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This page is about the representation of natural numbers as diagrams. In particular the focus is on a specialized way of constructing the natural numbers in axiomatic set theory. Sounds sort of grandiose for what was originally just some doodling on course notes... I'd kept the scribbles in my pile of 'recmath' papers and thought it might be interesting to translate them into a web page.
Peeling away some of the layers of representation involved here:

The diagrams below represent the natural numbers from 0 to 6  that's probably enough to get the picture. The significance of the two colours white and blue is only to help in distinguishing rectangles within rectangles  each successive nesting has colours reversed.
The number 0 is represented by an empty rectangle. 1 is a rectangle containing a single empty rectangle, 0. 2 contains two rectangles, 0 and 1. The outer blue rectangle of each number's diagram represents the 'container' set for that number. Each outer blue rectangle contains a number of white rectangles  the elements or members of the set.
For the natural number 3, the blue outer container has three white rectangle 'members' which represent the natural numbers 0, 1 and 2. The set representation for the natural number 3 is { {} { {} } { {} { {} } } }.