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Sierpinski counters

By David, 24 February 2016 Activity

You will need lots of black and white counters.

You will need

• About 100 black counters
• About 100 white counters
• If you can’t find any counters, look for glass vase fillers in a discount store, or use the pieces out of a Go or Reversi set

What to do

When there are two black counters above, put a white counter.

1. In this activity, you will make a triangle out of counters. Start by putting a black counter on the ground. This will be the top of your triangle.
2. Make a row of two black counters underneath the first counter. This should make a small triangle.
3. For your third row, put a black counter, a white counter and finally a black counter.
4. For each row afterwards, follow these rules:

• Each row will have one more counter than the last row did
• Start each row with a black counter
  

  Keep following the rules and a pattern will appear.

• If there are two black counters directly above your counter, make it a white one
• If there is one black and one white counter above your counter, make it a black one
• If there are two white counters above your counter, make it a white one
• End each row with a black counter

If you keep following these rules, eventually a pattern will emerge!

What’s happening?

You’ll end up with a triangle made of triangles!

The shape that you make in this activity appears very complicated. There are many patterns in the colours, with repeating triangles and arrangements of dots. Although it has a lot of patterns in the shapes, it all comes from only a few simple rules.

As you made the triangle line by line, you might have noticed other patterns too. The seventh row alternates between black and white counters – in this case black, white, black, white, black, white, black. The next row is all black, and the row after that is almost all white, with just one black counter on each end. This pattern involving three rows repeats through the triangle too, first turning up in the third row, and appearing again in the fifteenth.

Real-life maths

There are many different ways of making complicated patterns using simple rules. A lot of them create striking patterns such as this one. Some of them make patterns that look like living things. For example, you can come up with a pretty good picture of a tree using some simple rules:

• Start with a stalk (a line coming out of the ground)
• The end of every stalk (line) grows three new stalks (lines)

If you start out with long stalks and add shorter and shorter stalks, these rules make a reasonable looking tree. However, the tree might get a bit messy after branching off several times, as it ends up growing a very large number of branches.

If all trees followed these rules, then they would all look very similar, but real trees, even of the same species, can look wildly different. When a tree is growing, some branches will get eaten, or will die because they don’t get sunlight, while others get a lot of sun and are more successful. Little changes like these can cause a huge variety in the shape of trees.

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