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Art and culture · Visual arts and geometry

Symmetry and pattern: the geometry of Seljuk stars

From the dome of the Karatay Madrasa in Konya to the Kubadabad tiles, Anatolia's geometric patterns are built with compass and straightedge. Learn reflection, rotational and translational symmetry, and find out how many turns bring a star pattern back to itself.

  • 18 min
  • 2 simulations
  • Mini quiz

Updated:

Preview

The question

Anyone who looks up at the dome of the Karatay Madrasa (a medieval college) in Konya sees hundreds of stars, made of turquoise and black tiles, wheeling like the night sky. In this dome, built in 1251, not a single star is placed at random. How did the craftsmen set out such orderly patterns without computers? And when we call a pattern "orderly", what do we actually mean?

Try it yourself

No account needed: change the values and check your answer. Some simulations ask for a prediction before they start; it doesn't matter whether it's right or wrong.

Perspective and symmetry

The symmetry of an eight-pointed star

Choose a number and press Rotate: a transparent copy of the pattern turns by 360° divided by that number. If the copy lands exactly on the pattern, that rotation is a symmetry. Try different numbers, then choose your answer and press Check.

Prediction

Predict first

What is the smallest angle you can turn an eight-pointed star by so that it looks the same again?

It doesn’t matter whether you’re right; the simulation opens once you choose.

Perspective and symmetry

Do colours change the symmetry?

Choose a number and press Rotate; see whether the transparent copy, colours included, matches the pattern. Choose your answer and press Check.

Prediction

Predict first

If you colour the points of a twelve-pointed star in two colours in turn, what happens to its rotational symmetry?

It doesn’t matter whether you’re right; the simulation opens once you choose.

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Paste this code into the HTML of your blog or school site. Visitors try the simulation without an account; no progress is saved.

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Frequently asked

5 questions

Is the order of rotational symmetry always equal to the number of points?

Often, but not always. If colours or small details alternate, the order drops. If the points of an eight-pointed star are coloured gold and purple in turn, only 90° turns keep the colours in place, so the order falls from 8 to 4.

What is the difference between mirror symmetry and rotational symmetry?

With mirror symmetry, a pattern stays the same when reflected across a line; with rotational symmetry, it matches itself when turned about its centre. A propeller or pinwheel motif matches itself when turned but faces the other way in a mirror: it has rotational symmetry but no mirror symmetry.

Why does a shape with no rotational symmetry have order 1?

A full turn of 360° brings any shape back onto itself. If no other turn makes the shape match itself, only that full turn is left, so its order is 1.

How did craftsmen build such regular patterns without computers?

With a compass and a straightedge. The craftsman draws a circle, divides it into equal parts and joins the points by a set rule to get star polygons. Then the stars are placed on a grid and the gaps filled with other shapes; the pattern spreads across the wall by translational symmetry.

Does the "Seljuk star" belong only to the Seljuks?

No. In Türkiye the eight-pointed star is often called the "Seljuk star", but the motif is common across the Islamic world and in earlier art as well. The Seljuks were especially fond of it and used it a great deal.