Algorithmic Art: How Geometric Star Patterns Are Generated

What makes art algorithmic?

A rule becomes an artistic tool when its behavior matters to the result. Choosing the rule is part of the composition; so is deciding which of its results deserves to become a finished piece. An algorithm can be deterministic, with the same inputs always producing the same image, or use controlled randomness. Neither requires an AI image generator.

Geometric patterns make the rules unusually visible. You can follow a single strap through the design, see where it crosses its neighbours and notice how a star sits in the middle of a larger tile. That is part of their appeal: the order is something you can investigate, not only admire.

How a star pattern is built

The construction most generators use goes back to E. H. Hankin, who described it in 1925 as polygons in contact; Craig Kaplan formalised it for computers in 2005. It has three steps:

1. Start with a tiling of polygons: squares and octagons, hexagons and triangles, or the larger polygons of a girih set.
2. Find the midpoint of every tile edge. Two straight lines leave each midpoint, angled into the tiles on either side.
3. Extend the lines until they meet the lines from neighbouring edges. Inside each polygon they close into a star or rosette; across the whole tiling they join into continuous straps.

Because every strap leaves an edge at its exact midpoint, neighbouring tiles always agree on where the line crosses their shared edge. The pattern joins up by construction rather than by careful drawing, which is what lets it grow across any area.

One number changes the pattern

The angle at which the lines leave each midpoint is called the contact angle. It is the most powerful setting in the whole construction. Keep the tiling the same, change only that angle, and the pattern changes completely.

At a shallow angle the straps stay close to the tile edges and the design reads as a lattice. As the angle steepens, the lines reach further into each polygon and the stars grow sharper and larger. Designers in the historical tradition made the same choice with a compass and straightedge; an algorithm makes it possible to explore the whole range before committing to one.

Patterns that repeat, and patterns that don't

Most star patterns repeat like wallpaper: one rectangle of the design, copied edge to edge, covers any area. Mathematics limits what such a repeat can do. A periodic pattern can only have two-, three-, four- or six-fold rotational symmetry overall; this is the crystallographic restriction. A periodic pattern can still contain ten- or twelve-point stars, but the pattern as a whole cannot turn by a tenth of a circle and land on itself.

Some patterns escape the restriction by never repeating. A Penrose tiling has ten-fold order and no translation that maps it onto itself, and decorating it with straps gives a star pattern that stays fresh however far it extends. In 2007, Peter Lu and Paul Steinhardt showed that girih designs on the Darb-i Imam shrine in Isfahan (1453) come remarkably close to this kind of quasicrystalline order.

Between the two sit periodic approximants: patterns that keep the look of ten- or twelve-fold order but do repeat, after a much larger rectangle. They are useful when a design has to tile, on fabric, wallpaper or a print-on-demand product, but should not look like an obvious grid.

The over-and-under weave

Drawn as single lines, a star pattern is flat. The tradition usually renders it as interlace: each strap passes alternately over and under the straps it crosses, as if the design were woven.

Alternation is a global puzzle. Choose over or under at one crossing and the choice propagates along every strand through the whole pattern; a single inconsistency shows up as two overs in a row. My engine solves the assignment across the entire pattern at once and counts any crossing it cannot satisfy. Across the thousands of crossings in the patterns on this page, that count is zero.

The colors follow the geometry too. Each space between the straps is grouped by its shape, so every twelve-point star shares one color, every small kite another, and the palette reads as a deliberate scheme rather than random fills.

Where AI fits in this work

I use coding assistants to help write and test the pattern engine. I choose the tilings, set the angles, judge the results and decide which designs become finished pieces. The images on this page come from explicit geometry and rendering, not from a text-to-image model, so every line is where the construction puts it.

A coding assistant can also make a plausible mistake, so the geometry is checked separately from the picture: closed strap networks, alternating weaves, and seamless tiles that genuinely repeat. A beautiful render is not proof that a pattern is correct.

Explore a rule for yourself

Mandala Studio generates printable mandalas from the same kind of rules: rings, sectors and repeated motifs. Change one setting and watch the design respond. For geometry beyond the plane, the guide to 4D shapes shows what happens when the rules move into four dimensions.