ANDALUSIAN COURSE · MODULE 3

3. Sacred Geometry & Girih Tilings

The profound mathematics of Arabic geometric patterns: the 5 canonical Girih tiles, Topkapi scroll analyses, quasicrystalline symmetries, and step-by-step compass-and-straightedge star rosette constructions.

Geometry
Girih Strapwork & Tilings
Canonical Tiles
5 Shapes: Decagon, Pentagon, Diamond, Bowtie, Hexagon
Symmetry Order
8-Fold, 10-Fold, 12-Fold Star Rosettes
Modern Discovery
500-Year Anticipation of Penrose Tilings
A geometer constructing decagonal star polygons with ruler and compass.
Sacred Geometry — the five canonical Girih tiles, self-similarity, and quasicrystalline symmetries.

1. The Mathematics of Girih Strapwork

Girih (Arabic: جِرِيح, Persian: گره, meaning "knot") refers to the complex interlacing geometric strapwork adorning Arabic architecture across Andalusia, North Africa, Persia, and the Levant. While appearing dauntingly intricate, the entire system is governed by rigorous mathematical principles.

The 2007 Breakthrough by Peter Lu and Paul Steinhardt
In 2007, Harvard physicists Peter J. Lu and Paul J. Steinhardt published a landmark paper in Science demonstrating that by 1200 CE, Arabic artisans had moved beyond simple compass-and-straightedge drafting to a modular set of five self-similar polygon tiles (the Girih tiles). These tiles allowed artisans to create infinite, non-periodic quasicrystalline patterns exhibiting 10-fold rotational symmetry—more than 500 years before mathematician Roger Penrose discovered Penrose tilings in the 1970s.
— Peter J. Lu & Paul J. Steinhardt, Decagonal and Quasi-crystalline Tilings in Medieval Arabic Architecture, Science 315, 2007

2. The Five Canonical Girih Tiles

Tile Shape Internal Angles Internal Line Decorations Structural Role in Tessellation
Regular Decagon Ten angles of 144° 10-pointed star rosette formed by intersecting strap lines Primary central hub; anchors 10-fold rotational symmetry nodes.
Regular Pentagon Five angles of 108° 5 internal strapwork segments radiating from midpoints Surrounds decagons to mediate rotational progression.
Diamond (Rhombus) Angles of 72° and 108° Crossed internal strapwork connecting opposite edges Fills intermediate seams and creates star points.
Bowtie (Non-Convex Hexagon) Four angles of 72° and two reflex angles of 216° Interlacing butterfly pattern connecting edge centers Locks decagons together at non-periodic boundary intervals.
Elongated Hexagon Four angles of 144° and two angles of 72° Continuous horizontal and diagonal ribbons Extends patterns linearly across architectural friezes.

3. Step-by-Step Compass Construction of an 8-Fold Star

1

The Circumscribed Circle & Perpendicular Axes

Draw a circle with radius $R$. Draw perpendicular horizontal and vertical diameters through center $O$ to establish the primary cardinal points $N, S, E, W$.

2

Bisect Quadrants for the Diagonal Axes

Bisect each 90° quadrant using compass arcs from $N, E, S, W$ to generate the 45° diagonal axes ($NE, NW, SE, SW$), dividing the circumference into 8 equal arcs of 45°.

3

Inscribe Two Overlapping Squares

Connect alternating division points to draw two identical squares rotated by 45° relative to each other (the Khatam / Eight-Pointed Star of Al-Andalus).

4

Extend Rays & Apply Strapwork Width

Parallel offset each line segment by $\pm w/2$ to create physical ribbons. Alternate "over" and "under" crossings at every intersection to render continuous interlaced depth.

All course illustrations are original works created for The House of Geometry.

Previous: 2. Geometry, Algebra & Optics Next: 4. Architecture & Muqarnas