"INSIDE OUT"
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.
** karl west ** ---science news--
SCIENCE DIGEST, NOV 89*
