Pi is defined as the ratio of the circumference of a circle to its diameter in Euclidean geometry. It is a constant. Its value does not change, irrespectively of what geometry is being investigated, in the same way that 4 or root 2 don't change values.
What you mean is that in certain structures the ratio of circle's circumference to its diameter is not equal to the ratio of a circle's circumference to its diameter in Euclidean geometry, and therefore not equal to pi.
Originally posted by treborbassettWell, we are now discussing the definition of words.
Pi is defined as the ratio of the circumference of a circle to its diameter in Euclidean geometry. It is a constant. Its value does not change, irrespectively of what geometry is being investigated, in the same way that 4 or root 2 don't change values.
What you mean is that in certain structures the ratio of circle's circumference to its diameter ...[text shortened]... circle's circumference to its diameter in Euclidean geometry, and therefore not equal to pi.
If pi is defined 3.14... in Euclidian geometry, then what is it defined in non-Euclidian geometry? I would say that the value of pi defines how non-Euclidian the space really is. When pi is 3.14... the geometry is flat, as Euclidian space should be.
In Euclidian geometry, of course pi is a constant. We agree on that.
The fun is that this means that we now have a tool to measure the geometry of universe itself: By measure a circle large enough, like a diameter of billions of light years, and measure its circumference, then we can deduce exactly how non-Euclidian the Universe actually is. Is it spherical? Or is it hyperbolic? The answer is at hand. Or not. We have no methods to measure circles this large.
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Originally posted by FabianFnasYou're quite right: we are discussing the definition of words. In particular, we're discussing the definition of the word 'pi'. But I'm afraid your definition is quite simply wrong (in the sense that it disagrees with what mathematicians mean by 'pi'😉. It is not defined as the function that accepts a circle (and perhaps a geometry, both suitably defined) as input and yields as output the real number that is the ratio of that circle's circumference to its diameter. 'Pi' simply refers to a number, in the same way that '1' and '2' do. The number to which it refers does not change.
Well, we are now discussing the definition of words.
Of course it's turned out that the geometry of the universe (if we take plausible interpretations of geometrical primitives - 'point', 'straight line', etc.) is non-Euclidean. This does not mean that the reference of 'pi' has changed. It remains the same, even though it may not be the ratio of a circle's circumference to its diameter in the geometry of our universe.
Originally posted by FabianFnasWhat you are saying doesn't work. Think about the area enclosed by a circle on a 2-sphere. The distance from the pole is theta and the area element is Sin(theta) (Assume the sphere has unit radius). When you work out the area enclosed by the circle it comes to 2 pi (1 - cos(theta)). If theta is small then Cos(theta) = 1 - theta^2/2 + ignored terms. So we get the area of the enclosed (small) area as pi theta^2, as we'd expect having taken the Euclidean limit. circumference = 2*pi*sin(theta) which again gives us 2 pi theta in the limit that theta is small. In non Euclidean geometries the circumference of a circle is not as simple a function of radius (in this case theta) as in Euclidean geometries, you'd need a different value of pi for each circle radius and it loses it's general meaning. pi is ratio of a circles circumference to it's diameter - in the limit that the diameter is zero in all geometries.
Well, we are now discussing the definition of words.
If pi is defined 3.14... in Euclidian geometry, then what is it defined in non-Euclidian geometry? I would say that the value of pi defines how non-Euclidian the space really is. When pi is 3.14... the geometry is flat, as Euclidian space should be.
In Euclidian geometry, of course pi is a constan ...[text shortened]... it hyperbolic? The answer is at hand. Or not. We have no methods to measure circles this large.
Originally posted by bloodyboyTo get back on topic;
If you are traveling at the speed of light would you be invisble? because the light could not catch up to you casue you would be going faster. but then would you be visible from the direction that you are moving becasue you would run into the light rays which would then reflect off of you.???
No you wouldn't be invisible. Light still bounces off your forehead into my eyes. As for light not catching up to you, you wouldn't be able to tell. Aside from being dead, light still travels with the speed of light, even if you reference system is moving at that speed.
Originally posted by FabianFnasI was quoting an american movie, dumb and dumber when the main character asks what his chances are of dating this beautiful woman. She replies one in a billion, and he says so you mean there is a chance!
Oh, you think it is actually possible to reach light speed? Not only near that speed?
Well, I don't.
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Even if m = 0, c^2 - c^2 = 0, and you can't divide by zero...ever. So shouldn't photons be limited by the speed of light as well?
You can't apply the formula for mass dilation to something with no rest mass...
Even if you could, 0/0 is not the same as 1/0 or 2/0. Consider the value of (2x)/x as x goes to 0. Even though the denominator is going to 0, the value doesn't go to infinity.
Originally posted by joe shmoActually it was 'one in ten million'. OH, so I HAVE A CHANCE. she gives him a funny look, one of the best scenes in the movie.
I was quoting an american movie, dumb and dumber when the main character asks what his chances are of dating this beautiful woman. She replies one in a billion, and he says so you mean there is a chance!
I know what is faster than the speed of light:
Thoughts.
That is some of the fastest energy that I know of.
You know when someone is about to say something
and you know what they're going to say?
Or...better yet, you start singing a song and
then you get in your car and turn on the radio
and there is that song right on the same word
as you.
There is something there.
Faster than light.
Instaneous.
Originally posted by joe shmoAh, I didn't know that reference, but now when you say it, a remember that line vaguely. 😀
I was quoting an american movie, dumb and dumber when the main character asks what his chances are of dating this beautiful woman. She replies one in a billion, and he says so you mean there is a chance!
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Originally posted by treborbassettWhen I was a boy they told me that you can't take the square root of a negative number. I took that as a truth. Later they told me that in R it is impossible, but in C it is and they gave me a wonderful new toy to play with.
But I'm afraid your definition is quite simply wrong (in the sense that it disagrees with what mathematicians mean by 'pi'😉.
Newton told us the wonders of physics and everyone thought that all makes sense. Then came Einstein that said, yes in ordinary life this works, but in high gravitation and in high velocities, Newton is not enough. And he gave the world new toys to play with.
Mathematicians thought since the days of Euclides that the parallel axiom was the truth, and pi was 3.14... and this was told to all students over the world. Later someone (who?) created a totally new branch of mathematics where the foundation was that the parallel axiom wasn't the truth, that it was possible to create a non-Euclidian mathematics. In this mathematics an area of an circle has to be recalculated, the circumference, the volume of a sphere, about everything in geometry had to be redefined. Is pi really 3.14... in another geometry than the Euclidian?
Now, I don't put the neck in the guillotine for this matter. If you say that pi is exactly 3.14... in all weathers, I say so too, but I finish my sentence (as Galileo (?) did: "but still she moves" ) with "but only in Euclidian geometry".
Originally posted by FabianFnasIn any geometry the ratio of the circumference to the diameter of a circle is pi in the limit that the radius is zero. Non-Euclidean geometries are locally flat so you can recover the c = pi d rule as a limiting case. In non Euclidean geometries the ratio of the circumference of a circle to it's diameter is not a constant so trying to define pi as a simple ratio doesn't work unless you take the limit that the diameter goes to zero when you get the same answer as in Euclidean space.
When I was a boy they told me that you can't take the square root of a negative number. I took that as a truth. Later they told me that in R it is impossible, but in C it is and they gave me a wonderful new toy to play with.
Newton told us the wonders of physics and everyone thought that all makes sense. Then came Einstein that said, yes in ordinary l ...[text shortened]... tence (as Galileo (?) did: "but still she moves" ) with "but only in Euclidian geometry".