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Dynamic fractals

Dynamic fractals are objects that are scale-symmetric in time as well as space, so the object is composed of smaller versions of itself which also run at a smaller time scale. They are seen frequently in nature.

3d twisting tree 

This is a dynamic tree version of a single oscillating line. 
Notice that the natural look, and lack of artificial looking symmetry in the tree is a result of the subtrees operating on different time scales, so getting out of alignment, no random offsets have been used anywhere.

So a very symmetric dynamic fractal can appear completely asymmetric and natural when seen at any snapshot in time. 
Universe fractal 

This dynamic fractal looks a lot like a solar system. The smaller objects spin at proportionally smaller time scales.   
Boiling mud dynamic fractal  

Boiling liquid is also a dynamic fractal, since the small bubbles come and go proportionally more frequently than the large bubbles. The entire animation can be composed of several copies of itself shrunk in size and in time scale, and placed nearer the surface.  
Fractal brownian motion  

A dynamic fractal version of Brownian motion. The large objects change direction less frequently than the smaller lighter objects.   
growing circles fractal  

Although this example seems unnatural, it it similar to the life cycle of may things, take for example mushrooms.
Many only grow to a small size before dying, but a few manage to survive and grow large. The sizes and frequencies follow a power law.  
2-form tree sponge  

A swaying tree like the one above is a dynamic fractal, but by category it is still just a tree, since its self connectivity doesn't change. This abstract example demonstrates a swaying tree with hitting branches, which gives the tree short periods of time in a sponge fractal form, hence it is categorised as a two-form tree-sponge.  
Sierpinski dynamic fractal  

Another abstract example shows how a Sierpinski triangle can rotate into something like Cantor dust and back, so it also has two forms, rather than just being a moving, vibrating Sierpinski triangle (which is a sponge form).