What is a star polygon {n/k}?

A star polygon {n/k} is drawn by placing n equally-spaced vertices on a circle and connecting every k-th vertex with a straight line. When gcd(n,k)=1 this traces a single closed star; otherwise it produces gcd(n,k) separate components. The notation was introduced by Schläfli and is used throughout Islamic geometric art.

How does the simulator generate the tiling?

The simulator places a {n/k} star at each node of either a square or hexagonal grid. For each star, it computes n vertex positions on a circle of radius r = scale × 0.45, then draws a polyline connecting every k-th vertex (the star polygon construction). A background n-gon fills the connector tile region to create the interlocking mosaic effect seen in real Islamic architecture.

What is the difference between n (star points) and k (stellation)?

n is the total number of vertices on the circle (equivalently the number of points if gcd(n,k)=1). k is the step size: how many vertices you skip before drawing the next edge. A 5-pointed star is {5/2} — 5 vertices, skip every 2nd. A Star of David is {6/2} — two overlapping triangles. Larger k produces more deeply indented (stellated) stars.

Why does gcd(n,k) matter?

If gcd(n,k)=1 the star polygon is a single connected closed path visiting all n vertices. If gcd(n,k)=g>1, the polygon breaks into g separate closed components, each visiting n/g vertices. For example {6/2} has gcd=2, producing two separate triangles (the Star of David). The simulator draws each component separately.

Can I save the generated pattern?

Yes. Click the 'Save PNG' button in the controls panel to download the current canvas at full resolution as a PNG file. The filename includes the {n/k} notation for easy reference.

What historical patterns do the presets represent?

Star of David {6/2}: two interlocked triangles, universal in Islamic and Jewish geometric art. 8-Star Window {8/3}: the classic 8-pointed star seen in countless mosque windows from Morocco to Central Asia. Girih {10/4}: the 10-pointed star central to the Girih tile system and the Darb-i Imam shrine in Isfahan (1453 CE). Ottoman {12/5}: the 12-pointed star common in Ottoman architecture such as the Blue Mosque and Topkapi Palace.

What symmetry groups appear in Islamic patterns?

Islamic geometric patterns exploit all 17 wallpaper groups (the crystallographic restriction theorem limits periodic tilings to 2-, 3-, 4- and 6-fold rotational symmetries). The most common are p4m (square grids with 4-fold symmetry), p6m (hexagonal grids with 6-fold symmetry), and p4g. Non-periodic Girih tilings have 10-fold local symmetry, exceeding the crystallographic limit.

What is the difference between square and hexagonal grid modes?

In square grid mode, stars are placed at the vertices of a regular square lattice (spacing d). In hexagonal mode they are offset on an alternating row pattern with vertical spacing d·√3/2, creating a denser packing suitable for patterns with 6-fold or 12-fold symmetry such as the {6/2} Star of David or {12/5} Ottoman star.

Is Islamic geometric art connected to modern mathematics?

Yes, deeply. The 17 wallpaper groups that classify all periodic plane patterns were first formally proven complete by Fedorov (1891) and Pólya (1924), yet Islamic craftsmen had empirically discovered and used all of them by the 13th century. The quasi-crystalline Girih tilings discovered in 2007 predate Penrose's mathematical description of aperiodic tilings by 500 years, and the underlying mathematics connects to quasicrystals discovered in 1984 (Nobel Prize 2011).

How does the shader draw the star shape without pre-made artwork?

For every pixel it converts position to polar coordinates, folds the angle into one wedge, then measures distance to two mirrored rays and two circles at radii 0.28 and 0.46. Where that distance drops below Line width, the pixel is painted as strapwork - a distance-field trick that redraws instantly as sliders move.

Home▸Generative Art & Algorithmic Patterns▸Islamic Geometric Patterns

✦ Islamic Geometric Patterns

Generate interactive Islamic geometric star patterns. Choose the number of star points (n) and stellation parameter (k) to create {n/k} star polygon tessellations on square or hexagonal grids. Inspired by Girih tiles, Alhambra mosaics, and Ottoman geometric art. Save PNG.

Generative Art & Algorithmic Patterns3DEasy60 FPS

About this simulation

This generator draws Islamic star-and-strapwork patterns live in a WebGL shader using a signed-distance-field technique, not pre-rendered tiles. Each grid cell is folded by its dihedral symmetry — 4-fold on the square lattice, 6-fold on the hex lattice — then a rosette of mirrored rays and rings traces an n-pointed star, echoing the girih strapwork of the Alhambra and Persian shrines.

🔬 What it shows

Each cell's angle is folded into one symmetric wedge, then mirrored rays plus an inner ring (radius 0.28) and outer tip (radius 0.46) trace the star outline. Cells alternate between two palette colours in a chequerboard fill, with strapwork drawn wherever the distance field crosses the Line width threshold.

🎮 How to use

Pick 6, 8, 10 or 12 star points and a Square or Hex lattice via the toggle buttons. Scale/zoom (1–9) sets tile density, Line width (0.01–0.12) thickens the strapwork, Rotation speed (0–0.3) spins the pattern. Choose from six palettes, drag/scroll to pan and zoom, or click Reset for the default.

💡 Did you know?

In 2007 physicists Peter Lu and Paul Steinhardt showed that 15th-century craftsmen at the Darb-i Imam shrine in Isfahan used girih tiles to build patterns equivalent to Penrose tilings — nearly five centuries before Penrose described them in 1974.

Frequently asked questions

What do the 6, 8, 10 and 12 star-point buttons change?

They set n, the points folded into each cell. This changes the wedge angle (360° divided by n) the shader mirrors around — 12 gives finer, more spiked rosettes than the bolder 6-point hexagram.

What's the difference between the Square and Hex lattice modes?

Square mode repeats stars on a plain grid. Hex mode interleaves two offset grids with vertical spacing d × √3÷2 — the standard trick for tiling hexagons from staggered rectangular lattices.

How does the shader draw the star without pre-made artwork?

Each pixel converts to polar coordinates, folds its angle into one wedge, then measures distance to two mirrored rays and two circles at radii 0.28 and 0.46. Below Line width, it's painted as strapwork.

What do the six colour palettes represent?

Islamic Gold and Moroccan Terracotta echo ochre tilework, Alhambra uses Nasrid-palace teal, Ottoman Blue reflects Iznik ceramic blues, Mughal White gives marble-inlay monochrome, and Neon Night is a modern high-contrast option.

Is this an accurate reconstruction of real Islamic tilework?

It's a faithful model of dihedral symmetry, star polygons and girih-style strapwork, not a copy of any single historical tiling, which often mixes several star orders and irregular grids.

Frequently Asked Questions

Why did Islamic art focus on geometric patterns rather than figurative imagery?

While the Quran does not explicitly prohibit figurative art, Islamic tradition cautioned against creating images that might lead to idolatry or compete with God's creation. This led artists to develop abstract geometric and calligraphic traditions instead of representational art, although figurative painting was practised in secular contexts across the Islamic world.

What mathematical properties do Islamic patterns exhibit?

Islamic patterns display all 17 wallpaper group symmetries — every possible 2D periodic tiling pattern appears somewhere in Islamic art. They commonly use 4-fold, 6-fold, 8-fold, and 12-fold rotational symmetry, along with reflection and glide-reflection symmetries. Some aperiodic patterns exhibit 5-fold and 10-fold symmetry, which cannot occur in periodic tilings.

What are girih tiles?

Girih tiles are a set of five decorated polygons (a decagon, a hexagon, a bowtie, a wide rhombus, and a narrow rhombus) whose edges are marked with strap lines at specific angles. When assembled, their strap lines automatically generate complex Islamic star patterns without individual construction. 15th-century craftsmen appear to have used these tiles to construct quasi-crystalline patterns.

How are Islamic geometric patterns drawn with compass and straightedge?

The process begins by establishing a regular grid based on circles or polygons. Lines are drawn connecting specific points on the grid at prescribed angles, typically multiples of 15° or specific fractions of the circle. The intersections of these lines define the vertices of star polygons. Skilled craftsmen developed sets of template shapes (girih tiles) to achieve consistency without individual calculation.

Are Islamic geometric patterns related to modern quasicrystals?

Yes. Roger Penrose discovered aperiodic tilings with 5-fold symmetry in 1974; Dan Shechtman found the first physical quasicrystal in 1984 (earning the 2011 Nobel Prize in Chemistry). Peter Lu and Paul Steinhardt's 2007 Science paper demonstrated that Islamic craftsmen constructed equivalent aperiodic Penrose-like tilings at least 500 years earlier, suggesting remarkable intuitive geometric understanding.

⚙ Under the hood

Build star polygons and girih tilings from compass-and-straightedge rules, generating the interlaced patterns of the Alhambra and Ottoman workshops.

Islamic geometrystar polygontessellationgirih tilesAlhambraOttomangenerative artn-k star

3D · Three.js / WebGL renderer · 60 FPS target · runs fully client-side, no install