Semi-dimensional shapes

[Tglad]

I realised the other day that there is a quite systematic way to make fractals with exactly half dimensions. Here is a 1.5D curve in 3D space with no intersection:



you grid the space and divide each length by a quarter, then build your replacement shape from 4^(X.5) pieces (which is integer for integer X).

Here is a 2.5D surface in 3D with no intersection:




and its 2D equivalent is a 1.5D curve in 2D space:

http://tglad.blogspot.com.au/2016/11/semi-dimensional-shapes-and-other-curves.html

I tried to make a 2.5D curve in 3D space without intersection... but it is pretty tricky 

[ciric50]

Can you explain to me how that shape is exactly half-dimensional? It's not obvious to me.

[Tglad]

Hi ciric,
 for the 2.5D surface example, you start with a flat square (iteration 0) then grid it up into a 4x4x4 grid. You then want to replace the one big square with 32 of the smaller (1/4 scale on each axis) squares. so from:

to:

(the darker shade here show upside-down squares).
..and then you re-apply this replacement for each of the smaller squares, and so on, forever.

The resulting shape is exactly 2.5D because 4^2.5 = 32.
For the curves you are replacing 1 line with 8 lines of one quarter the size. This is exactly 1.5D because 4^1.5 = 8. I hope that makes sense.