April 15, 2014

Many a child has wondered if the face in the mirror might have a different mind lurking behind her eyes.  The symmetry of a mirror is different than the symmetry of a frog.  It is a total symmetry, a surface where two isometric worlds are glued along a surface.  Why give the mirror world its own identity?  For sake of amusement.  Why take away its identity?  For sake of simplicity, Occam’s razor.  The same cannot be done with the right and left sides of the frog.


(concept credit to Daniel Johnston)

because one side is closer to that leaf, the other is half submersed in water.  Context of a point of view from which the two sides enjoy their own identity necessitates the separate identities.  We cannot form an orbifrog simply by quotienting the frog by \mathbb Z_2.  At least the quotient doesn’t extend, to the frog’s innards or to the world around.

The total symmetry of the mirror means that the actors in that world have duplicates in our world.  The totality allows us to quotient the conceivable double world, and form the orbiverse in which we live.

When we quotient by all symmetries we end up with an essential, and asymmetric space, (or perhaps a point, what symmetries does it enjoy?).  Yet there is the mirror, still, the evidence of a universe with symmetry.

My thesis, today:  a fundamental group is a symmetry much like the symmetry of a mirror.  It is a total symmetry which allows for a redundant, anecdotal universe:



Here, Castor, faced with the humdrum of a thoroughly repetitive life, dreams one day that he comes home not to the home that he left that morning, but to a home with differences ever too subtle to discern (indeed, so subtle they never effect divergent large-scale events).  Work the next day has suspicious but subtle differences to work yesterday, and at the end of the second day he is back home to tell his original (or even) family about the world he’s sure he inhabits.  Here is Castor’s world:

ImageAll the while a twin of Castor is both leading and following in his footsteps.

To recap: a fundamental group is a symmetry of a space as a mirror is a symmetry of a space.