Introduction E9F5FC

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Symmetry

Symmetry is invariance under transformations. In other words, symmetry is when you take an action but the result is that nothing happens. Such actions may, however, be distinct. In that case, there must be some features which distinguish the actions but not the resulting state. The actions can then be composed. They are the elements of a symmetry group.

Choice templates

There are four choice templates:

• {$ \prod_{i}(\varnothing + \leftrightarrow_{i}) $} Simplexes.
• {$ \prod_{i}(\varnothing + (\leftarrow_{i} + \rightarrow_{i})) $} Cross-polytopes.
• {$ \prod_{i}((\leftarrow_{i} + \rightarrow_{i})+\leftrightarrow_{i}) $} Cubes.
• {$ \prod_{i}(\leftarrow_{i} + \rightarrow_{i}) $} Coordinate systems.

Simplexes

Each choice is between a labeled choice and an unlabeled choice.

This yields a set of labeled choices.

We can swap between two different labels.

Choosing between two unmarked opposites is choosing between two equals. Labeling them is treating them equally. Consider two unmarked opposites, such as the two imaginary roots of -1. Suppose we want to give them labels 0 and 1, or {$\leftarrow$} and {$\rightarrow$}. Then there is only one way to do that because it doesn't matter how we label them for they are distinct but indistinguishable. However, subsequently, since they are distinct, they have become distinguishable. But there is never a distinction between 0 and 1. Choosing all 0s is the same as choosing all 1s from the point of view of the unmarked opposites themselves. Choosing 01101 is thus the same as choosing 10010.

How does for a cube the whole keep track and break the symmetry?