Friday, 18 December 2009

Hex "Fractal Carpet"

Hex Fractal Carpet, Eric Baird 2009
It snowed here today! Wheee!

In honor of the White Hexagonal Fluffy Stuff, here's a nice fractal carpet made of hexagons that illustrates how an infinite number of copies of a shape can converge on a larger fixed-area version of the same shape.

This one's generated from about five and a half thousand hexagons, but obviously, you can keep going infinitely far.

The construction rule's simple. You start with one hexagon (with sides of length "one"), and then add half-size hexagons to any free corners. Then repeat, infinitely (with sides of length "one half", "one quarter", and so on).

What the process converges on is a larger completely-filled hexagon with sides of length "three", so the final area is exactly nine times the original.

Hex Fractal Carpet: Total areaIf you wanted to get even more recursive, you could try copying the entire hexagonal shape into every hexagon that you used to draw it (and then repeat that). Which would look rather cool. But take rather longer to calculate.

2 comments:

Heath Rezabek said...

Hello. This is a fascinating fractal, and the first I have seen quite like it. Is the iterative formula for this one of your own or drawn from elsewhere?

I ask also because I am putting together a symposium session, and would very much like to use an image of this; I'm wondering if it is yours, how I should credit it, and if I could get a larger iteration / rendering with the credit not overlapping the shape.

I've looked around a bit for a better way to contact you here and failed, so please feel free to get in touch if you think something could be arranged. My thanks,

- Heath

Heath Rezabek said...

Hello. This is a fascinating fractal, and the first I have seen quite like it. Is the iterative formula for this one of your own or drawn from elsewhere?

I ask also because I am putting together a symposium session, and would very much like to use an image of this; I'm wondering if it is yours, how I should credit it, and if I could get a larger iteration / rendering with the credit not overlapping the shape.

I've looked around a bit for a better way to contact you here and failed, so please feel free to get in touch if you think something could be arranged. My thanks,

- Heath