I have a truly marvelous mathematical proof that these variations of pi are concealed in the so-called "hidden dimensions" of an expanded Klein-Kaluza space.
Unfortunately at the moment I have neither the time nor is this post sufficiently capacious to allow delineation of the significant factors involved in the development of this astonishing mathematical result. However I do resolve to do so at a later time (that is, once April 1 passes).
Published for April Fools day 2009