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                                    INVERT1.ASC

                                 October 26, 1990
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                                   "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*


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