This fractal is known as the Koch curve

after the Swedish mathematician Helge von Koch

who was the first to come up with this construction shortly after 1900

I might sleep up and call it, pronounce it the American way, the Kotch curve but I'll try not to

OK, so our task is to come up with the self similarity dimension for this

so we'll use the regular formula

number of small copies is magnification factor to the D dimension

So first let's think about the number of small copies

So I'll look at the, I guess, the bottom figure

And I see 4 small copies, 1, 2, 3 and 4

So the number of small copies is 4

What's the magification factor?

Well the magnification factor is 3

I would need to stretch this 3 times

in order for this small piece

to be as long as that one.

And that's true for all these shapes.

This piece, I would need to pick it up, move it here and then stretch it 3 times to have it be the same length

You can see also 1, 2, 3

so the, uh, so then I would have to stretch it by 3

Alright so the stretch factor, the magnification factor is 3

that's raised to the D power.

So now, we use logarithms to solve for D.

Take the log of both sides

Use the exponent property of logs to bring D downstairs outside

Divide through by log3 and we get

So there's our answer

Sorry

The self similarty dimension D is log4 over log3

And we can get a number for this with a calculator

4 log divided by 3 log equals around 1.262

So D is approximately 1.262

So the self similarity dimension for the Koch curve is around 1.262