The maths of one line

A kolam is a ray of light

Every morning across Tamil Nadu, and all over the world this Diwali week, people press a grid of dots (pulli) into the ground with rice flour and loop a line around them. The hardest kind, the sikku or “knot” kolam, uses a single line that passes every dot and closes on itself. Mathematicians found a lovely way to think about it.

1 · Light between mirrors

The Mozambican mathematician Paulus Gerdes described these drawings as mirror curves. Imagine the line is a beam of light travelling at 45° between the dots. The edge of the grid is a mirror, so the beam bounces back in. Midway between any two neighbouring dots you may place a little two-sided mirror, or leave the gap open.

At each gap there are three choices:

Cross: no mirror. Two strands pass straight through and cross.
Turn: a mirror across the gap. Each strand bends back round its own dot.
Join: a mirror along the gap. The strands hug both dots together.

Tap the gaps on this 3 × 3 kolam. Different colours are different lines.

2 · Why 4 × 6 is never one line on its own

Start with no mirrors at all: every gap is a crossing, like a woven mat. How many separate lines do you get? The answer is the greatest common divisor of the width and height. A 3 × 4 grid is one line; a 4 × 6 grid falls into two; a 5 × 5 grid falls into five. Drag the sliders.

Dots across: 4

Dots down: 6

Teal cells are grids that are already one line with no mirrors, exactly when width and height share no common factor. The Chokwe storytellers of Angola used the same fact; they could remember a drawing as two numbers and a rule.

3 · The merge rule

Here is the secret that makes the puzzles solvable. Look at the two strands passing through any gap:

If they belong to two different lines, then any change at that gap fuses them into one. The number of lines goes down by one.
If they belong to the same line, one change cuts it in two and the other keeps it whole (just rerouted).

So you can always merge everything into a single line, one tap at a time, by tapping where two colours meet. The puzzles get hard when a symmetry brush makes one tap act in four or eight places at once: a merge on the left can be a split on the right.

Try it: this 4 × 4 grid starts as four lines. Join them into one by tapping only where two colours cross.

4 · Cousins across an ocean

Kolam has a relative in south-central Africa. Among the Chokwe people of Angola, storytellers smooth the sand, press in a grid of equidistant dots and draw a continuous line around them, called sona (singular lusona), to illustrate a fable. Gerdes showed many sona are mirror curves too, and UNESCO inscribed sona on its list of the Intangible Cultural Heritage of Humanity in 2023.

Kolam itself has drawn mathematicians and computer scientists for decades: Gift Siromoney and colleagues in Chennai studied kolam patterns with formal grammars, and when you remove the dots a sikku kolam is the shadow of a knot or link, which is why knot theorists like them.

5 · Draw a sikku kolam by hand

Press a square grid of dots, say 3 × 4 (no shared factor, so it works with no mirrors at all). Keep them evenly spaced.
Start at the middle of any edge. Move diagonally, slipping between two dots.
Every time you reach the edge of the grid, curve round the outer dot and head back in, like a beam hitting a mirror.
Keep going without lifting your hand. Cross straight over earlier strands.
When you arrive back at the start, every dot is enclosed and you have used one line. Now try 4 × 4, which needs at least one turn or join to stay a single line.

Traditionally the line is let out of a pinch of rice flour between thumb and finger. The flour is also food: ants and birds eat it during the day, which is why kolam is often described as a daily act of generosity.

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