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Compare also sorted multiway systems on page 937 .)

Mathematical Constants The last few sections [ 2 , 3 , 4 ] have shown that one can set up all sorts of systems based on numbers in which great complexity can occur.

Growth of Plants and Animals

And what one sees is that even though the same very simple underlying model is used, there are all sorts of visually very different geometrical forms that can nevertheless be produced.

Space, Time and Relativity

The rule effectively just sorts elements so that black ones come first, and yields the same causal network regardless of what updating scheme is used.

The Phenomenon of Gravity

And indeed if space has appropriate curvature one can get all sorts of paths, as in the pictures below.

Note (e) for General Notes

And particularly if there are few colors arranged in ways that are not visually familiar it is typical for this effect to make all sorts of spurious patterns appear.

Note (d) for Implications for Mathematics and Its Foundations

Junctional calculus Expressions are equivalent when Union[Level[expr, {-1}]] is the same, and this canonical form can be obtained from the axiom system of page 803 by flattening using (a ∘ b) ∘ c  a ∘ (b ∘ c) , sorting using a ∘ b  b ∘ a , and removing repeats using (a ∘ a)  a .

Note (c) for Elementary Particles

But particularly in making contact with existing physics it is almost inevitable that all sorts of technical formalism will be needed—and to maintain balance in this book I have not included this here.

Note (b) for Implications for Mathematics and Its Foundations

The pictures below show what happens if one computes x m and y n for many x and y , sorts these values, then plots successive differences.

Note (a) for Recursive Sequences

Ulam sequences Slightly more complicated definitions in terms of numbers yield all sorts of sequences with very complicated forms.