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→Structure and symmetry
* The group <math>S^1</math>, which acts on the first coordinate by rotating the base circle.
* The group of order two which reflects the whole configuration about the <math>xy</math>-plane
All these can be extended uniquely to isometries of <math>\R^3</math>.
In fact, this is precisely the group of isometries of <math>\R^3</math> which fix the origin and the preserve the <math>z</math>-axis (though they may not preserve direction).
