From: Dickey, Michael F (michael_f_dickey@groton.pfizer.com)
Date: Mon Mar 11 2002 - 14:24:57 MST
-----Original Message-----
From: Ken Clements [mailto:Ken@Innovation-On-Demand.com]
Sent: Monday, March 11, 2002 2:08 PM
To: extropians@extropy.org
Subject: Re: Pi (was Carl Sagan's Contact)
"I thought I explained this rather carefully, but I guess I was not as
helpful as I would have liked. Perhaps I was not clear about the difference
between physical space-time and mathematical space. I will try again."
After re-reading what you had said, I think I understand what the
disagreement is, since we were both saying the same thing yet drawing
different conclusions...
"Dickey, Michael F" wrote:
>
> Ken - "Pi is a constant that does not change no matter where you are, how
> much you weigh, or how fast you are going. It says some things about flat
> Euclidean space, and to the extent your observations of the space around
do
> not match, pi helps you figure out the nature of the curvature"
>
"Be sure to read the rest of the post; I went into the spherical case."
I did read you original comments about a sphere, and I basically parallel
what you said. Here is your post...
"Here is an example: suppose you are at the south pole of the earth and you
have 100 m of steel wire. You can put one end of the wire at the pole and
stretch it out so as to walk a circular path around the pole at a constant
distance of 100 m. If you divide the distance you walked by 200 m, you will
get a number very close to pi. Next, get a boat and travel around
antarctica at a constant distance of about 3000 km. If you divide the
distance you go around by 6000 km, you will find the result is very much
lower than pi. If you keep going north to the equator and try this, the
ratio of circumference to diameter will be exactly 2, and if you keep going
to within a couple meters of the north pole, this ratio get very close to
zero. So, pi did not change, but the ratio of the circumference to diameter
of a path through the locus of points equidistant to a fixed point on a
spherical shell is dependent on the length chosen for the radius, although
the curvature of the space is constant"
You specifically state that if the curvature changes, the ratio of the
circumference to the diameter changes BUT PI does not. (???) Isnt PI the
ratio of the circumference of a circle to its diameter? (dont answer yet,
see below)
>
> Assuming that PI is the ratio of the diameter of a circle to its
> circumference, then altering the curvature of space time would alter that
> ratio, this is a fundamental aspect of general relativity is it not?
>
> Imagine a circle drawn on a piece of paper ... The curvature of the
> sphere the circle is drawn on is analogous to the curvature of space time,
> Thus PI changes with the curvature of space time.
"I covered the spherical case in my post. Yes, the ratio changes with
curvature, but pi does not change. Again, the ratio is simply no longer
equal to pi."
If PI *is* the ratio of the circumference of a circle to its diameter, and
that ratio changes with curvature, then wouldnt pi change? You said that
ratio may change so that it is no longer PI. I am a bit confused here, I
suppose this depends if you define PI as the ratio of the circumference of a
circle to its diameter or define PI as the number 3.14159 etc. etc (see next
part)
"You can refer to the ratio of the circumference to the diameter of all
circles in all kinds of mathematical spaces. Just because the ratio may
exist in some given mathematical space, that does not mean its value is pi.
Pi does equal this ratio when we are talking about a flat two dimensional
Euclidean space. If you start curving that space, the ratio changes so it
is no longer equal to pi."
Well here is the quip obviously, is PI's definition "the ratio of the
circumference of a circle to its diameter IN FLAT SPACE"
or is PI's definition "the ratio of the measured circumference of a circle
to its diameter" If the former, the you are correct, PI never changes, if
the latter, then I am correct, and PI changes. This seems to merely be a
disagreement in the definition of PI.
"scerir posted this link:
http://math.ucr.edu/home/baez/physics/Relativity/GR/pi.html be sure to take
a look. "
I did, I didnt understand it, but now I do. And from that page is "Pi is a
mathematical constant __usually__ defined as the ratio of the circumference
of a circle to its diameter in Euclidean geometry" So there you have it, pi
is constant depending on your definition.
> In the absence of a
> gravitational field, I believe there is a net curvature of space time (is
> space time curved or flat?) If that is the case, the the geometry of the
> space time curvature can be altered to change the ratio of a circle to its
> radius and thus hide a number in it (can it be arbitrary?)
>
"It's the other way around. In the absence of a gravitational field it is
believed there would be *no* curvature, which is not the case because there
is a gravitational field. This is physical space-time, not a constructed
mathematical space where there are circles."
This question relates to whether space-time is curved at all, as far as I
can tell this question is, as of yet, unanswered in cosmology. It may not
be curved, or it may be positively curved or negatively curved. It is
reasonable that one could take the average space-time curvature from the
distribution of matter in a universe and determine the ratio of the
circumference of a circly to its diameter. If that average space time
curvature is different, so is that ratio. If you define PI as the measured
ratio, and not the ratio in eucledian space, the it too changes. Would the
universe be curved in the absence of matter? If it is, than that curvature
could be altered to change the value of the ratio, if PI is defined as the
measured value of that ratio, then it too could change. If you wanted to
change the value of that ratio, you could increase or decrease the
curvature, or increase or decrease the density of matter, or increase or
decrease it's affects on the curvature of space-time.
>
> "Damien, the problem is that, in the book, Ellie does not go out and
measure
> some aspect of this universe. Instead, she uses computers to look into
the
> mathematical expansion of pi for a hidden message of Creation."
>
> If Elly was figuring this out on a computer, perhaps she modeled a
> mathematically perfect circle and compared its circumference with its
> diameter with the already known curvature of space time, thus making the
> calculation one that is representative of the geommetry of space time, and
> not merely an abstract mathematical construct. Altering such a curvature
> would probably not effect any organisms evolving in the space-time unless
> the curvature was severe.
>
"A circular path may exist in physical space-time, but a circle does not. A
circle *is* a mathematically perfect circle, and in pane geometry its
circumference to diameter ratio is pi. "
I suppose it all depends on your definition of a circle. If your definition
is an absolute mathematical one, and there is no curvature of the surface
the circle exist on, then you are right. If you definition of a circle is
something that is drawn on a piece of paper, then the ratio of its
circumference to its diameter would vary with the curvature of space time.
I think this discussion could delve into the metaphysical nature of
mathematics and discussions about objective reality, something I am not
capable of intelligently discussing.
"Your suggestion would have been better than what was in the book, as it
would have been based on a physical measurement, your "already known
curvature." But Carl did not write that. Had he picked that one, it still
would not have worked because the message appears "kilometers downstream of
the decimal point" which means the measurement would require at lest several
hundreds of thousands of orders of magnitude in resolution. The whole
Universe has less than 200 orders of magnitude from top to bottom. (By the
way, we know pi to over 200 billion places which, if you wrote down on paper
at 4 mm per digit, would run over 800,000 km. Long enough to go from the
earth to the moon and back.)"
Well, in any case thanks for helping to clear that up, it seems we both said
the ratio changes with the curvature of space time, but differed in our
interprations of the definition of pi, and thus had a different conclusion
to the question. Im my google searches I could not find a definition that
was absolute either way, but I guess definitions are arbitrary social
constructs anyway. Not being a formally trained mathematician, I couldnt
say either way what the *official* definition is.
Michael
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