Are there any definitions of (geometric) complexity of shapes that do not rely upon quantity of information?

Question

I am interested in finding a measure for the complexity of geometric objects. I am focused primarily on closed figures composed of lines, curves and surfaces. Therefore, any measure of the complexity must relate to these components and their interrelations.

I intuit that, for example:

• The least complex object would be a point. A circle has the least complexity of the 2D shapes.
• Pentagons have more complexity than triangles. Triangles have more complexity than circles.
• It is likely that fractal curves (e.g. the Mandelbrot set), are the greatest-complexity 2D shapes.
• Cubes have greater complexity than squares.
• Otherwise comparing 3D shapes and 2D shapes is challenging, but should be possible.
• The great stellated dodecahedron has greater complexity than the regular dodecahedron (and of course the cube).
• Naturally, the shape of St. Paul's Cathedral in London has far more complexity than a simple polyhedron.
• It is likely that a torus has more complexity than a sphere.

I would like to identify a measure to quantify this topological / geometric complexity, which:

1. Produces a rank ordering of complexity for geometric objects which corresponds to intuition
2. Allows the comparison of the 'relative complexity' between objects, which also corresponds to intuition
3. Is rotationally and translationally invariant
4. Does not incorporate concepts of 'order' such as entropy or symmetry

A particularly intelligent approach I have found is described in the paper Shape complexity based on mutual information by Rigau et al. The authors propose a technique to use continuous mutual information to measure the complexity of 2D and 3D shapes, including compound polyhedra and Von Koch fractals. However, the approach conflate orderliness and complexity, breaching rule (4) above. I believe that the otherwise-promising measures of Topological entropy / Haussdorf Dimension suffer from the same issue.

My suspicion is that one can define a far-simpler, and more general, complexity measure using something like (vaguely) 'the density of discontinuities' or the 'average differentiability'. My question is, does such a definition exist already? Do similar attempts exist in the literature which I may have missed?