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Sometimes the group-theoretic concept of conjugation is a useful one in defining calls. In particular, "conjugate X by Y" means "do Y, then do X, then do the inverse of Y". There are a number of ways to define "inverse":

Different ones are useful in different contexts, but "rewind mirror" is probably the most broadly natural (and it's the flavor I'll assume by default, unless otherwise specified).


ping pong

The motivating example for the ping pong concept is the "definition" of plusping pong circulate as "backnose-conjugate circulate by extend (but smoothly)". That is, extend, do an all 8 circulate, then "un-extend". This is kind of like thinking of a distorted 2x4 in which there are "ping pong lines" (O and X) and "ping pong boxes" (red and green):

They're very different, though. For example, "ping pong trade the wave" is just "centers trade the wave, ends trade" with the distorted formation definition. But with the conjugation definition, it ends in the same spots as "ping pong circulate, new centers trade the wave". (To be correct, the original ends should actually trade after passing the centers but before making their wave.)

You could argue that the distorted formation definition is actually better, because it's easier and does nicer things. For example, from a quarter tag, it allows defining C1jaywalk as "ping pong slide" and A1double cross as "ping pong slither", whereas the conjugation definition makes the first one "do nothing" and the second one illegal.

Anyway, the coolest thing about the conjugation definition is that from right hand waves, "ping pong little" is the same as "rewind mirror little" (i.e. the last part of C1plenty). There's no good reason, it just is.