Symmetry and dance

If the dancers are allowed only 3 positions of the arms and two positions of the legs, there would be 35 different possible positions. Five of these are symmetrical so if the dancers wear masks on the backs of their heads, rotating through 180 degrees produces a mirror image of the same position. In other words, the dancers have not changed shape.

Each of the 15 asymmetrical positions produces a new one when rotated through 180 degrees. So the dancers would appear to have changed shape.
Moving from one position to any other requires a real change of shape (i.e. a move through at least one other dimension).

Multi-dimensional space - and dance

In Flatterland* the 'Spacehopper' explains that a bicycle has seven dimensions:


The Spahehopper nudged the bicycle handlebars through a small angle. "There - a fourth dimension!"
"But you just moved it in three dimensions."
"Yes, but it has changed shape - if it had stayed the same shape, you could make it fit its current position by rigid motion."
"So if it has moved through a new dimension, in which direction does that dimension point?"
"The turn-the-handlebars direction." said the Spacehopper "Different from the turn-the-front-wheel direction and the turn-the-back-wheel direction and the turn-the-pedals direction. That makes seven dimensions in all."
So every time we change shape, we enter a new dimension. Dance is a truly multi-dimensional art-form. 
* Flatterland (Like Flatland only more so) by Ian Stewart