Conference rusure::math

	I wonder if that's Bill Gosper, the mathematician I used to work
	with at MIT's Artificial Intelligence Lab during college summers.

	He as an amazing genius.  I think he went to Stanford after MIT.

	Here's one of the things that really impressed me about him:

	It was the days of John Conway's cellular game called Life that
	people liked to program on the computer.

	First Mr. Conway offered a monetary prize for anyone that could
	come up with a life pattern whose population grew without bound.

	Bill Gosper won this prize by coming up with the glider gun,
	a population-46 creature that every 30 generations spit out another
	glider.  Glider looks like this:

			* * *
			    *
			  *

	Gosper went on to create some populations he called puffer trains.
	These creatures move along, leaving debris behind.  Such creatures
	also may grow without bound, depending on how the debris behaves.

	Well, Conway offered another prize, this time for anyone that can
	prove or disprove that a life population can grow at the fastest
	possible rate, which is proportional to t**2.  This is maximum, since
	we're dealing with a 2-dimensional model, and only spots adjacent
	to cells can ever possibly be born.

	Well, I was at the lab watching the round DEC340 display that Gosper
	was programming in PDP6 assembly language, using a KSR35 terminal.

	What he came up with on the display was truly amazing.  It was
	a puffer train whose debris was glider guns !
	Viewed from a distance, it looked like a small fish with an
	ever-growing triangular tail.  The end of the tail was widest, because
	that was the glider gun that had been born longest ago and hence
	had spewed out the most gliders.

	Since the number of glider guns spewed out the back of the train
	was proportional to t, and since the contribution to the population
	BY EACH GUN is proportional to t also, the entire population grew
	proportional to t**2, and hence Gosper again won the prize!

	You might have to fool around with the game of life to appreciate
	the accomplishment his puffer train invention was.  (I'm sure
	life is discussed elsewhere in this notes file, perhaps someone
	can share the cross-reference ?)

/Eric

    Yes, it's the same Bill Gosper...

    I also remember him demoing his life programs and the displays you
    mentioned.  He also showed a truly monstrous puffer train that took
    over a second to refresh on the display, and had proved that it was
    actually a puffer train.

    He also had a configuration that had gliders tracing out the sides of
    a diamond shaped figure, as well as the glider-gun puffer train.

    There were a bunch of awsome people around that lab - Rick Schroeppel
    was also pretty astounding.

    Gosper holds the record for the longest non-regular continued fraction
    expansion for PI, using a transformation of a series due to Ramunjun.
    He did it on a Symbolics in LISP.  The series gives something like
    10 decimal digits per term [compare that to the Liebnitz series :-)]

    - Jim

	Just saying that each term of Gosper's non-regular continued fraction
	for pi produced 10 digits is a meaningless statement without saying
	how large the terms were.

	Here's a single term that give us 20 digits of e:

		71827182845904523536/100000000000000000000

	Not very impressive !

	Here's a single term that gives quite a few digits of pi:

		355/113

	More impressive.

	So, I'd be curious to know what those terms of Gosper's continued
	fraction looked like ?  How many digits in the terms ?

	By the way, Rich Shroeppel was also at the lab when I was.  He
	had long straight hair back then, almost down to his waist.

/Eric

    Re .-1

	The series is in HAKMEM, I believe; see also "Pi and the AGM"
    by Borwein and Borwein.  It is not too complicated.  Ramunjun gave
    quite a few such series in a paper co-authored with Hardy.

	10 digits is very impressive, and the crossover point where
    an AGM algorithm starts to win is extremely high.  Of course, if you
    can do better than Gosper's nonregular continued fraction approximation
    for PI with another method, have at it!

	- Jim

    FWIW, statistically each term of a continued fraction expansion for
    a uniformly chosen real number (on (0,1) say) is about 1 decimal
    digit per term of the continued fraction.  This is not really true for
    a transcendental number like PI, but assuming it's not too far off,
    we get a ballpark estimate of about 10 CF partials per term of
    Ramunjun's series.

    - Jim