Koch Curve

Koch CurveI drew another fractal today, the Koch Curve (click on the graphic for full size). The Menger Sponge took more time, and by rights I should be more satisfied with it, but the Koch curve is somehow nicer to me, it's simply elegant.

The curve is formed by starting with a line, and in the centre third, creating an equilateral triangle. This is repeated for every one of the four lines we now have, and so on.

The fractal dimension is around 1.26, this is because to make curve we have had to use four smaller copies, each is a third the linear dimension. I.e. To increase the size 3 times we need four copies.

3dimension=4, so dimension=ln(4)/ln(3).

Like the Menger Sponge I drew the koch curve freehand.

today's sketchesIt was just one of several sketches I drew today. Of the 'arty' ones, I'm quite pleased with the water.

Menger Sponge

Menger spongeThis is a close up of the sketch in my moleskine (click on the picture to enlarge) which shows a Menger sponge. A Menger sponge is a fractal shape, and so an accurate rendition is not possible. I've gone to 'level 4', with 'level 1' as a cube.

To make a Menger sponge, start with a cube, and make a square tunnel through each side. Each face is 8/9th the area it started with. This can be thought of as eight squares in a ring. In the centre of each of these squares, remove another square tunnel. Wash, rinse, repeat.

When taken to infinity, we end up with a very holey solid. It has a fractional dimension, it's a fractal.

It actually has a dimension of 2.72683. What's this mean? Well, imagine a line, double it in size. It gets twice as big. I.e. you need two original lines to make the new one. That's a change of 21. I.e. this has one dimension.

Take a square, double it in size, it's area increases four times. I.e. you need four original squares to make the new one. That's a change of 22. I.e. this has two dimensions.

Take a square, double it in size, it's volume increases eight times. I.e. you need eight original cubes to make the new one. That's a change of That's 23. I.e. this has three dimensions.

Now, to make a larger menger sponge, we need to increase it in linear size three times. That's not a problem, with cubes we'd need 33 cubes (27 cubes), the dimension is still 3.

With menger sponges we'd need 8 for the top and bottom layer, and 4 for the inner layer, so that's 20 smaller spongers to make one larger sponge.

This means that 20=3dimension, so ln(20)=dimension*ln(3)

In turn this means that the dimension of the sponge is ln(20)/ln(3) or approximately 2.72683. It's more solid that a flat surface, e.g. paper, but less solid than a solid, e.g. a cube.

For more information on this topic, I can highly recommend 'Flatterland' by Ian Stewart. The classic prequel is out of copyright and available online as well as a bound edition.

Geometric Shapes and Hatching

Geometric ShapesThese are a series of geometric shapes, the idea was to give practice with shading and hatching. At the top left are a series of hatching practices, followed by a sphere (not too hot), some cones (I like), a cube, a cube done with a putty eraser, a menger sponge, a möbius strip and some hatched cones.