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