congratulations, it will be an interesting read

Thanks mclarekin and lkmitch!

Here's another one from the first chapter:



You might recognise it, I called it a semi-dimensional shape because it is exactly two-and-a-half dimensional.
As 3D shapes approach 3D they fill more space, but they are in a sense less rough because they become more like a solid block with a few cracks in it.

So the best examples of fractals are (in my opinion) half way between their topological dimension and embedding dimension. The above one is a surface (topological dimension 2) in 3D (embedding dimension 3) so half way is 2.5D.

It is simple to make, you just replace every square plane

with 32 square planes of one quarter the width


The dark squares here indicate a backwards facing plane. These 'inwards' planes mean the resulting fractal is self-avoiding.