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INRM
8th July 2008, 05:48 AM
I was thinking.

Does anything in nature fit the profile for an irrational number such as Pi? It sounds kind of strange, but I've kind of thought about it, and Pi seems to be one of those "ideal" numbers that works with "perfect circles"

However, in nature I don't know if there are such things. Earth isn't perfectly circular or spherical. The earth is larger in diamater across it's equator than it's pole (due to our rotation), and we have mountains, and canyons and valleys and stuff. The sun, while lacking mountains have some areas higher than others (prominences and flares), and also has probably some flattening across the the pole than the equator as it rotates too.

I've even been told that some black holes look like rings.

So, do any real irrational numbers (non ending, non repeating) actually exist in nature, not just in hypothetical?

INRM

GoodGuysEatPie
8th July 2008, 05:55 AM
So, do any real irrational numbers (non ending, non repeating) actually exist in nature, not just in hypothetical?
INRM

I'm not sure what you mean by this. If you want examples of other 'natural' irrational numbers, try e or the "golden ratio".

~ggep~

Ocelot
8th July 2008, 06:06 AM
I would suspect that most numbers found in nature are irrational in that you can contiue measuring them to ever greater precission.

leon_heller
8th July 2008, 06:08 AM
PI is more than an irrational number, it's transcendental. It's quite ubiquitous in mathematics, which could mean that it exists in nature, apart from it's being associated with the circle.

Leon

ddt
8th July 2008, 06:15 AM
Does anything in nature fit the profile for an irrational number such as Pi? It sounds kind of strange, but I've kind of thought about it, and Pi seems to be one of those "ideal" numbers that works with "perfect circles"

However, in nature I don't know if there are such things. Earth isn't perfectly circular or spherical. [...]

So, do any real irrational numbers (non ending, non repeating) actually exist in nature, not just in hypothetical?

Do (integer) numbers exist in nature at all? Or didn't you mean it that philosophical?

As you rightly note, things aren't perfect in nature. Lines when drawn on a sheet of paper do not have breadth 0 and points are not dimensionless. Circle-like shapes, whether drawn on paper or otherwise occurring in nature, are not perfectly circular.

We don't even know if space/time is a continuum; maybe only rational coordinates are valid in "real nature". But the existing real/complex-number model we have of nature gives a great model with astounding accuracy.

leon_heller
8th July 2008, 06:58 AM
There are plants whose leaves grow in a Fibonacci series, which is a property of the integers.

Leon

Stir
8th July 2008, 07:11 AM
So, do any real irrational numbers (non ending, non repeating) actually exist in nature, not just in hypothetical?

INRM

IMHO, a meaningless question without very precise definitions of "exist" and "in nature" ... and such definitions are either very very difficult or completely impossible

quarky
8th July 2008, 08:48 AM
A nut falling into a still pond will cause circles of disturbance that might be accurate enough for some pi to come into it. Aren't some electron orbits pretty damn circular?

INRM
8th July 2008, 11:59 AM
I'm not sure what you mean by this. If you want examples of other 'natural' irrational numbers, try e or the "golden ratio".

~ggep~

What's a golden ratio?

PI is more than an irrational number, it's transcendental. It's quite ubiquitous in mathematics, which could mean that it exists in nature, apart from it's being associated with the circle.

Leon

Transcendental? Huh?

TheDon
8th July 2008, 12:02 PM
Am I right in thinking that there are infinitely more irrational numbers than rational numbers? As there are infinitely many points between any two arbitrarily close rational numbers.

ddt
8th July 2008, 12:03 PM
What's a golden ratio?

Transcendental? Huh?

Golden ratio (http://en.wikipedia.org/wiki/Golden_ratio)

Transcendental (http://en.wikipedia.org/wiki/Transcendental_number)

When you've read them and understood, here is a quiz question: is the golden ratio transcendental?

ddt
8th July 2008, 12:09 PM
Am I right in thinking that there are infinitely more irrational numbers than rational numbers? As there are infinitely many points between any two arbitrarily close rational numbers.

Yes, but there are also infinitely many rational numbers between two rational numbers; so that's no argument.

But indeed: the set Q of rational numbers is countable, i.e. "as big as" the natural numbers. The set R of real numbers is supercountable. The irrational numbers - the difference between the two - then logically is again supercountable (because the union of two countable sets is again countable).

The same holds, btw, for the set A of algebraic numbers and its complement, the transcendental numbers.

lumos
8th July 2008, 01:19 PM
I was thinking.

Does anything in nature fit the profile for an irrational number such as Pi? It sounds kind of strange, but I've kind of thought about it, and Pi seems to be one of those "ideal" numbers that works with "perfect circles"

However, in nature I don't know if there are such things. Earth isn't perfectly circular or spherical. The earth is larger in diamater across it's equator than it's pole (due to our rotation), and we have mountains, and canyons and valleys and stuff. The sun, while lacking mountains have some areas higher than others (prominences and flares), and also has probably some flattening across the the pole than the equator as it rotates too.

I've even been told that some black holes look like rings.

So, do any real irrational numbers (non ending, non repeating) actually exist in nature, not just in hypothetical?

INRM

There is no obvious perfection in objects existing in the physical world, simply very, very close with a very tight tolerance. There will always be errors and imperfections if you measure at a fine enough level. Maybe a neutron star is pretty close to a perfect sphere. Maybe quantum particles and atoms are perfect, but I doubt it.

Mathematics is used to simply model the physical world. Yes, the math model is perfect and hypothetical, but can be very close to reality. The ratio of the circumference of a perfect circle to its diameter is exactly pi, but try to measure that. (really, try it, use a measuring tape and try to get pi.)

drkitten
8th July 2008, 01:25 PM
So, do any real irrational numbers (non ending, non repeating) actually exist in nature, not just in hypothetical?

Let me turn this around.

"Yes, I have a photograph of a perfect circle produced by dropping a rock into the pond near my house."

Disprove me.

Given that any finite approximation to an irrational number is by definition rational, and given that all our measurements are finite and error-limited, how could you ever demonstrate an irrational number in nature? How could you ever disprove it?

fuelair
8th July 2008, 08:12 PM
A nut falling into a still pond will cause circles of disturbance that might be accurate enough for some pi to come into it. Aren't some electron orbits pretty damn circular?No. The "orbits" (orbitals) are areas the electron is most likely to be found in based on it's energy level. Think of the orbital shape as a distribution pattern most likely to include one or two electrons most of the time they are moving in the space around the nucleus - but, one or both may not be present in that location/area at any specific time. Each specific orbital is limited to two electrons with opposite spin.

Ocelot
9th July 2008, 01:47 AM
No. The "orbits" (orbitals) are areas the electron is most likely to be found in based on it's energy level. Think of the orbital shape as a distribution pattern most likely to include one or two electrons most of the time they are moving in the space around the nucleus - but, one or both may not be present in that location/area at any specific time. Each specific orbital is limited to two electrons with opposite spin.

In fact most electron shells are anything but spherical.

http://core.ecu.edu/phys/flurchickk/AtomicMolecularSystems/electronShells/electronshells.html

Puppycow
9th July 2008, 02:31 AM
Duh. People make pies. Oh, that pi

Pi is used in physics to describe laws of nature.

E.g. Inverse-square law

Hence I would say that yes, there is something natural about Pi.

If, indeed any numbers can be said to exist in nature at all. They exist in our descriptions of nature and natural laws, not in nature itself.

Scazon
9th July 2008, 05:22 AM
This thread belongs under Religion, because Piism is the One True Religion. Pi is the Holy Number, it makes the Universe go round. It is infinite (in digits), transcendental, and normal. A binary representation of Pi contains all possible information, all books that have been written, will be written, and all that never will be written. It also includes a complete mathematical description of all the approximations to circles produced by dropping stones into ponds.

Fall on your knees and worship the Holy Pi.

quarky
9th July 2008, 10:53 AM
No. The "orbits" (orbitals) are areas the electron is most likely to be found in based on it's energy level. Think of the orbital shape as a distribution pattern most likely to include one or two electrons most of the time they are moving in the space around the nucleus - but, one or both may not be present in that location/area at any specific time. Each specific orbital is limited to two electrons with opposite spin.

Yes, but
When an electron isn't being a wave, don't you suspect that the particle is spherical?
And that its volume can be expressed in terms of 4/3 Pi r2?

The smallest particles have no room for dirt, bumps and dips.
Perfect spheres, in my mind.

jj
9th July 2008, 10:55 AM
Look into the "needle drop experiment", why don't you?

9th July 2008, 12:01 PM
A nut falling into a still pond will cause circles of disturbance that might be accurate enough for some pi to come into it. Aren't some electron orbits pretty damn circular?

No. Electron orbits are probabilistic, and unmeasureable due to the uncertainty principle.

As far as a circle, it is, indeed, a perfection in nature. A drop of any liguid or a bubble will assume a perfectly spherical shape as long as there are perfect conditions - no rotation, no net exterior gravity field, no movement through a media. There has been some cosiderable experimentation about what that means in orbit; the ability to create perfectly spherical ball bearings which are a very valuable commodity, for instance, might be enhanced in microgravity, which is about as close as we can come for the time being. Balanced gears and other rotational parts require circularity; the various telescopes require perfectly parabolic surfaces (a circle is a certain kind of parabola). Except those that we have fitted for spectacles.

I saw a piece on a TV documentary once about a guy who built a wheel that, when rolled 1000 times, rolled the distance of the edge of the great pyramid of Giza. Then he measure the distance from the base to the apex and found it was some neat number (like 500 or 1000) of the wheel's diameter. He thought it was a miracle. Heh.

INRM
9th July 2008, 02:11 PM
How did mathematicians arrive at the number Pi? There are so many circular objects and none are perfect circles...

Might sound silly but it's a legit question.

BTW: To whoever said this should belong in religion, I would strongly disagree -- it belongs in mathematics and science.

leon_heller
9th July 2008, 02:19 PM
How did mathematicians arrive at the number Pi? There are so many circular objects and none are perfect circles...

Might sound silly but it's a legit question.

BTW: To whoever said this should belong in religion, I would strongly disagree -- it belongs in mathematics and science.

Leon

ddt
9th July 2008, 02:41 PM

Golden ratio (http://en.wikipedia.org/wiki/Golden_ratio)

Transcendental (http://en.wikipedia.org/wiki/Transcendental_number)

When you've read them and understood, here is a quiz question: is the golden ratio transcendental?

How did mathematicians arrive at the number Pi? There are so many circular objects and none are perfect circles...

Might sound silly but it's a legit question.

As leon said. Or look up on wiki. It sounds indeed awfully silly. You're either trolling, or unwilling to put in some basic research yourself.

fuelair
9th July 2008, 04:11 PM
Yes, but
When an electron isn't being a wave, don't you suspect that the particle is spherical?
And that its volume can be expressed in terms of 4/3 Pi r2?

The smallest particles have no room for dirt, bumps and dips.
Perfect spheres, in my mind.I believe that this is called repositioning the goalposts. We were discussing (as in you specifically mentioned) the orbits - not the electrons themselves and, I addressed the orbits. And, I am not actually in this discussion because it seems to be turning on points of philosophy - I only wrote to clear up the point of your specific post.

quarky
10th July 2008, 07:35 AM
Thanks.

(I thought the op was dorky enough to go ahead and add more dorky stuff.)

ddt
10th July 2008, 08:14 AM
(I thought the op was dorky enough to go ahead and add more dorky stuff.)

What happened to the OP anyway? Did he abandon his own thread?

Hellbound
10th July 2008, 10:37 AM
Yes, but
When an electron isn't being a wave, don't you suspect that the particle is spherical?
And that its volume can be expressed in terms of 4/3 Pi r2?

The smallest particles have no room for dirt, bumps and dips.
Perfect spheres, in my mind.

Just to add a bit to this, electrons are thought to be point particles, or so close to them that it doesn't matter (radius of less than the Planck length). This means their volume is zero (or indeterminate). They aren't spheres, because spheres have definite radii and spacial extension.

If you subscribe to one of the various superstring theories or M-theory, then electrons are vibrating strings, not spheres at all.

Just more food for thought :)

Soapy Sam
10th July 2008, 10:46 AM
There are only two numbers, both of them imaginary.

This is my "New, New Math", which is concise & elegant, therefore kewl and fits on a T-shirt, therefore potentially profitable.

And it's easier to learn than the old stuff, or the new stuff.

I once read that Pi has a tendency to crop up in places like actuarial statistics, which I find a bit spooky, frankly. But then, six probably makes the odd appearance too and there's nothing transcendothingy about six.
Sooner we stop this sort of thing , the better, I say.

Make mathematics great again. Bring back Real Numbers!

ddt
10th July 2008, 11:09 AM
There are only two numbers, both of them imaginary.

This is my "New, New Math", which is concise & elegant, therefore kewl and fits on a T-shirt, therefore potentially profitable.

And it's easier to learn than the old stuff, or the new stuff.

I once read that Pi has a tendency to crop up in places like actuarial statistics, which I find a bit spooky, frankly. But then, six probably makes the odd appearance too and there's nothing transcendothingy about six.
Seeing the density function for the normal distribution is
f(x) = 1/sqrt(2*pi) * e^(-x^2/2)
I'm not surprised.

Sooner we stop this sort of thing , the better, I say.

Make mathematics great again. Bring back Real Numbers!
Real numbers are for quiche eaters. Real men use integers.

jj
10th July 2008, 11:21 AM
How did mathematicians arrive at the number Pi? There are so many circular objects and none are perfect circles...

Might sound silly but it's a legit question.

BTW: To whoever said this should belong in religion, I would strongly disagree -- it belongs in mathematics and science.

Once again, look into the needle drop experiment.

cosmic
10th July 2008, 11:43 AM
This is actually a very deep question to my mind.

Many fundamental equations of physics include pi. Of course any accepted theory must be backed up by experiment and data. Yet any theoretical prediction from an equation that includes pi will often be an irrational/transcendental number. So a perfect match between any test/data and the prediction is inherently inaccessible. A measurement, consisting of a finite number of digits, can not provide an irrational number.

Of course we can define/demand any level of precision when checking prediction vs. measurement (barring the uncertainty principle). Yet it seems an interesting point that irrational numbers are in essence beyond measure, and yet our best theories often must incorporate them. So our best theories are in some sense fundamentally beyond exact comparison. It's not just a matter of practical difficulty.

An aside: The golden ratio does not seem to have near the ubiquity in physics (or any) theoretical equations. It is often cited in flowers, seashells, etc...but I have not seen one convincing case that phi is fundamental in nature-- the closer we measure these entities the more we see deviation-- unlike the equations that include pi.

69dodge
10th July 2008, 04:51 PM
Many fundamental equations of physics include pi. Of course any accepted theory must be backed up by experiment and data. Yet any theoretical prediction from an equation that includes pi will often be an irrational/transcendental number. So a perfect match between any test/data and the prediction is inherently inaccessible.

And if the theory predicted a rational number as the result of an experiment? We couldn't verify that any more easily. The problem isn't the irrationality of the prediction; the problem is the limited precision of the measurement.

Soapy Sam
10th July 2008, 04:57 PM

Actual data being absent helps.

SirPhilip
10th July 2008, 05:00 PM
However, in nature I don't know if there are such things. Earth isn't perfectly circular or spherical. The earth is larger in diamater across it's equator than it's pole (due to our rotation), and we have mountains, and canyons and valleys and stuff. The sun, while lacking mountains have some areas higher than others (prominences and flares), and also has probably some flattening across the the pole than the equator as it rotates too. An even more hypothetical question arises: does nature permit what can be mathematically defined as perfect order. This question came to mind when considering the actual existence of evolving mathematical symmetry in nature, of which we're a product.

fuelair
10th July 2008, 05:26 PM

Actual data being absent helps.
Thats why I don't tend to play with it a lot.:)

ddt
11th July 2008, 12:44 AM
Many fundamental equations of physics include pi. Of course any accepted theory must be backed up by experiment and data. Yet any theoretical prediction from an equation that includes pi will often be an irrational/transcendental number.
Let me rephrase that: many fundamental equations in mathematics include pi. Its ubiquity in physics is just an outgrowth of that, as physics uses mathematical models to describe the world.

It's quite remarkable that two fundamental constants that are ubiquitous in mathematics - e and pi - are transcendental. I don't think anyone has ever offered a reasonable explanation for that.

An aside: The golden ratio does not seem to have near the ubiquity in physics (or any) theoretical equations.
Same explanation: the golden ratio is, mathematically speaking, just a side show.

jj
11th July 2008, 12:46 AM

ei pi+1=0

What else do you need to know?

ddt
11th July 2008, 12:54 AM

ei pi+1=0

What else do you need to know?

I was planning on including Euler's equation but then I forgot. But does it give an explanation why both are transcendental?

Soapy Sam
11th July 2008, 03:49 PM
ei pi+1=0

A kludge, if ever I saw one! :D

Vorpal
12th July 2008, 09:25 PM
Many fundamental equations of physics include pi. Of course any accepted theory must be backed up by experiment and data. Yet any theoretical prediction from an equation that includes pi will often be an irrational/transcendental number. So a perfect match between any test/data and the prediction is inherently inaccessible. A measurement, consisting of a finite number of digits, can not provide an irrational number.
69dodge is completely right here. If an astronomer says that a certain star is 38,000 light-years away, you can bet there's an implicit "give or take 5,000 ly or so" clause. Well, distance measurements in astronomy may be particularly imprecise, but the point remains true for any kind of measurement in every kind of science: measurements are always tainted by uncertainty, making them more akin to "smears" of possible values. This makes rational values, either predicted or real, also "inherently inaccessible" due to measurement uncertainties and likewise "beyond measure".

It's quite remarkable that two fundamental constants that are ubiquitous in mathematics - e and pi - are transcendental. I don't think anyone has ever offered a reasonable explanation for that.
I don't understand the issue. Mathematically, an 'explanation' of π or e being transcendental would amount to a proof of such, which was done. If not that, what would be the criteria for something to be an explanation? I don't know the following is an explanation or not, but as a kind of meta-probabilistic argument, a circle is a limit of a sequence of polygons, so π can be likewise generated as a limit of a rational sequence. 'Most' such sequences that are convergent at all converge to a transcendental number, so we "should expect" this to turn out transcendental. The 'most' remains true for several senses other than measure--cardinality of equivalence classes, being topologically meager.

cosmic
14th July 2008, 12:39 AM
And if the theory predicted a rational number as the result of an experiment? We couldn't verify that any more easily. The problem isn't the irrationality of the prediction; the problem is the limited precision of the measurement.

I agree that the constraints imposed by the errors on measurement limit how well we can verify. It still seems that irrational numbers are less accessible that rational ones.

If we had some way of giving absolute precision on a measurement, wouldn't an irrational number remain inaccessible? (I'm now not certain this is true) But, if so, it seems to indicate that many theoretical predictions are in some real sense inaccessible even to ideal measurement.

Vorpal
14th July 2008, 01:03 AM
If we had some way of giving absolute precision on a measurement, wouldn't an irrational number remain inaccessible? (I'm now not certain this is true) But, if so, it seems to indicate that many theoretical predictions are in some real sense inaccessible even to ideal measurement.
No. If you have an exact theoretical prediction for a problem, and if furthermore if your conditions and measurement can be made to exactly match the problem, then there would be no problem at all in verifying whether theory matches experiment, rational or irrational.

Perhaps you're thinking of non-computable real numbers... the computable (recursively enumerable) numbers are those which have a recursive algorithm for calculating them for to given accuracy. Obviously, all non-computable numbers are irrational, and in fact transcendental (but not every transcendental is non-computable). If the values of an observable can take on any real value, then there is no guarantee that this value is computable. This would be in some sense "inaccessible" to theory. But nevertheless, whenever you have a theoretical answer, exact conditions and measurement would verify it regardless of (ir)rationality.