133 lines
3.6 KiB
Plaintext
133 lines
3.6 KiB
Plaintext
|
|
|
|
|
|
|
|
(word processor parameters LM=1, RM=70, TM=2, BM=2)
|
|
|
|
Taken from KeelyNet BBS (214) 324-3501
|
|
Sponsored by Vangard Sciences
|
|
PO BOX 1031
|
|
Mesquite, TX 75150
|
|
|
|
INVERT1.ASC
|
|
|
|
October 26, 1990
|
|
--------------------------------------------------------------------
|
|
|
|
"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*
|
|
|
|
|
|
Page 1
|
|
|
|
|
|
|
|
|
|
|
|
--------------------------------------------------------------------
|
|
|
|
If you have comments or other information relating to such
|
|
topics as this paper covers, please upload to KeelyNet or
|
|
send to the Vangard Sciences address as previously listed.
|
|
|
|
Thank you for your consideration, interest and support.
|
|
|
|
Jerry W. Decker.........Ron Barker...........Chuck Henderson
|
|
Vangard Sciences/KeelyNet
|
|
--------------------------------------------------------------------
|
|
If we can be of service, you may contact
|
|
Jerry at (214) 324-8741 or Ron at (214) 242-9346
|
|
--------------------------------------------------------------------
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Page 2
|
|
|
|
|