On Aug 30, 2:18 am, wrote:
I apologize in advance for my ignorance.

Whereas pi is defined as the ratio of the circumference of a circle to
it's diameter, it is constant and irrational on a euclidiean plane.
It seems logical to me to suppose that the value of pi would be
different on curved surfaces (like a sphere or hyperbola). This led
me to suppose that there were curvatures where the value of pi is
rational. From there I wondered if there were some characteristics of
those surfaces wherein pi ir irrational that were shared in common.

I trivially came up with the example that the value of pi on a surface
in which all points are adjecent is 1, but my knowledge of math is
nowhere near enough to do anything more.

I'd be interested in your thoughts.

Thanks,
Ben

The ancient Greeks discovered that the ratio of a circles
circumference to its radius is of fundamental importance. They called
this _ratio_ "pi". The "number pi" is the Pi in Euclidean geometry.

There are infinite geometries which have Pi = 1. The easiest ones to
imagine is that for a (suitably chosen) cone.

Visualize it like this
[1] Draw a circle on a rubbery sheet.
[2] Poke the center of the circle until a cone-like structure is
formed.
[3] Keep poking until the diameter of the circle equals the
circumference.

Since it is possible to have infinite deformations in the underlying
manifold which don't affect diameter, then there are infinite possible
manifolds giving Pi=1. Those geometries over those manifolds would all
need to be hyperbolic, so I suppose that is the broadest answer to
your question (without getting into the math). If you go even more
abstruct there's probably semi-riemannian manifolds which could do the
same, but doing that you the geometric meaning from the "ancient"
point of view.

NOTE:

In you keep poking the center of the rubbery sheet it means the
circumference will be SMALLER than the diameter.

This means if you shot a light-beam through the diameter you ran
around the (smaller) circumference you could conceivably detect your
own light-beam - giving you a time-machine.

On Aug 30, 5:05 pm, wrote:

[...]

So...we should treat the pi in G_uv = 8pi*T_uv as a variable that is a
function of the geometry?

On Aug 31, 2:23 pm, Sam Wormley wrote:
wrote:

The ancient Greeks discovered that the ratio of a circles
circumference to its radius is of fundamental importance.

Not the Greeks.

The ratio itself probably dates back to Zulu shaman (as does spacetime
by the way). I think the Greeks recognized its fundamental importance
more than anyone.

This means if you shot a light-beam through the diameter you ran
around the (smaller) circumference you could conceivably detect your
own light-beam - giving you a time-machine.

No time machine, no more than if you were to shine a light on a mirror
and observe the reflected ray.

On Aug 31, 2:41 pm, Don Stockbauer wrote:
This means if you shot a light-beam through the diameter you ran
around the (smaller) circumference you could conceivably detect your
own light-beam - giving you a time-machine.

No time machine, no more than if you were to shine a light on a mirror
and observe the reflected ray.

Observing the rays of WW2 is a time-machine. Observing the reflected
rays of WW2 (via stored video for example) is not a time-machine.

wrote:

Observing the rays of WW2 is a time-machine. Observing the reflected
rays of WW2 (via stored video for example) is not a time-machine.

I was about to argue with you, but I think you are right! I was
misled by the popular idea of a time-machine where one goes back and
changes the past etc. But obviously that idea is incorrect. The past
is fixed once it passes out of the present. Therefore, the best one
can do is to just OBSERVE the past, but cannot change it. Hence if
one were to accept the notion that the entire universe is a large
"hypersphere" there exists the possibility that the rays of any
particular time are still traveling on along the sphere. Hence
eventually all past events may circle around the spherical universe
and could be "read" as visions from the past. In that sense the entire
universe would be a "time-machine".

The necessity for the universe being spherical has to do with the
"edge" problem. By the flatland analogy, one would have to ask what a
"flatlander" would find if he traveled as far as possible in his
"universe"? One would have to imagine some massive God-like clamps at
the edge of Flatland. Clearly such a solution is not very "elegant".
A much more elegant possibility would be to make flatland actually
sphereland! If the Sphere were large enough, flatlanders would not be
able to detect the spherical geometry with their instruments (such as
sum of angles of a triangle).

My calculation shows the universe to be a hypersphere about 500
million light years in diameter.

wrote:
On Aug 30, 2:18 am, wrote:

In you keep poking the center of the rubbery sheet it means the
circumference will be SMALLER than the diameter.

This IS an interesting concept! Especially given the theory of a
spherical universe (see my other post). The end result is that given
a spherical geometry of space, like your conical one, mathematical Pi
unlike the common assumption in science would NOT be the true value of
the ratio of a circumference to the diameter of a circle. In actual
fact spherical geometry would apply. The point being that these huge
computer calculations of Pi to n places really is only correct for a
FLAT Euclidian space and the the spherical manifold comprising the
universe means that in our world Pi and the mathematical pi could be
different. Although the very large diameter of the spherical universe
means that the two values would be quite close and the differences
impossible to actually measure.

The measurement of an actual circle circumference to it's diameter in
our physical space and compare it to Euclidean Pi would be an
interesting thing, especially given the Einsteinian interest in space-
time "warps". Extreme accuracy would make the experiment difficult.

On Aug 31, 6:05 am, wrote:
On Aug 30, 2:18 am, wrote:



I apologize in advance for my ignorance.

Whereas pi is defined as the ratio of the circumference of a circle to
it's diameter, it is constant and irrational on a euclidiean plane.
It seems logical to me to suppose that the value of pi would be
different on curved surfaces (like a sphere or hyperbola). This led
me to suppose that there were curvatures where the value of pi is
rational. From there I wondered if there were some characteristics of
those surfaces wherein pi ir irrational that were shared in common.

I trivially came up with the example that the value of pi on a surface
in which all points are adjecent is 1, but my knowledge of math is
nowhere near enough to do anything more.

I'd be interested in your thoughts.

Thanks,
Ben

The ancient Greeks discovered that the ratio of a circles
circumference to its radius is of fundamental importance. They called
this _ratio_ "pi". The "number pi" is the Pi in Euclidean geometry.

There are infinite geometries which have Pi = 1. The easiest ones to
imagine is that for a (suitably chosen) cone.

Visualize it like this
[1] Draw a circle on a rubbery sheet.
[2] Poke the center of the circle until a cone-like structure is
formed.
[3] Keep poking until the diameter of the circle equals the
circumference.

Since it is possible to have infinite deformations in the underlying
manifold which don't affect diameter, then there are infinite possible
manifolds giving Pi=1. Those geometries over those manifolds would all
need to be hyperbolic, so I suppose that is the broadest answer to
your question (without getting into the math). If you go even more
abstruct there's probably semi-riemannian manifolds which could do the
same, but doing that you the geometric meaning from the "ancient"
point of view.

NOTE:

In you keep poking the center of the rubbery sheet it means the
circumference will be SMALLER than the diameter.

You mean SMALLER than the stretched diameter measured along the curved
meridian?

If instead you poke with a round and blunt pole applying distributed
pressure in place of a concentrated force the rubbery sheet would
deform to a dome shape whose circumference will be smaller than the
stretched diameter, creating an elliptic but not a hyperbolic
manifold.

One can explain this from Von Karman's equation of
_differential_(meaning changing with location on the sheet) non-linear
deformations and strains or Beltrami operators. But first avoiding
maths and visualizing physics in principle, it would help for a more
fundamental insight.

If in a flat rubbery sheet we

Stretch both diameter and circumference by 50 percent OR shrink both
diameter and circumference by 50 percent - Flat sheet remains flat,2-
D Euclidean.

Stretch diameter and/or shrink circumference by 50 percent - Flat
sheet becomes a dome, the 3-D elliptic manifold.

Shrink diameter and/or stretch circumference by 50 percent - Flat
sheet becomes a part of horn like catenoid, the 3-D hyperbolic
manifold.

HTH

Narasimham

On Aug 31, 6:05 am, wrote:
On Aug 30, 2:18 am, wrote:

I apologize in advance for my ignorance.

Whereas pi is defined as the ratio of the circumference of a circle to
it's diameter, it is constant and irrational on a euclidiean plane.
It seems logical to me to suppose that the value of pi would be
different on curved surfaces (like a sphere or hyperbola). This led
me to suppose that there were curvatures where the value of pi is
rational. From there I wondered if there were some characteristics of
those surfaces wherein pi ir irrational that were shared in common.

I trivially came up with the example that the value of pi on a surface
in which all points are adjecent is 1, but my knowledge of math is
nowhere near enough to do anything more.

I'd be interested in your thoughts.

Thanks,
Ben

The ancient Greeks discovered that the ratio of a circles
circumference to its radius is of fundamental importance. They called
this _ratio_ "pi". The "number pi" is the Pi in Euclidean geometry.

Lower precision pi was found even before.

There are infinite geometries which have Pi = 1. The easiest ones to
imagine is that for a (suitably chosen) cone.

Visualize it like this
[1] Draw a circle on a rubbery sheet.
[2] Poke the center of the circle until a cone-like structure is
formed.
[3] Keep poking until the diameter of the circle equals the
circumference.

stretched diameter along the now curved radius ?

Since it is possible to have infinite deformations in the underlying
manifold which don't affect diameter, then there are infinite possible
manifolds giving Pi=1. Those geometries over those manifolds would all
need to be hyperbolic, so I suppose that is the broadest answer to
your question (without getting into the math). If you go even more
abstruct there's probably semi-riemannian manifolds which could do the
same, but doing that you the geometric meaning from the "ancient"
point of view.

NOTE:

If you keep poking the center of the rubbery sheet it means the
circumference will be SMALLER than the diameter.

This means if you shot a light-beam through the diameter you ran
around the (smaller) circumference you could conceivably detect your
own light-beam - giving you a time-machine.

You mean SMALLER than the stretched diameter measured along the curved
meridian?

If instead you poke with a round and blunt pole applying distributed
pressure in place of a concentrated force, or better still apply high
air
pressure on one side, the rubbery sheet would deform to a dome shape
whose circumference will be smaller than the stretched
diameter,creating
an elliptic but not a hyperbolic manifold.

One can explain this from Von Karman's equation of _differential_
(meaning changing with location on the sheet) non-linear
deformations and strains or Beltrami operators. But first avoiding
maths and visualizing physics in principle, it would help for a more
fundamental insight.

If in a flat rubbery sheet we

Stretch both diameter and circumference by 50 percent OR shrink both
diameter and circumference by 50 percent - Flat sheet remains flat,
2-D Euclidean in 2-space.

Stretch diameter and/or shrink circumference by 50 percent - Flat
sheet becomes a dome, the 2-D elliptic manifold, jutting out into
3-space due to such strains.

Shrink diameter and/or stretch circumference by 50 percent - Flat
sheet becomes a part of horn like catenoid, the 2-D hyperbolic
manifold, jutting out into 3-space due to such straining.

Regards,

Narasimham

on aug 31, 6:05 am, wrote:
on aug 30, 2:18 am, wrote:

I apologize in advance for my ignorance.

Whereas pi is defined as the ratio of the circumference of a circle to
it's diameter, it is constant and irrational on a euclidiean plane.
it seems logical to me to suppose that the value of pi would be
different on curved surfaces (like a sphere or hyperbola). this led
me to suppose that there were curvatures where the value of pi is
rational. from there i wondered if there were some characteristics of
those surfaces wherein pi ir irrational that were shared in common.

I trivially came up with the example that the value of pi on a surface
in which all points are adjecent is 1, but my knowledge of math is
nowhere near enough to do anything more.

I'd be interested in your thoughts.

thanks,
ben

The ancient greeks discovered that the ratio of a circles
circumference to its radius is of fundamental importance. they called
this _ratio_ "pi". the "number pi" is the pi in euclidean geometry.

lower precision pi was found even before.

there are infinite geometries which have pi = 1. the easiest ones to
imagine is that for a (suitably chosen) cone.

visualize it like this
[1] draw a circle on a rubbery sheet.
[2] poke the center of the circle until a cone-like structure is
formed.
[3] keep poking until the diameter of the circle equals the
circumference.

Stretched diameter along the new curving radius ?

since it is possible to have infinite deformations in the underlying
manifold which don't affect diameter, then there are infinite possible
manifolds giving pi=1. those geometries over those manifolds would all
need to be hyperbolic, so i suppose that is the broadest answer to
your question (without getting into the math). if you go even more
abstruct there's probably semi-riemannian manifolds which could do the
same, but doing that you the geometric meaning from the "ancient"
point of view.

note:

if you keep poking the center of the rubbery sheet it means the
circumference will be smaller than the diameter.

this means if you shot a light-beam through the diameter you ran
around the (smaller) circumference you could conceivably detect your
own light-beam - giving you a time-machine.

You mean smaller than the stretched diameter measured along the curved
meridian?

If instead you poke with a round and blunt pole applying distributed
pressure in place of a concentrated force, or better still apply high
air
pressure on one side, the rubbery sheet would deform to a dome shape
whose circumference will be smaller than the stretched
diameter,creating
an elliptic but not a hyperbolic manifold.

One can explain this from Von Karman's equation of _differential_
(meaning changing with location on the sheet) non-linear
deformations and strains or Beltrami operators. But first avoiding
maths and visualizing physics in principle,it would help for a more
fundamental or unifying insight I think.

If in a flat flexible but inelastic(rubber is elastic so it regains
its original shape) sheet we

Stretch both diameter and circumference by 50 percent or shrink both
diameter and circumference by 50 percent - flat sheet remains flat,
2-d euclidean in 2-space.

Stretch diameter and/or shrink circumference by 50 percent - flat
sheet becomes a dome, the 2-d elliptic manifold, jutting out into
3-space due to such strains.

Shrink diameter and/or stretch circumference by 50 percent - flat
sheet becomes a part of horn like catenoid, the 2-d hyperbolic
manifold, jutting out into 3-space due to such straining.

The FlatLanders in Relativity would get to different embeddings into
their
surrounding space when deformations of the surface/speeds in the
manifold
changes.

Regards,

Narasimham