Pull the slider to create the braid and list the permutations.

Thanks to Bill Mill's idea for this visualization (http://billmill.org/permvis.html).[br][br]The braid is formed by exchanging the position of element "P" with its neighbor at each step. When "P" reaches the end of a row, the two elements at the opposite end exchange position in the following step. Then "P" starts to move back across the row.[br][br]How can we prove that this method creates each one of the n! permutations of n different objects in exactly n steps? It seems to work in this particular case.[br][br]Can you find another switching method for producing each one of the n! permutations of n different objects in exactly n steps? One step might involve more than rearranging just two of the elements. What does the braid of your method (algorithm) look like in the case of 4 elements?