Can a Shape Have an Edge That Never Ends?

Step by step

Each step does the same thing to every piece of the edge: one straight piece becomes four pieces, each a third as long. Our script worked out the first steps for the 9 cm triangle.

Step	Sides	Each side	Edge
0	3	9 cm	27 cm
1	12	3 cm	36 cm
2	48	1 cm	48 cm
3	192	3.3 mm	64 cm
4	768	1.1 mm	85.3 cm
5	3,072	0.37 mm	113.8 cm
6	12,288	0.12 mm	151.7 cm

Look at the edge column. From 27 to 36 is ×4/3: four pieces, each a third as long, are 4/3 as long as the one piece they replace. From 36 to 48 is ×4/3 again, and so on all the way down.

How far can it go?

Growing by a third each time is like the rice on the chessboard doubling in topic 1, only more slowly: slow at first, then unstoppable. Our script found the first step at which the edge of the 9 cm snowflake passes each length:

The edge passes	at step
1 m	5
10 m	13
1 km	29
100 km	45

At step 29 the snowflake still fits on the same page, but its edge is over a kilometer long, in 8.6 × 10¹⁷ pieces each about a ten-trillionth of a centimeter. No pencil could draw them, but the rule never runs out of steps.

And the area?

Now look at what each step adds. At step 1, three triangles are added, each with sides 3 cm, a third of the big triangle’s. A triangle with sides a third as long is a ninth of the area (nine of them tile the big one, three by three), so step 1 adds $3 \times \tfrac{1}{9} = \tfrac{1}{3}$ of the triangle.

At step 2, every one of the 12 sides gets a new triangle, each a ninth of the area of a step-1 triangle. That is 4 times as many triangles, each 1/9 as big: step 2 adds 4/9 of what step 1 added. And step 3 adds 4/9 of what step 2 added, and so on: 4 times as many triangles every time, each 1/9 as big.

Step	Triangles added	Area added (× the big triangle)	Area so far
0			1
1	3	1/3	1.333
2	12	4/27	1.481
3	48	16/243	1.547
4	192	64/2,187	1.577
5	768	256/19,683	1.590
6	3,072	1,024/177,147	1.595

The edge column ran away; this one is slowing down, and it looks as if it is heading somewhere a little above 1.5. Where, exactly? And how can we be sure it never runs away too?

An edge that grows for ever

Every step multiplies the edge by 4/3, and you can always take another step. Multiply any length by 4/3 enough times and it passes any length you name, just as the table showed: a thousand times as long by step 25, a million times by step 49. So if we imagine all the steps done, the edge isn’t just very long. Its length has no end: no number of centimeters, however big, is enough to measure it. Mathematicians say its length is infinite.

The shape with all the steps done, every one of them, is called the Koch snowflake, after the Swedish mathematician Helge von Koch, who described its curve in 1904 (see the story below). Its edge is the Koch curve.

An area that can’t run away

Press Show the hexagon in the widget. The hexagon has its six corners at the triangle’s three corners and the three outer tips of the star of step 1. Move the slider through every step: the snowflake never leaves the hexagon.

Here is why. Look at one side of the triangle. Everything ever built on that side stays inside a flat “tent” on it: a triangle whose base is the side and whose top is the tip of step 1’s triangle. At the next step, the side’s four pieces each get their own small tent, a third the size, and the four small tents fit inside the big tent (two of them just touch its sloping sides). The same is true of their pieces, and their pieces’ pieces, at every step. So nothing built on a side ever pokes out of its tent, and the triangle with its three tents is exactly the hexagon.

How big is the hexagon? Its three tents together are as big as the triangle. Join the triangle’s center to its corners, cutting it into three equal pieces: each tent is the mirror image of one piece across their shared side (step 1’s tip is as far outside the side as the center is inside), so it is a third of the triangle. So the hexagon is twice the triangle, and the snowflake’s area, at every step, is less than twice the triangle’s, however far the steps go.

Exactly 8/5

Now the exact answer. Call the area that will ever be added, at all the steps together, the future. It is what step 1 adds, plus everything after step 1. The hexagon showed that the future is a definite area, less than one more triangle, so we can name it and work with it like any number.

And everything after step 1 is the whole future again, only shrunk to 4/9 of it, because each step adds 4/9 of what the step before it added. Say the future is some area F, measured in triangles:

$$F = \tfrac{1}{3} + \tfrac{4}{9} F.$$

That is an equation you can solve (Grade 8’s linear equations). Take $\tfrac{4}{9} F$ from both sides: $\tfrac{5}{9} F = \tfrac{1}{3}$, so $F = \tfrac{1}{3} \times \tfrac{9}{5} = \tfrac{3}{5}$.

The whole snowflake is the triangle plus its future: $1 + \tfrac{3}{5} = \tfrac{8}{5}$ of the triangle. For the 9 cm triangle, whose area is about 35.1 cm², that is about 56.1 cm².

So both answers to the teacher’s question are true at once. The edge gets longer by folding into smaller and smaller wiggles, and wiggles take length but almost no room.

Look closer

What does the finished edge look like close up? Zoom in on one side of the snowflake, with every step done.

At ×3 you see exactly what you saw at ×1: the first third of the side, 3 cm long, is the whole side in miniature. At ×9 the same again, 1 cm across. However far you zoom in, the edge never flattens into a straight line. That is why its length has no end: the usual way to measure a curved line is to zoom in until it looks straight and add up the straight bits, and here that never happens.

A shape like this, which shows new detail however closely you look, and in which a small part looks like the whole, is called a fractal. When the small part is an exact copy of the whole, shrunk, the shape is self-similar. The Koch curve is both.

Here is another way to see that the Koch curve is a strange kind of line. Shrink a shape to a third of its size and count how many copies of the small one make the whole.

A line gives 3 copies; a square, 3 × 3 = 9; a cube would give 3 × 3 × 3 = 27. A line is one-dimensional, a square two-dimensional, a cube three, and the number of copies is 3 to that power. The Koch curve gives 4 copies: more than a line, fewer than a square. In this sense it is somewhere between a line and an area: too wiggly to be an ordinary line, too thin to cover any room.

The Koch snowflake keeps adding. Here is a fractal that keeps taking away. Start with a filled triangle. Join the midpoints of its sides and cut out the upside-down triangle in the middle: three triangles are left, each half as wide. Do the same to each of them, and again.

The middle triangle is a quarter of the triangle it is cut from (the four triangles you get by joining the midpoints are all the same), so each step keeps 3/4 of the area. Our script followed it:

Step	Triangles left	Area left	Holes
0	1	100%	0
1	3	75%	1
2	9	56.3%	4
3	27	42.2%	13
4	81	31.6%	40
5	243	23.7%	121
6	729	17.8%	364

After 17 steps less than 1% of the area is left, and taking 3/4 again and again, the area left goes to nothing. But the edges go the other way. Each step turns one triangle into three, each with half its edge, so the edges of all the triangles together grow 3/2 times at each step: for a 9 cm triangle, 27 cm, then 40.5 cm, 60.75 cm, past 1 m at step 4 and past 1 km at step 21. The finished shape, called the Sierpiński triangle after the Polish mathematician Wacław Sierpiński, who described it in 1915, has an area of zero, and its edges together are longer than any length you can name. It is the snowflake turned inside out.

Out of chaos

Here is a game that seems to have nothing to do with it. Mark the three corners of a triangle. Start with a dot anywhere. Choose a corner at random, and put the next dot halfway between your dot and that corner. Choose again, jump halfway again, and again, thousands of times.

You might expect a random smudge. Instead the dots draw the Sierpiński triangle. Why?

Take any dot inside the triangle and jump halfway towards the top corner. Wherever it started, it lands in the top half-size triangle, the one touching that corner: halfway to the corner shrinks the whole big triangle to half its size, towards the corner. The same for the other two corners. So after one jump the dot is in one of the three half-size corner triangles, never in the middle hole. After two jumps, it is in one of the nine quarter-size ones, never in a hole of step 2. After many jumps it is as close as you like to the Sierpiński triangle, wherever it began. (The very first dots, from the starting point, can be in the holes; the widget counts the dots in the big hole only after the first 12.)

And now the twist. With a square, halfway to a corner lands in that corner’s quarter of the square. But the four quarters cover the whole square: there is no gap between them for a hole. So nothing is ever left out, and the dots fill the square. The rule leaves holes, and makes a fractal, only when the shrunken copies leave gaps between them.

Real things are never true fractals: their copies inside copies stop after a few levels, when the pieces reach the size of cells or crystals. But many come close, because they grow by repeating one rule at smaller and smaller sizes.

Ferns are leaves made of leaflets shaped like small ferns. Rivers split into streams that split into smaller streams. At Science Playground, the Fractal-Like Tree Root shows large roots dividing into smaller roots, which divide again into finer ones.

People have built fractal-like shapes too, long before the word existed.

• How Long Is Nepal’s Border? (Grades 9–12) Real borders and coastlines wiggle at every scale too: why their length depends on your ruler, and fractal dimension worked out.
• Can You Add Halves Forever and Still Get One? Endless sums that add up to something, like the snowflake’s area.
• Why Do Sunflowers Count in Fibonacci Numbers? The spirals on a sunflower, a pine cone and a pineapple.
• How Much Rice Is on the Chessboard? Growth by multiplying, as the snowflake’s edge grows.
• How Many Ways Can You Walk to School? Pascal’s triangle, and the Sierpiński triangle hidden in its odd numbers.
• Where Did the Missing Square Go? The staircase that looks like the diagonal and isn’t as long.
• Grade 11 Computing, Fractals: pictures made of smaller copies: draw fractals with a program that calls itself.