Embedded Files
The basic symmetries of any shape are rotational, translational, reflective and scale symmetry. A general case shape will have an amount of all of these, here is a simple classification of these shapes based on how they self-connect.
In 3D there are seven classes, void is an absence of points (the empty set), its opposite is solid (the set of all points).
A cluster is a set of separated objects, with many tiny objects and few large ones, its opposite is foam a solid with a cluster of empty 'air pockets'.
A tree is defined to contain no loops and no concave areas, its inverse is a shell.
A sponge is a network of various thicknesses, it is its own inverse.
These class definitions are therefore non-rigid, more topological than geometric, so natural and artificial objects can be used as examples interchangeably.
A more specific classification is available by relating this list to itself as a matrix, here it is shown for 2D shapes:
In two dimensions there are only 5 classes in the list: void, cluster, tree, sponge and solid. In the table they exist as the diagonal elements (red outline).
The class is read off as column-row, such as cluster-solid.
The lower diagonal triangle represents critical cases, where the class a-b is b if borders are counted as connections and a if they aren't.
The upper-diagonal triangle represents non-critical cases where the class a-b generally means b composed of a.
The void column contains no area of solid, so each class here is a fractal. Conversely the solid row contains no area of gaps, so is solid minus a fractal set of points.
Whereas the lower off-diagonal elements represent an 'in phase' change from one diagonal class to another, the upper off-diagonal elements represent an 'out of phase' change. For instance, a tree where the branches all pinch off at the same time is a cluster-tree at that point, but if the branches pinch off at different times, then you get a Tree-cluster.
The sponge-cluster is a cluster of sponges and, being along the diagonal is the same class when inverted (when solid and air are swapped).
The most difficult to represent are the top row and the right column. These can be represented as the transition point between one structure and another. For example, the cluster void contains a continuous region of empty space, but also a scale symmetric cluster, this is achieved by being the edge of a cluster structure. Equivalently, for the right column, a solid-sponge contains a continuous solid in one direction, but the edge of sponge structure in the other direction. At the intersection, the solid-void is a solid half-plane (or more generally a solid infinite pyramid), being the complete separation of solid and void, it is the polar opposite of the void-solid, which is a homogeneous mixture of the two.
The above diagram is a bit artificial looking so here are some drawn examples, which show more natural versions of the structures:
Here is the equivalent table for 3D shapes, with transparency so you can see what's going on inside:
The shapes needn't be right-angled, this live shader code shows examples that use sphere inversion, so are curved: https://www.shadertoy.com/view/cslfWn
Here are some examples, excluding a couple of the void/solid rows and columns. The foggy ones are viewing the shape surface from inside the solid, as though it were semi-transparent. Click to see paper.
Another, more complete table. The foams and solids are mainly shown as cut through with the solid part see through.
The classes are braod enough that we can give natural examples as well. Here I found some examples for the lower diagonal triangle of classes, the standard versions of each class can be seen down the diagonal:
This classification idea is now explored in detail in Chapter 8 of my book Exploring Scale Symmetry published by World Scientific.
Page updated
Google Sites
Report abuse