Robert McNees
Swedish mathematician Niels Fabian Helge von Koch was born #OTD in 1870.
He established results in number theory, including one that relates the Riemann hypothesis to the asymptotic distribution of prime numbers. But he is probably best known for what we call the "Koch Curve" – one of the first examples of a fractal.
Robert McNees
A more familiar variant is obtained by joining together multiple Koch curves to produce a shape resembling a snowflake.
Robert McNees
The Koch curve is easy to construct.
Start with a line segment, then replace its middle third with the upper part of an equilateral triangle. You now have four connected line segments. Repeat the first step for each of those line segments.
Keep doing that to every straight line segment you see, ad infinitum. That’s the Koch curve.
Robert McNees
Each iteration replaces a line segment with 4 new segments, each with 1/3 the length of the original. After n iterations, the length of the curve is (4/3)ⁿ times the original length. Since 4/3 > 1, the curve becomes infinitely long as n → ∞.
But even though the curve becomes infinitely long, the area under the curve remains finite. You can see this right there on your screen! The curve always fits in that rectangle I used for the gif, so the area under the curve can't ever be bigger than that.
Robert McNees
If you start with a line segment of length L, then the first iteration adds one equilateral triangle with sides of length L/3. It has area (√3/4)(L/3)².
The nth iteration adds the area of another 4ⁿ⁻¹ equilateral triangles, each of which has sides of length L*(1/3)ⁿ. Adding the areas of all the triangles at that stage gives
(√3/4)L²(1/3)² + 4(√3/4)L²(1/3)⁴ + ... + 4ⁿ⁻¹(√3/4)L²(1/3)²ⁿ
As n → ∞ the infinite sum converges, and we find that the area under the Koch curve is
A = L² (√3 / 20)
Robert McNees
What a curious thing! This curve has infinite length, yet it fits in a finite region.
Koch's object packs more detail than would fit on something with finite length, so it seems like more than just a curve.
But it's not quite a surface, either. The points that make up the Koch curve don't cover an area like a surface would.
Robert McNees
One way mathematicians quantify this behavior is to assign a dimension greater than 1 (a curve) but less than 2 (a surface) to objects like the Koch curve.
This is referred to as the "fractal dimension" of the curve, a term coined by Benoit Mandelbrot.
For the Koch curve the fractal dimension is log 4 / log 3 = 1.262.
Old One Eye
@mcnees I appreciate learning from your posts. Thank you!
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