INTUITION TELLS YOU THAT TURNING A BALL INSIDE OUT WITHOUT CUTTING IT OPEN IS IMPOSSIBLE. UNLESS YOUR'RE A MATHEMATIAN.
SPHERE INVERSION IS A FAVORITE PREOCCUPATION AMONG THOSE WHO INDULGE IN TOPOLOGY, A SCIENCE THAT DESCRIBES GEOMETRIC SHAPES AND WHAT HAPPENS WHEN YOU DEFORM THEM.
THIRTY YEARS AGO, TOPOLOGIST STEPHEN SMALE DEVELOPED AN ABSTRACT THEOREM THAT PROVED IT WOULD BE POSSIBLE TO TURN AN UNPOPPED SPHERE INSIDE OUT IF TWO POINTS ON THE SURFACE COULD TEMPORARILY OCCUPY THE SAME POINT IN SPACE.
UNLIKE THE SURFACE OF A BEACH BALL, THE SURFACE OF A MATHEMATICAL SPHERE CAN BE PUSHED THROUGH ITSELF. HOWEVER, THE RULES BEHIND SMALE'S PROOF PROHIBIT THE MATHEMATICIAN FROM ALLOWING A CREASE TO FORM AT ANY POINT DURING THE PROCESS.
SMALE'S THEORETICAL PROOF DIDN'T END THE FASCINATION WITH SPHERE INVERSION; TOPOLOGISTS' SIMPLY BECAME OBSESSED WITH ILLUSTRATING WHAT THE TRANSFORMATION WOULD LOOK LIKE.
MORE THAN JUST IDLE AMUSEMENT, AN UNDERSTANDING OF SHAPE DEFORMATION HAS APPLICATIONS IN MOLECULAR BIOLOGY, PARTICLE PHYSICS, AND EVEN COSMOLOGY.
ONE OF THE MOST ELEGANT VISUALIZATIONs TO DATE WAS DEVELOPED RECENTLY BY JOHN HUGHES AT BROWN UNIVERSITY. HUGHE'S SPHERE, BEFORE IT'S DEFORMED, RESEMBLES TWO GEODESIC DOMES ATTACHED AT THEIR RIMS-- A SORT OF SOCCER BALL. HUGHES WORTE ALGEBRAIC EXPRESSIONS TO DEFINE KEY STAGES IN THE INVERSION AND ESTIMATED THE VALUES FOR THE PHASES IN BETWEEN.
THESE EQUATIONS, DERIVED WITH THE HELP OF ONE COMPUTER PROGRAM, WERE THEN PLUGGED INTO ANOTHER PROGRAM TO PRODUCE AN IMPRESSIVE VIDEO OF A CLOSED SPHERE TURNING INSIDE OUT. HUGHES IS NOW DETERMINED TO SIMPLIFY THE ELABORATE EQUATIONS HE USED TO DESCRIBE THE PROCESS.
