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#Post#: 97--------------------------------------------------
The game of Bagatelle
By: johnfree Date: May 12, 2012, 12:30 pm
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Here is a CHALLENGE for all of you!
I love the game of Bagatelle for here the ball bounces crazily
around inside a ring of posts. Its path is utterly PRE-DESTINED
yet CANNOT be predicted!
Suddenly it will escape, totally unexpectedly and nobody can
foretell when or where.
In this game we have say 39 perfectly elastic immovable posts
hammered vertically into a smooth horizontal plane
They are equally spaced on a circle each 1" apart from 2
closest-neighbours
Each post is 0.1" diameter
So between the gaps between posts a 0.8* diameter ball COULD
pass.
But our ball is frictionless, perfectly elastic and 0.75"
diameter.
We start it rolling within the circle of posts in no
specially-chosen way.
Let V(n) be the name of the portion of the path that starts at
the nth collision and ends at the next collision.
I maintain there is no formula for V(n)
The path is soon chaotic and the place and time of escape is not
computable.
The only way to find out is to actually compute the path impact
after impact in turn, from the beginning.
And no computer is accurate enough for that!
For the "bounce off" new direction depends CRITICALLY on where
on the post the ball strikes. And that depends SUPER-critically
on where the PREVIOUS path began and its direction.
It is a nightmare of chaos and instability. Like a house of
cards built with each card more wobbly than the previous card.
The bouncing from convex and concave mirrors is especially
interesting in the deign of LASERS, image-producing optics and
paths (numbers) that are as near as is possible to "random"
In fact the only situations that any computer can predict are
limited to those where the situation is STABLE. Otherwise the
smallest errors get magnified and the results soon meaningless.
(Soon means after a dozen or few dozen calculations).
But if the situation is STABLE the errors can get corrected -
like a ball rattling down a U-section slide or down a valley.
#Post#: 101--------------------------------------------------
Re: The game of Bagatelle
By: axlyon Date: May 15, 2012, 12:57 pm
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wow. that's a complicated idea. you might want to make it have
several patterns it loads one from the patterns at random.
#Post#: 102--------------------------------------------------
Re: The game of Bagatelle
By: johnfree Date: May 18, 2012, 5:03 am
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The people hundreds of years ago (thousands?) were VERY smart
Those who never played bagatelle missed a lot of FUN - only
slightly revisited in recent games like Pinball.
The whole thing is fascinating.
For a ball will rattle safely down a sloping U-section valley
(child's slide), while a togoggan on the Creata Run will also
survive PROVIDED the errors do not take it higher than the
walls!
So the two things are utterly different:-
!. How safely CONSTRAINED you are for SMALL errors (e.g of
calculation)
2. How BIG an error and you fly off over the side!
In the game of bagatelle there is always another pin to bounce
off if you miss one!
But in other cases (Pascal's Triangle) there may not be.
Even in Bagatelle you CAN (and do rarely) escape (the same way
you got in: by chance!)
It would be FUN to know how LONG you are likely to remain inside
the ring. This depends how SMALL is the gap between pins through
which the ball, diameter D, must pass.
I can write a programme that calculates this.
BUT
The errors (of computation) grow - each one an increase of the
earlier error - until the result is meaningless! Indeed,
eventually you get out but NOT EVEN between the correct two
pins!
The "whole new interesting thing" is this:-
WHY are some problems not calculable due to error growth
HOW to tell if your problem DOES forgive errors and how BIG an
error will be forgiven?
The best idea is to think of the U-section slide and find the
answers by experimental trials.
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