built in hexagonal cells. The traditional explanation — Darwin's, in *Origin of Species* (1859), and the mathematical-bee tradition before him — credited the bees with geometric construction. The modern reading is more careful. B. L. Karihaloo, K. Zhang, and J. Wang showed in 'Honeybee combs: how the circular cells transform into rounded hexagons' (*Journal of the Royal Society Interface*, 2013) that bees build cylindrical cells initially, and that surface tension at the triple junctions where three wax walls meet — operating on wax softened by the bees' body heat — pulls the cylinders into hexagonal cross-section. The hexagon is the equilibrium shape that surface tension imposes on a dense packing of equal cylindrical cells; the bees do not calculate hexagons, the physics does. The same mechanism produces the hexagonal cells in basaltic columnar jointing (Giant's Causeway in Northern Ireland, Devil's Postpile in California — cooling lava contracts and cracks at 120° angles for the same surface-tension-minimization reason), in dried-mud polygonal cracking, and in convection cells (Bénard cells in heated fluid layers). Snowflakes carry six-fold symmetry from the crystallographic structure of ordinary ice (Ih structure, space group P63/mmc): water molecules form a hexagonal crystal lattice through tetrahedral hydrogen bonding, and the macroscopic six-fold symmetry of the snowflake inherits from that underlying crystallography. Graphene — a single-atom-thick sheet of carbon — is a hexagonal lattice of carbon atoms held in place by sp² hybridization of their orbitals.

*The Islamic-artisan case.* From the 10th century onward, hexagonal-lattice tessellation becomes one of the foundational substrates of Islamic geometric pattern. The compass-and-rule construction of the regular hexagon from a circle is the simplest in the regular-polygon repertoire — set the compass to the circle's radius, step it around the circumference, and the six step-points are the hexagon's vertices. This construction may be the most-drawn figure in the artisan-scroll tradition. From the hexagonal lattice the artisan can derive six-fold rosettes, three-fold patterns (the triangular sub-tiling of the hexagon), twelve-fold rosettes (overlaid hexagonal grids offset by 30°), and the elaborate compound patterns at Bukhara, Isfahan, and the Alhambra.

The three cases converge on the same form for genuinely different reasons. The mathematician converges on the hexagon because it is provably optimal. The bee and the basalt converge on it because surface tension at triple junctions forces 120° angles, and 120° angles tessellate the plane into hexagons. The Islamic artisan converges on it because the construction is the easiest exact construction in compass-and-rule geometry and because the resulting symmetries support an indefinite range of derivative patterns. None of the three derives from the others. They are independent paths to a structural minimum.

The Satyori reading: the hexagon is what happens when a plane has to be divided into equal cells under any pressure — geometric, physical, or constructive. Mathematics derives it from optimality, nature derives it from surface tension, the artisan derives it from the simplest construction with circle and compass. The Islamic tradition's hexagonal work is part of a much wider phenomenon: the form is the destination of multiple converging paths. Where Islamic ornament does its specific work is in elaborating the hexagonal substrate into the registers of six-fold rosette, twelve-fold star, and three-fold derivative pattern that the lattice supports — and in transmitting the construction methods publicly through artisan scrolls (Topkapı, Tashkent) from generation to generation.

Connections

Hexagonal tessellation is the lattice substrate for several other Islamic geometric forms on this site. The [twelve-fold star](/sacred-geometry/twelve-fold-star/) is built on a hexagonal lattice with twelve-fold rosettes at the lattice nodes — the hexagonal grid supplies the 6-fold rotation that the crystallographic restriction theorem permits, and the rosettes double it visually to twelve. The [eight-fold star](/sacred-geometry/eight-fold-star/) and [octagram star](/sacred-geometry/octagram-star/) work on square (4-fold) lattices rather than hexagonal. The [girih tile](/sacred-geometry/girih-tile/) set includes both hexagonal and decagonal substrate options. The single-cell [hexagon](/sacred-geometry/hexagon/) entity covers the geometric properties of the hexagon as an individual figure independent of its tessellation behavior.

The cross-tradition resonances are real and structurally specific. The Jewish [Star of David](/sacred-geometry/star-of-david/) (Magen David) is built from two overlapping equilateral triangles forming a hexagram inscribed in a regular hexagon — the same six-fold symmetry derived from the same triangular sub-tiling of the hexagon. The Hindu [Shatkona](/sacred-geometry/shatkona/) is the same hexagram with a different ritual reading. The European medieval hexagonal-vault construction (visible in Cosmati floor work and in some Cistercian chapter-house plans) shares the lattice without the rosette elaboration. The [flower of life](/sacred-geometry/flower-of-life/) construction is a hexagonal grid of circles whose intersections produce the hexagonal lattice as the dual figure.

The natural-world parallel — honeycomb, basalt, snowflake, graphene — is treated in the *Occurrences in Nature* field rather than as a cross-tradition resonance. The convergence of the artisan, the bee, the basalt, and the carbon atom on the same six-fold form is one of the strongest cases on this site for the form being a destination rather than a cultural invention.

Further Reading

• Hales, Thomas C. "The Honeycomb Conjecture." *Discrete & Computational Geometry* 25, no. 1 (2001): 1-22.
• Karihaloo, B. L., K. Zhang, and J. Wang. "Honeybee Combs: How the Circular Cells Transform into Rounded Hexagons." *Journal of the Royal Society Interface* 10, no. 86 (2013): 20130299.
• Grünbaum, Branko, and G. C. Shephard. *Tilings and Patterns*. New York: W. H. Freeman, 1987.
• 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.
• 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.
• Thompson, D'Arcy Wentworth. *On Growth and Form*. Cambridge: Cambridge University Press, 1917 (rev. 1942).
• Lu, Peter J., and Paul J. Steinhardt. "Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture." *Science* 315, no. 5815 (2007): 1106-1110.