What makes a pattern a pattern?

I've been intrigued by the use of analogies, metaphors and schemas in our everyday thinking, in mathematics, in physics, and in engineering. Granted that we resort to all the above all the time in order to build our thoughts. Many philosophers of the modern age already proposed that all thinking, and consequently nature itself, were reducible to a set of few simple notions that upon some algebraic combination could form all of the available knowledge and nature laws. I already wrote about Leibniz and Descartes proposing such views. Today we may feel that those views were unwarranted extrapolations of some cases-where a given field rests upon few notions-into the generality.

However, there cases where the tenet holds! Now, circumscribing ourselves to those cases, I had this question: What makes a pattern a pattern? By pattern I mean that conceptual construct that will be resorted to frequently by us in order to explain, and/or build more elaborate pieces of reasoning.

Geometry is a fertile field for understanding this. A pattern might have these three characteristics:
  1. Regularity: sequences that repeat at constant/regular/predictable intervals of time, space, structure, etc.
  2. Simplicity: must be simple, short. Occam's razor applies here.
  3. Symmetry: the structural coherence upon some axis.
Any figure that doesn't have the above three cannot be a pattern. The most common geometrical figures-line, circle, triangle, square- are patterns. They have layers of pattern-ness: the first is the most obvious, the visual. There's another which is the association of straight lines, and even others as the relationship among figures.

There are more patterns, but that themselves are built upon simpler and more elementary patterns. They are related through laws of association-such as combination, harmonization, equation, parallelization-e.g. the platonic solids.

The Platonic solids:  tetrahedron, cube, octahedron, dodecahedron, icosahedron.

Figure Tetrahedron Cube Octahedron Dodecahedron Icosahedron
Based on: Triangle (3) Square (4) 3 Pentagon (5) 3
Coinciding lines (Assn. rule) 3 3 4 3 5
Faces 4 6 8 12 20

The number of faces is one of the most characteristics. But the other features contribute to the pattern. 

Thus the patterns are the fundamental cognitive categories that the mind resorts to to interpret all type of phenomena. Hence cognition comes to the use of analogies and metaphors to "make sense" of the observed. Therefore, "structuring" is the act of associating proto-patterns, or patterns among themselves to fit an observation.

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