Scale symmetry

Symmetry of scale remains an unfamiliar feature of geometry in general; whereas a hexagon or a grid are familiar concepts, a Sierpinski gasket is either unheard of or considered peculiar. 
But since scale symmetry completes the set of simple symmetries almost all symmetric objects must have some amount of scale symmetry. So on the contrary it is objects with scale symmetry of exactly zero (such as spheres, cones, circles etc) which are the peculiar cases. 

"Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." Benoit Mandelbrot

I created this Mandelbox fractal. Since it is a Mandelbrot set (not to be confused with the Mandelbrot set), it is an image of many Julia sets, so it has a large amount of variation, allowing exploration of many different fractal shapes. Some appear like gothic architecture, some like trees and some like metal structures similar to the Eiffel tower.  

It has featured in films Dr Strange (article) and Suicide Squad for the 'block like Mandelbulb used in the machine'.

Euclidean geometry has been well documented and categorised, such as the list of platonic solids, families of polyhedra etc. 
However, scale symmetric geometry doesn't seem to have this, only a list of very specific examples, such as Menger sponge, or by algorithm type such as IFS. 

So I created this categorisation which is based on the self-connectivity of objects, so bending or stretching does not change the category of the object. This recursive shape classification categorises objects in either a list or a table depending on the precision of categorisation that you want. 

The 3d table above is built by expanding the categorisation to moving objects. The scale symmetric version of a moving object is what I call a dynamic fractal. Small features appear to jitter quickly and large features only move slowly. The videos here show how many examples can look natural when keeping to this scale symmetric form.  

Since almost every symmetric object should exhibit some scale symmetry, we should expect to see it commonly in nature, and we do. It can be seen in clouds, hills, lightning, trees, the cratered moon, the asteroid belt, rivers, the list goes on and on... even in the paper texture under this text. So why do modelling packages just present you with boxes, cones and spheres as primitives?  

This Elementary plugin was written to allow people to generate scale symmetric objects easily by selecting from the table of categories and modifying other parameters like width. The picture is a particular example from the void-sponge category.

Another area where scale symmetry has been ignored is cellular automata (such as Conway's life and image filters). The scale symmetric equivalent of operating on a grid is to operate on a grid tree.

This Automata finder uses user selection to search among the huge space of possible 'fractal automata' to find interesting results.
Since the rules are scale symmetric, the results are often dynamic fractals, some appearing like fire, or growing bacteria or liquids.   
These Mobius maps have varying shape, a bit like the Mandelbrot set, but use Mobius transforms so could work in 3d or higher.

Shapes with adjustable fractal dimension can be as useful in measuring natural environments as the line, square and cube are in making Euclidean measurements.

Unlike Euclidean geometric shapes, scale-symmetric shapes can have their own colour, simply as a result of their shape. This is called structural colouration and is a result of the shape having geometric details at the scale of the wavelength of light. Here I look at the colours of the simplest such shapes; the Koch, Levy, dragon and random curves in 2D.