(This proof is essentially the same as the necklace-counting proof given above, simply viewed through a different lens: one may think of the interval [0, 1] as given by sequences of digits in base "a" (our distinction between 0 and 1 corresponding to the familiar distinction between representing integers as ending in ".0000..." and ".9999..."). "T""a""n" amounts to shifting such a sequence by "n" many digits. The fixed points of this will be sequences that are cyclic with period dividing "n". In particular, the fixed points of "T""a""p" can be thought of as the necklaces of length "p", with "T""a""n" corresponding to rotation of such necklaces by "n" spots.