  Trigonometric and trigonometric-algebraic entropies are introduced.
Regularity increases the entropy and the maximal entropy is shown to result
when a regular $n$-gon is inscribed in a circle. A regular $n$-gon
circumscribing a circle gives the largest entropy reduction, or the smallest
change in entropy from the state of maximum entropy which occurs in the
asymptotic infinite $n$ limit. EOM are shown to correspond to minimum perimeter
and maximum area in the theory of convex bodies, and can be used in the
prediction of new inequalities for convex sets. These expressions are shown to
be related to the phase functions obtained from the WKB approximation for
Bessel and Hermite functions.
