Although I don't know which is which, I understand that a certain type of curved space yields pi to be less than 3.14. . . and a certain type of curved space that yields pi to be greater than 3.14. . . To anyone's knowledge, has anyone looked into the specific curvature that would yield pi to be exactly 3? Am I mistaken to think that said curvature would hold some sort of significance?
Pi is allways 3.14159... . There exist spaces where the ratio between the circumference of a circle and its diameter is larger than Pi' and there exist spaces where this ratio is smaller than Pi.
First you claim that the circumference/diameter ratio of a circle is always 3.14... but then you acknowledge that different geometries yield different ratios for the circumference/diameter of a circle; some greater than 3.14... and some less than 3.14... I must assume that what you meant was that pi is always 3.14... within the context of euclidean flat space. Otherwise, it cannot ALWAYS be 3.14... and sometimes NOT BE 3.14... at the same time. So, my question still stands: has anyone investigated the space inwhich the ratio of the circumference and the diameter of a circle is equal to exactly 3? What is the nature of this specific curvature and does it hold any significance in the physical world, what with relativity and string theory and all that non-euclidean-geometry-jazz?
It's important to remember that pi is used for many things that don't relate to circles. For example, it's used in the probability density function in statistics and Heisenberg's uncertainty principle in physics. The ratio between a circle's circumference and diameter is just one of the many places where pi appears. Switching to a non-Euclidean space can change the ratio of a circle's circumference to diameter, but it wouldn’t change many of the other things that pi is used for, so in general people say that pi doesn't change in non-Euclidean spaces. Even if you only want to talk about circles, I don't think that pi could ever exactly equal 3. It's been a while, but I'm pretty sure that even in a non-Euclidean space either the circle's circumference or diameter will always be an irrational number. This would make it impossible to have the circumference/diameter ratio ever work out to a whole number like 3. Maybe someone who's had a math class a little more recently could confirm this?
Where did you see that I claim this? Again, I remind you that Pi is the number which equals 3.1415926... The ratio circumference/ratio is not the definition of Pi. In Euclidean geometry this ratio equals 3.1415926... (i.e. the ratio equals Pi) However ther are geometries where this ratio is larger than 3.1415926... (It is larger than Pi), and some geometries where this ratio is smaller than 3.14159265... (i.e. it is smaller than Pi).
Don't get me wrong, I am no expert, but if there is a continuous transformation which maps the ratio C/D from 3.141.... in Euclidian space to some number < 3 in non-Euclidian space, then there must exist a space for which C/D =3. Yes you are probably mistaken. Humans like nice symmetrical relationships involving rationals numbers and the like. I don't think it actually has any physical significance.
Hmm...that makes sense, but it doesn't seem intuitively right that a circle's circumference and diameter could ever both be rational numbers, in any sort of space. I know that isn't any sort of argument, but it just strikes me as wrong…
in spaces with nonzero curvature, the ratio of the circumference to the diameter depends not only on the curvature, but also on the size of the circle. for example, if you take a small circle on the surface of the earth, you will measure approximately 3.14 for the ratio. if you choose an entire hemisphere for your circle, you will measure 2. as you can see, the number certainly can be rational. i don t think it has an special significance.
No. As the size of the circle increases it's the diameter that increases more rapidly than the circumference. Your example is a perfect case. If you choose an entire hemisphere for your circle, the diameter will be exactly one half of the equator, which will equal exactly one half of the circumference of the circle. The ratio will be 2, not 4. But that's a nit. You're right about a rational ratio obviously being possible. (What a funny thing to see written. A "rational ratio" ought to be a tautology. Language is amusing.)
oops! you're right! i divided by radius instead of diameter! lemme fix it. thanks for keeping me honest.
Trivial note: On a sphere's surface, the ratio of circumference to diameter will be 3 when the angle subtended by the circle's diameter at the center of the sphere is about 0.93486 radians (53.56 degrees). The exact answer is not computable, being the solution of: Pi.cos(x/2) = 3x If the Earth was a perfect sphere, this would be at a latitude of about 63.22 degrees.
Thank you all for answering my question in detail. One interesting note was that pi comes up in plenty of non-circular relations, such as the uncertainty principle, et al. I am sorry that I misinterpreted what 1100f said. So although the curvature of space is different at different locations depending on the material context of that region of space, Euclidean pi is still what is used in such things as the uncertainty principle? I see now that pi=3 would have no more significance than any other whole number outcome. On a slight tangent, how many TOTAL (known) constants are there in nature and what are they? I can think of only 3 right now: pi, Planck's, and the gravitational
there are 26 physical constants in todays current modern model. Planck's constant and Newton's constant are not on that list of 26 constants, because, since they are dimensionful constants, their values depend on your choice of units, and are therefore not really constant. as for mathematical constants like pi, there are infinitely many. for example, every integer is one.
WOW Now THAT is interesting. Planck's and Newton's constants are not true constants because they exist only because we use arbitrary systems of measurement? In the hypothetical case that we convert to some system of measurement which is more objective, those constants would no longer "exist"? Is pi one of the 26? I've never looked at interger values as constants before, but I'll be damned! That is very interesting indeed. Could someone recommend to me some good books to get me more aquanited with the 26 constants (nuts-n-bolts/no fluff). . . and if anyone has the time, I'd love to see them in a concise list!
exactly. the system of units where it works like this is called Planck units. some scientists advocate replacing the metric system with Planck units. you can read some stuff about Planck units here the metric system, as well as the English system of units, is roughly based on the size of a human being, which is not a fundamental concept. no, pi is not considered a physical constant. a physical constant is one which must be measured, and cannot be calculated. since the digits of pi can be calculated, it is a mathematical constant. mathematically, you can calculate all the digits of 1, 2, 3, 4, etc. basically any number is a constant, pi just happens to be a rather useful constant. but 2 is also a useful constant, along with 3, 4, 10, 24, e, Euler's constant, Feigenbaum's constant, etc. there is no particular reason to consider pi any more fundamental than any other mathematical constant. the fine structure constant is one of the most famous ones. its value is 1/137, and it is unitless. since it is unitless, its value does not depend on what system of units you choose to use (which is, recall, completely arbitrary). this number cannot be calculated with current theories, it can only be measured in the lab. you can read more about the list of fundamental constants here
Can't thank you enough for the explanations and links. Can the 26 empirical constants be considered arbitrary in a totally different sense than the metric or English system? "The 26" (for short) are not subjectively arbitrary like our man-sized units of measure are, but are still arbitrary in the strict sense that they could have any other way. Doesn't the abstract math and geometry exist for countless self-sustaining universes which would be totally alien and divorced from ours in structure and dynamics, making our particular constants an arbitrary subset of all possible constants? Or is there something necessarily essential about our universe's constant's?
Yes, today, these 26 constants are arbitrary. However, the theoretical physicist's "wet dream" is to find a theory of everything which will explain (or predict) the values of these constant. One of the hopes is that indeed, the universe is self-consistent only with the interractions that we know and with the actual values of these constants.
yes, i think that is a fair assessment. i imagine it like this: in the master control room for the universe, there are 26 dials, and god can tune those unitless numbers to any value he wants. of course, if he doesn't get them right, life is not possible, and we will not be here to discuss it, but it would still be a mathematically consistent universe. in such a universe, the only answer to the question "why does the universe look the way it does?" is "because we are here to see it. if it looked otherwise, we would not exist to comment on it". this explanation of the universe is known as the Anthropic Principle. many physicists reject this explanation out-of-hand. in other words, we know of know way to determine the values of these constants, and no explanation why they have the values they do. all we can do is measure them with our crude instruments. on the other hand, god does not have a dial for pi. his hand is forced, mathematical consistency requires pi to have the value it has. same for all the mathematical constants. the values of these numbers are based on consistent logic, not measurement. however, all this is only based on the current modern models of physics. it is conceivable that one day we will have a theory of everything, with which all constants can be mathematically calculated. if we find such a theory, and it turns out to be an accurate description of our universe, then we will realize that those constants were not physical constants, but rather mathematical constants. we just didn't realize it because of our ignorance of the universe. consider for example, the relative heat capacity of, say, lead. 200 years ago, this was an apparent fundamental constant, inherent to the element, and there was no way to predict its value. it was one of a list of fundamental unitless constants, which was apparently just an innate property of the universe, with no possibility for explanation, only a chance for measurement. and there will a lot of numbers like this, one such number associated with every element, and in fact every chemical composition. plus electrical conductivity, thermal conductivity, absorptivity of light, density etc etc etc. hundreds and hundreds of "fundamental" physical constants. then along comes quantum mechanics, and it turns out that the heat capacities of all the elements can be mathematically calculated (at least in principle), which just a knowledge of quantum mechanics, the fine structure constant, and the mass of the electron. voilà! where once there were hundreds of constants, now there are only 26. the hope is that eventually, we will realize that there are no physical constants. rather that, in fact, god's hand was forced for all the properties of the universe. he did not fine tune any dials, rather our universe is the way it is, because it could not look otherwise. this was Einstein's hope, and the hope of most physicists seeking after a theory of everything. these days, if you believe that string theory is the theory of everything, there is some evidence that this position will have to be abandoned. string theory cannot predict a unique universe, but rather allows for many many inequivalent universes. then, to answer that question of all questions, "why are we here?" you have to fall back on the Anthropic Principle: "We are here because we are here" not all string theorists believe that we will have to fall back on the Anthropic Principle (many view it as unscientific, including me), but some due. like Lenny Susskind, one of the founding fathers of string theory. i hope that we do get a theory of everything that makes unique predictions for all the constants, and abolishes the Anthropic Principle. if string theory cannot be that theory, then perhaps we should keep looking. time will tell though. the point of the story is, what we consider a fundamental constant depends on what our fundamental theory is. right now, our fundamental theories are the standard model plus GR. the standard model lagrangian has 25 constants in it, and the GR lagrangian has one (the cosmological constant). we know that these can't truly be fundamental, since they are incompatible. they will have to be replaced with something more fundamental. whether the replacement fundamental theory will have fewer constants is the real question. (please note: i do not take this concept of a god with dials for tuning the constants literally. it is just a metaphor)
Oh, and by the way CTEBO, i have to say, your thread here is like a breath of fresh air: 1. you ask well-posed interesting answerable questions 2. you read peoples responses, and understand them 3. and then ask interesting followup questions 4. you are polite, and admit mistakes when they are pointed out in short, i like your questions a lot. i wish there were more of these kinds of discussions here.
These are exactly the types of discussions I expected when I first saw that forums like these existed, but they are, unfortunately too few and far between. I must take this opportunity to thank you for your very understandable and thought-provoking answers. So, we have 26 dials (figuratively speaking, of course), each with an entire numberline of possible settings. Some permutations or combinations (I can't think of the right word) of these possible settings yield self-perpuating universes. Now what I think I read you saying is that only a subset of those "EQ settings" are capable of yielding self-sustaining universes which are themselves capable of producing life, and then perhaps an even smaller subset of constant-combos for universes to produce conscious agents. But if different constants yield different laws of physics, then shouldn't the limitations or restictions on possibility in any given universe be specific to that universe, and so our particular universe's "necessarities for the existence of life (let alone consciousness)" would be different from some other universe's "necessities of life"? Can we say with any certainty that certain other "possible" universes would not be able to sustain life? Our conceptions of reality may be "universal" but not (I'm stretching here, I know) "omni"-versal? I think I also understood you to say that, should we devise a theory that reduces all of our physical constants (apparently arbitrary) to mathematical constants (apparently necessary), then we will have shown that our "particular" universe is not particular at all, but instead, that no other universe is possible. But doesn't even the existence of different geometries (constants aside) show that there are more possibilities of existence than the phenomena which make up our universe? It seems that all that is important is that a system remain self-constistent, meaning: to be free from the restraints-on-possibilty of other systems.
