Some Pi Questions. Easy on the mathematics please folks!

1. As Pi is a transcendental number does it follow that the centre of a perfect sphere is:

- transcendental, a limit rather than a location?
- necessarily always measurably larger than a mathematical point
- a geometric impossibility
- itself a sphere
- a limit on our ability to calculate related to Planck lengths (limits of measurement etc)
- other (please specify)

2. Is it known why Pi has to be transcendental number (why it is inevitably transcendental)?

3. What difference would it make to geometry and mathematics if Pi was a rational number.

none of these makes any sense. none is true.
well, there are proofs that pi is transcendental. i don t know if you would be satisfied with a proof as an answer to a "why" question.

wait, i have an idea: a finite degree polynomial has the property that if you take the derivative enough times, you will end up with zero. cosine does not have this property, if you take its derivative, you get a cosine back. this means that cosine is not a finite polynomial. since pi is a root of cosine, then pi is transcendental (hmm... there is a flaw in this explanation.)

but to understand why pi is transcendental, i think it makes more sense to look at why cosine is transcendental. it is an infinite degree polynomial (speaking loosely), and therefore it is only natural to expect it might have transcendental roots, since the definition of transcendental is a number that is not the root of any finite polynomial. it doesn t constitute a proof, but it makes it plausible, at least.
the laws of mathematics and arithmatic would have to be drastically altered for this to happen. there seems to be no logical way. you might have to throw logic out the window, or at least take on a whole new kind of logic. i have no idea what it would look like.

The center of a sphere is <i>defined</i> to be a point, is it not?




The postulates of geometry contain the notion of a point. The center of a sphere is defined to be a point.




Probably a sphere of zero radius.




I don't think that the transcendental nature of a number has anything to do with QM. Math may be used to describe QM (and yes, logic has been adjusted to address QM situations), but QM puts limits on what math we use to meaningfully describe it; I don't think that it is the other way around.




Probably a fundamental discretization.

Hmm. In hindsight the question does look a bit daft. I was trying out a strange way of looking at it. (It seemed like a good idea at the time). Something like this seemed to follow from the fact that the length of circumference and its radius must have an irrational mathematical relationship. I wondered if that meant that we are wrong to consider that it is possible to define a circle, a radius, and a centre point in the way that we do. I.e. whether there was a mismatch between our concepts of circumference, radius and infinitessimal centre point. There's still something that bothers me about this.

I suppose I was asking what is about our concept of a circumference and a radius that must inevitably must make one immeasurable in terms of the other. It seems to me to be a sort of geometric uncertainty principle.

By asking why Pi is irrational I meant what is about circumferences and radii that makes their relationship inevitably irrational, whether it has to be like this because of some underlying fact of life.

I mentioned the Planck length thing because it occured to me that it's impossible to construct a sphere with a radius of 1 P-length. This means that if you start with a perfect sphere then reduce the length of all its radii at every surface point progressively and equally the sphere progressively shrinks while remaining perfect. However eventually the sphere becomes so small that it bumps up against the limits of quantisation.

So although the original sphere is perfect the shrinking sphere onto which it maps eventually becomes imperfect, since some circumferences are not allowable.

I know I'm muddling up the model with real thing here, the geometry with the ontology, but I like trying to do that.

Yes, but should it be?
Ditto

OK, but this is a mathematical entity, not a real thing. If one were to define this point in terms of its distance from the circumference it would never reduce to a point. The point would be a unreachable limit.

But the limits of math also place a limit on what we can describe, and perhaps even on what we can think, seeing as we think mathematically.

Mathematically it is impossible to fully define a perfect sphere by reference to a point at its centre. This makes me wonder is a perfect sphere with a precisely specified centre is an entirely logical concept.

I'm afraid I don't know what that means. To put the question better I meant is it possible to create a Euclidean type geometry in which c & r have a rational mathematical relationship, or does the rest of the system become irrational.

It is? What is wrong/impossible about the definition that a sphere is the set of all points at some specified distance from some specified point?




What I meant at first was something that you mentioned in your most recent post. The relationship between the radius and the circumference of a sphere is irrational. I was blaming the continuity of the space on this aspect of the space. If, for instance, we had a 2-D discrete space that was the set of all points at the corners of adjacent equilateral congruent triangles, then a circle would be a regular hexagon with circumference equal to six times the radius.

In light of your most recent post, I am not sure that this would maintain rationality for all other relationships in the space.

there should be nothing bothersome about the radius and the circumference being noncommensurate (this is the term for what you are describing.) it did bother the greek geometers, but that was before the discovery of real numbers. with real numbers on hand, there is nothing mysterious about this.

are you deliberately using quantum mechanical concepts here? i wish you wouldn t.

in the real world, it is impossible to measure anything with infinite precision. therefore you can never measure a value to be 1 exactly. you can never measure the circumference to be 2*pi exactly either. but in both cases, you can get as close the the number as you like, by using better equipment. there isn t really any difference between the rational and the irrational case.

for the rational, every more accurate measurement will return the same digit (or perhaps a repeating sequence). for the irrational, the sequence won t repeat. but neither one is inherently more "measurable" than the other.

and anyway, a mathematical circle is an ideal object, so it is foolish to ask about measuring it. it only really exists in your mind, a mathematical abstraction. in that context, both numbers are exactly calculable, and calculation is just a form of measuring things in abstraction.

the more i think about it, the more i like my previous argument. pi is irrational because of the perfection and periodicity of the circle. (it s still not a proof though)
who told you that the planck length is the smallest length? i see that notion a lot on this board. there have been people who toyed with the idea. there is some evidence that area might be quantized (and some experimental evidence that it is not), but by and large, this is an unproven fact. people are always mentioning the planck length to solve Xeno s paradox. the planck length is that scale at which quantum gravity is expected to be important. that is all it is. it does not imply the discretization of spacetime.

since a circle or a sphere is a mathematical object, it is impossible to create one of any radius. however, you can write down mathematical descriptions of an ideal sphere of any radius, including the planck length.

again, i am going to suggest that you leave quantum theory out of this. either you want to talk about pure math, or you want to talk about physics in the real world, that can be measured. one things got nothing to do with the other. well, not nothing. the mathematical idealization is a good approximation in a lot of cases. but it doesn t mean they are the same thing.

if space were quantized, in a naïve way, then there would be no such thing as an irrational number, and no such thing as a circle, as currently defined, anyway, except in our imaginations.

all circumferences are allowable.

A lot has been said here so I will only point out that there is GEOMETRICALLY (i.e. thinking of numbers as points on a number line) no difference at all between algebraic numbers and transcendental numbers. I think you are looking for relationships where none exist.

Hmm. Perhaps I asked a stupid question. Sorry about the P length thing - I thought it was considered to be a limit on the 'smallness' of things. Is spacetime a continuum then?

I don't see the harm in using QM concepts outside of their usual context. There must be a connection between all of these things. Still, I can see that it's dangerous.

Something still intrigues me about the fact that circumferences and radii must be in principle incommensurable in all possible universes. I don't know why, it doesn't seem to bother anyone else.

I apologise for improperly mentioning the uncertainty principle. However there does seem to be something reminiscent of it in the fact that for a circle one can know the precise length of its radius or its circumference, but not both.

Anyway thanks for the comments.

Canute

did you even read my post? the point of my post is that this is false. the circle "knows" both its radius and circumference precisely.

Why? It's also true that the length of a diagonal of a square is incommensurable with the length of its side. Does that "intrigue" you? (It did intrigue the Pythagoreans!) Do you have a problem with irrational numbers in general?

By the way, what do you mean by "all possible universes"? I don't believe that's a common mathematical phrase. If you ARE talking about mathematics, you might want to reflect on the fact that in elliptic geometry the circumferences of SOME circles ARE comensurable with their radii!

Not only is it not a proof, it's mathematical nonsense. The circle is NOT "periodic" unless you are talking about measuring around it circumference repeatedly, something that is not part of the definition of a circle. I don't know of any mathematical definition of "perfection" that would fit here!

mathematical nonsense?? sloppy language perhaps, but i don t think it is nonsense, and might even be useful, although that is certainly open for debate.

the circle is parametrized by (cos t,sin t). that is periodic. in fact, any paramterization of the circle is periodic. the circle can also be described as the real line modulo the equivalence relation x &sim; 2&pi;. again periodic. i think of the circle as the paradigm of periodicity.

now, my point is this: pi is defined as the root of cosine. cosine is defined as the coordinate of a circle. because of the periodic nature of the circle, the cosine function must have nonzero derivatives of all degrees. therefore, written as a polynomial, it is of infinite order. a transcendental number (which is necessarily irrational) is, by definition, one which is not the root of a finite order polynomial. it can be realized as the root of an infinite degree polynomial. it is therefore natural to expect that the root of cosine would be irrational (although not required. an infinite polynomial might have only rational roots, though that would be rather surprising, since rational numbers are so few on the real line).

I didn't read it as saying that.

I said that the relationship intrigued me, as it has many others. Sorry if that's a problem.

OK. All possible geometries in flat spacetime or whatever - all possible occurences of perfect circles in non-elliptical circumstances - whatever.

This wasn't my comment (perfection and periodicity of a circle) - but I liked it. It showed an understanding of what I was actually asking, and was rather a good answer - relating the problem to the calculus rather appropriately.

ok, fair enough. anyway, what i was trying to say is that just because a number is irrational, that doesn t make it any easier or harder to measure. so i don t really think any kind of uncertainty principle applies here.

Do you mean this the other way around?

Got you. But I don't really see measurement as an issue here.

right. neither do i. which is why i am asking you to leave off talking about things like the Uncertainty Principle in this discussion. they simply don t apply, since measurement is not relevant to purely mathematical objects.

yes. whoops, thanks.

That seems a bit touchy. Measurement isn't the only issue relevant to the uncertainty principle either. I wasn't suggesting that the same mechanism was at work, I was just musing on a another example of paired variables that behave awkwardly.