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Rationalize Pi

Good enough for the likes of anyone
  (+2, -14)(+2, -14)(+2, -14)
(+2, -14)
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against]

If you recall your junior high math classes, a big deal was made about pi and the fact that it's irrational. Well, maybe so, but only in isolated theory. Here's the problem, though: Once you throw physics into the mix, you soon realize that can't possibly be true.

39 digits of pi is reportedly sufficient to calculate the volume of the universe to a margin of error of one atom. There must then be some (surprisingly small) number of digits that allows you to calculate the diameter of any possible circle to a margin of error of less than one Planck length. Beyond this, further precision is literally impossible, as there's no way to measure any length smaller than that. So it's meaningless to say that pi is any more precise than that, because there's no way it could be within the confines of our universe.

All we have to do is calculate that number of digits, and fix pi at whatever that works out to be (maybe throw in a few extra digits to be safe). And instantly, pi goes from a magical, "irrational" number to just a plain old constant with a lot of decimal places. Much easier to grasp for students, much easier to deal with for engineers and scientists. The only people who are going to have a hissy fit about this are geeks who love memorizing pi to thousands of decimal places, and that lobby is not as powerful as it once was.

Oh, and mathematicians, I suppose, but they're the ones who got us into this mess in the first place. Just hand them a physics textbook and tell them to get a REAL job.

ytk, Jun 16 2012

The bible makes it easy! http://www.extremel...en/Bible_errors.php
12. Pi equals exactly three. [Klaatu, Jun 16 2012]

Mathematical Infighting Promotes Tau instead of Pi http://tauday.com/tau-manifesto
"No, really, Pi is wrong!" [zen_tom, Jun 17 2012]

Considering n-dimensional Spheres http://en.wikipedia...me_and_surface_area
Yup, when working out the volume/circumpherence, surface area etc, pi works in pretty much all the dimensions. [zen_tom, Jun 19 2012]

Reduced Plank Constant = h/2*Pi http://en.wikipedia...iki/Planck_constant
Hmm, so the plank length itself is defined in terms of Pi! Your move [ytk] [zen_tom, Jun 19 2012]

Attempt made to rationalize pi in 1897 http://www.straight...-saying-pi-equals-3
There is also a way around the biblical link provided by [Klaatu] [AusCan531, Jul 21 2012]

vi hart pie https://www.youtube...watch?v=5iUh_CSjaSw
[pashute, Jan 21 2016]

vi is still wrong https://www.youtube...watch?v=jG7vhMMXagQ
[pashute, Jan 21 2016]

Field's metal https://en.wikipedi...iki/Field%27s_metal
for [Cuit_au_Four] [notexactly, Jan 21 2016]

[link]






       So, not a rant, then?   

       Yes, for most practical purposes you only need pi to so many places (although there are some real- world functions that involve high powers of pi, for which you'd need more digits to get a reasonable accuracy in the result). I am pretty sure that carpenters, engineers and architects use pi to just a few digits.   

       So, what exactly is your point? Is it to define an approximation of pi which is good enough for practical porpoises? If so, this is widely baked.   

       Also, who wants things to be such that e-to-the-i- pi= 1.000000000000000000000037?
MaxwellBuchanan, Jun 16 2012
  

       My vast ignorance leads me to ponder inexpertly on the Nyquist- Shannon sampling theorem and wonder if you need a few more decimal places, and also that it may not fit exactly into multiples of ten, thereby leading to some weird recurring decimal or something.   

       Wouldn't it be easier to define other numbers as ratios of pi and make them all approximate?
nineteenthly, Jun 16 2012
  

       //within the confines of our universe//   

       When did we get so provincial?
pertinax, Jun 16 2012
  

       Not to mention all the silly lopsided circles we'd end up with.
Alterother, Jun 16 2012
  

       A mathematician told me yesterday, "Physics and mathematics are fundamentally different. In math we use definitions." Hm...
sqeaketh the wheel, Jun 16 2012
  

       // e-to-the-i- pi= 1.000000000000000000000037//
I believe you mean e-to-the-i- pi= -1.000000000000000000000037
Or were you simply 'rounding off' the minus sign?
sqeaketh the wheel, Jun 16 2012
  

       I think I put the minus sign between the e and the to.
MaxwellBuchanan, Jun 16 2012
  

       While our Observable Universe (at least what we observe so far) appears to be a large finite object, such that this Idea could be practical as far as real-world uses are concerned, there remains the possibility of some sort of infinite expanse outside the boundaries of what we currently observe.   

       If we ever obtain some hints that there is more Out There than just the so-far-Observable Universe, especially if there is an infinity of "more", then this Idea becomes wrong.
Vernon, Jun 16 2012
  

       If your Observable Universe appears to be finite, you've clearly only been observing it in three dimensions.   

       Even if you wish to continue hindering yourself in such a manner, I'd say the fact that we're constantly expanding the scope of our observation is a pretty solid indicator that there is indeed more Out There, and our well-established inability to observe a point of origin speaks to its probable infiniteness.
Alterother, Jun 16 2012
  

       Hmmm...I think this new Pi needs a slight name change:   

       Crusty Pi; maybe Baked Pi.   

       On another point, if we were to count in Base Pi or something like that*, would the irrational numbers be removed?   

       * Completely not thought through.
Ling, Jun 16 2012
  

       //So, not a rant, then?//   

       Hardly. What I'm proposing is nothing less than the unification of the hardworking, proletarian sciences like physics with the aristocratic, ivory tower discipline of mathematics, thereby improving both. Math that doesn't reflect and acknowledge the fundamental reality of our existence is meaningless and completely arbitrary.   

       //Also, who wants things to be such that e-to-the-i- pi= 1.000000000000000000000037?//   

       Simply define e in terms of pi such that the equation works out nicely. This has the beauty of unifying theory and practice, as it would define e to the same degree of precision as pi, which is to say "more than enough".
ytk, Jun 16 2012
  

       Never did understand 'what' pi was until I saw the gif about a year ago.
It was just some random number we had to memorize to make the circle math work.
  

       Sad.   

       // Never did understand 'what' pi was //   

       Don't feel bad; Euclid didn't get it, either.
Alterother, Jun 16 2012
  

       When in doubt, consult the bible. <link>   

       1 Kings 7:23 gives the absolute value if pi.
Klaatu, Jun 16 2012
  

       //The bible makes it easy!//   

       Yes, yes, we've all seen this and it's ever so amusing. But it's easily explainable by the fact that the vat was unlikely to be a perfect cylinder. If it sloped outward like a tumbler then the diameter measurement would have been taken at the top but the circumference would have been taken at the bottom. In fact, the ancients knew that you could calculate the circumference of a circle from its diameter and vice- versa, so the only reason to provide this information would in fact be to describe the vat's shape as an inverse truncated cone.
ytk, Jun 16 2012
  

       // truncated cone //   

       "Frustum" (q.v.)
8th of 7, Jun 16 2012
  

       //7:23 gives the absolute value//   

       7:23 is 3.28571429 - wayy out.   

       [ytk] either you have both tongues in your left cheek, or you are missing the point of pi.   

       One of the beautiful things about pi is that it expresses something fundamental about circles, regardless of where they are or even whether they exist or not. The fact that so simple and fundamental a number cannot be written down is, to me, also very beautiful and strange.   

       What I find even stranger about pi is to ask if it could be different, in the same way that (say) the charge on an electron could be different, in a different universe.
MaxwellBuchanan, Jun 16 2012
  

       I've always found that 355/113 was close enough for most purposes.
csea, Jun 16 2012
  

       // if it could be different, in the same way that (say) the charge on an electron could be different, in a different universe. //   

       In his novel 'Eon', Greg Bear proposed that changes in the local value of pi would indicate singularities and dimensional wormholes.
Alterother, Jun 16 2012
  

       //changes in the local value of pi would indicate singularities and dimensional wormholes.// That's interesting, albeit complete poppywash. Carl Sagan did something similar in Contact, where data had been built into the value of pi.
MaxwellBuchanan, Jun 16 2012
  

       Yes, that's why I made sure to mention the word 'novel'. I suppose I meant it as some sort of evidence supporting the natural artistry of pi. Something like that.
Alterother, Jun 16 2012
  

       //you are missing the point of pi//   

       It comes right after the "3".   

       //One of the beautiful things about pi is that it expresses something fundamental about circles, regardless of where they are or even whether they exist or not.//   

       The problem is that this is only actually true if you apply mathematics as separated from absolutely all other scientific knowledge. So, it's really more accurate to say that it expresses something fundamental about circles that couldn't possibly exist. But just like the discovery of quantum physics forced us to rethink what we knew of the universe based on Newtonian physics, it is time to revise our understanding of mathematics based on our enhanced knowledge.   

       In our universe, there is only so much precision possible. Take Zeno's paradox of Achilles and the tortoise as an example. Prior to the discovery of the Planck length as the smallest possible measurement of distance, Zeno's conclusion (that motion was impossible) was nothing more than a reductio ad absurdem of his entire philosophy. And yet, with our current understanding of quantum physics, we can say that perhaps he was on to something after all. His paradox was conditioned on the fact that there is no minimum length. Once we've shown that there is, in fact, such a minimum distance, the paradox neatly resolves itself. Since there is no longer an infinite number of points between Achilles and the tortoise, he /can/ in fact overtake the tortoise in a race. Without realizing it, Zeno effectively proved the existence of the Planck length long before anyone knew what it was. And today, we can no longer take for granted what we once assumed to be fundamental truths about physics.   

       So it must also go with mathematics. There is a point beyond which one simply cannot make a more precise measurement. I grant that our perception of the size and scope of the universe may be flawed, and so it may be impossible to make an exact calculation based on our limited knowledge, but the principle still holds that pi /must/ actually be some rational number in our universe. Simply put, our understanding of mathematics, like Zeno's understanding of physics, was incomplete.   

       You make the point yourself when you ask //if [pi] could be different, in the same way that (say) the charge on an electron could be different, in a different universe//. It certainly could. In some other universe where the speed of light or the gravitational constant is different, the Planck length, and therefore pi, must be different as well. Once we accept that mathematics is not necessarily multiversal, we have to accept that what we know of mathematics only applies within our own universe. And we therefore have to apply the rules of our universe to mathematics. Trying to prove that there are an infinite number of decimal places to pi is as futile as trying to prove the existence of God.
ytk, Jun 17 2012
  

       Pi could be different if certain non-Euclidean geometries applied, for instance hyperbolic geometry would entail that it was a variable rather than a constant, although it would still be a limit of some kind.   

       Concerning the Bible, this probably reflects the lifestyle and the resultant approach to mathematics. Decimal fractions were unknown at the time, so pi could at best only be approximated as a ratio. The Bible also claims insects have four legs, which is a similar phenomenon.
nineteenthly, Jun 17 2012
  

       //Take Zeno's paradox of Achilles and the tortoise//   

       The argument that the Planck length solves this paradox is not really true. The paradox resolves even if there is no smallest possible length: it resolves through pure mathematics, which shows that the infinite series (1+1/2+1/4...) does not sum to infinity.   

       The more general point you raise is whether pi should represent the mathematical limit of a circle's circumference, or the nearest physical value that can be attained in our own (finite and granular) universe.   

       However, an approximation to "fit" our reality will differ on a case by case basis. If my circle has a diameter of 1 femtometre, then its value of pi will differ from the abstract pi by quite a large amount (either more or less than the abstract pi). If it has a diameter of 1 billion light years, then its value of pi will be much closer to the abstract value.   

       In other words, the value of "physical pi" depends on the size of the circle, whereas the "abstract pi" does not.   

       Regarding the dependence of pi on G or on the speed of light: this applies only to the "physical pi". (Also, to calculate the dependence of the "physical pi" on G or C, you need to use the "abstract pi".)   

       Regarding [19thly]'s point that non-Euclidean geometries will change pi, again this applies to a physical pi but not to an abstract pi, which is defined for a flat geometry.   

       Finally, it's worth pointing out that "abstract pi" crops up in many non-circle contexts (as a simple example, it's the limit sum of several series).   

       Anyway, I guess my point is that I don't see your point. Pi exists as a mathematical construct, just as the square root of 2 exists as a mathematical construct. You're arguing that for practical purposes we just need an approximation (which might vary depending on circumstances), which is fine but so what?
MaxwellBuchanan, Jun 17 2012
  

       // The Bible also claims insects have four legs, which is a similar phenomenon.// To be fair, spectacles hadn't been invented at the time.
AbsintheWithoutLeave, Jun 17 2012
  

       And, insects generally do have four legs. Plus a couple extra, sure.
pocmloc, Jun 17 2012
  

       // To be fair, spectacles hadn't been invented at the time.// Yeah, but some pretty big beetles had.
MaxwellBuchanan, Jun 17 2012
  

       If the bible included just sixteen digits of pi, the atomic numbers of a single element, or an accurate description of earth's place in the universe, I would be more inclined to respect it as a document of divine and transcendent inspiration. Clearly if it is a divine document of super human knowledge, it would have at least a few helpful hints thrown in, for skeptics sake. Also, apparently god needs and editor in the worst kind of way.
WcW, Jun 17 2012
  

       //If the bible included just sixteen digits of pi, the atomic numbers of a single element, or an accurate description of earth's place in the universe//...   

       ... then Occam's razor would guide one to the simplest interpretation, namely that humans will invent time travel one day, and will have written the bibble as a joke.
MaxwellBuchanan, Jun 17 2012
  

       look, that's magic enough for me. Time travel is a good enough excuse for a religion. The bible as a the wrote word of an all knowing deity is like kind of time traveling prank that you might get if the pranksters were from a future where all that remained of society was illiterate subsistence shepherding, functional time machines, and complicated unfunny pranks.
WcW, Jun 17 2012
  

       //the value of "physical pi" depends on the size of the circle, whereas the "abstract pi" does not//   

       Not quite. The ratio of one particular circle's diameter to its circumference may vary slightly, but the constant pi remains the same throughout the universe.   

       //Anyway, I guess my point is that I don't see your point. Pi exists as a mathematical construct, just as the square root of 2 exists as a mathematical construct. You're arguing that for practical purposes we just need an approximation (which might vary depending on circumstances), which is fine but so what?//   

       In our universe, there must be some possible largest circle, and we know there's some smallest measurement. While we can, in theory, calculate what pi /would/ be if this were not the case, that makes no sense to do in our universe. It is as meaningless as talking about half a Planck length. While you can theoretically conceive of such a thing, it simply cannot exist. So, there must be some fixed, rational value "pi" that is sufficient for use in any calculation that is possible in the universe. I'm not arguing for an approximation—this number /is/ pi in our universe. Anything more "precise" would, in fact, be an approximation, not the other way around. Insisting that pi is nevertheless infinite as a "mathematical construct" is tantamount to a denial of current scientific knowledge. It would be like a Newtonian physicist simply denying the existence of quantum physics.   

       What's the point of all of this? Well, I believe we should teach our students with an eye towards a holistic model of the universe rather than supplying them with building blocks that are inadequate to progress to further levels of scientific understanding. Ask any grade school student whether the Sun revolves around the Earth, and you will get an emphatic “No, it's the other way around!” But this is, of course, wrong: The correct answer is that without specifying a frame of reference, either view is valid. But this incorrect “knowledge” has been so deeply embedded in the minds of students that significant effort is required to unlearn it in order to progress to a higher level of understanding of physics; most students never do so, and remain in ignorance throughout their adult lives.   

       In the same way, telling students that, under our old theoretical understanding of mathematics, pi was considered irrational, but given our knowledge of quantum physics we can show pi to be rational, and even know approximately what that rational value is, is not only more truthful in that it presents a broader view of the sum total of human scientific knowledge, but it opens up the door to a greater understanding of the universe at an earlier age. And all we have to give up is our outdated, security blanket belief that we can have such things as “mathematical constructs” apply to a universe that has a physical form.
ytk, Jun 18 2012
  

       //this number /is/ pi in our universe//   

       Well, no it isn't. It would be the practical value of pi for the largest possible circle in the universe of the size we currently know it to be.   

       If our measure of the universe were slightly wrong, then the value of your pi would be wrong.   

       If the universe is expanding, your pi would also be changing.   

       If the universe is not spherical, then your value of pi will depend on the orientation of the circle (since this will determine its diameter).   

       If you are interested in circles smaller than the diameter of the universe, your value of pi will again be different.   

       In short, your pi will be a variable, not a constant.   

       If it makes you happier, then of course we can define a new value of pi (which we might call, say, chi to avoid confusion), defining chi as the ratio of the circumference to the diameter of a circle which just fits into the universe as we understand it today, where both measures have been rounded to the nearest planck length before computing their ratio. Would that do?   

       Perhaps it would clarify things for me if you could explain how and where this new value of pi/chi will be used, by means of an example.
MaxwellBuchanan, Jun 18 2012
  

       The Greeks had an issue with irrational numbers as well - but that problem only occurs when you cling too strongly to the idea that number is the same thing as cardinality.   

       The problem with doing this is that you're going to have to provide a whole bunch of ugly standardisation lists for all the other irrational numbers, root2 (and all other non-perfect square roots) e (and an infinite number of other logarithms) - that means you've got an infinite list of irrational numbers to standardise, but then if you're going to avoid any references to infinity, then you're going to need to round off and effectively outlaw an infinite number of numbers, which since you're outlawing the concept of infinity, puts you in somewhat of a sticky predicament.
zen_tom, Jun 18 2012
  

       [Max]: None of that matters. We may not know exactly what the value of pi is at any given time. We may not ever know. But we do know that it is some rational constant, that depends on the size of the universe. If the size of the universe changes (assuming it ever does—we aren't sure about that yet), the value of pi changes. But it's not a variable, because its precise value is entirely dependent on the size of the universe at any given time. I believe you are confused about one thing, though. Pi doesn't change based on the size of the circle you are working with. Pi is simply the appropriate value for the largest circle possible. In real life, the ratio of a circle's circumference to its diameter simply isn't precisely tied to pi. The “mathematical abstraction” is wrong, and it's even more wrong than the actual value of pi (although neither one can be exactly precise all of the time).   

       So again, the point of this is that we shouldn't teach people something wrong just because it's easier than teaching something right. When we discovered quantum physics, we didn't just say “Oh, let's just keep piling on new theories of Newtonian physics to explain the universe, because classical physics should remain ‘pure’ in some way”. Why should we do so with mathematics? A theory that is demonstrably wrong (even if we can't necessarily demonstrate that to be the case with today's technology, a simple gedankenexperiment will suffice) has no place or purpose in our scientific body of knowledge.   

       Regardless of what it actually is at any given time, or whether its precise value changes over time, pi /is/ rational. Believing otherwise requires stubbornly clinging onto an outdated view of the universe. Just like we no longer teach about the aether, it's time to consign the irrationality of pi to the dustbin of scientific history. For what purpose, you ask? The advancement of knowledge is a goal unto itself.
ytk, Jun 18 2012
  

       //The Greeks had an issue with irrational numbers as well//   

       I'm not making any statement here about irrational numbers in general. Just that pi, as we understand it, simply /cannot/ be irrational when taken in context with the totality of human scientific understanding. There's no reason why root 2 couldn't be irrational, since it's not based on any geometric property of the universe.
ytk, Jun 18 2012
  

       //There's no reason why root 2 couldn't be irrational, since it's not based on any geometric property of the universe.//   

       Oh yes it is.   

       Take a right-angled triangle (i.e. one with one of the angles equal to Pi/2) with the two sides either side of the right-angle with a length 1. The slopey side will then, quite physically, have a length of root 2.   

       If you fiddle about with one irrational number, you have to be prepared to mess around with all of them. And since there's an infinite number of them, and all of them need to "add up" and continue to interrelate with one another, that's a lot of work you're proposing. By the time we've gotten around to filing off the tail ends of all those numbers, the universe will have been long-gone.   

       The same thing goes for all the other irrational numbers - there will be a physical analogy for each of them, picking on Pi alone isn't an option.
zen_tom, Jun 18 2012
  

       No, because root 2 is not defined in terms of geometry. If you perform the division in question in our universe, you arrive at some rational number that is /approximately/ root 2, which is irrational. It's fine for numbers to be irrational, except with regards to physical measurements of distance. Since pi is defined by measurements of distance, it cannot be irrational.
ytk, Jun 18 2012
  

       If the Bible included sixteen digits of pi, Hebrew numerical notation would have to be completely different. It can't express irrational numbers and even has a problem with integers because it avoids accidentally spelling out the name of God with the letters it uses to write them by expressing fifteen and sixteen without using the letter for ten. To express sixteen digits worth of accuracy for pi, it would have to describe it as a ratio in words, and i'm almost sure it couldn't express numbers that high. I see its expression of pi as an integer as quite appropriate since its expression of morality is also wildly inaccurate, at least in the Old Testament.
nineteenthly, Jun 18 2012
  

       Again, this “the Bible says pi is 3” nonsense has been debunked. I'm no fan of religion, but this gratuitous Bible- bashing irks me.
ytk, Jun 18 2012
  

       //No, because root 2 is not defined in terms of geometry. //   

       It depends on how you choose to define Pi or Root 2. But if you wished, you could choose to define Pi as the limit of an infinite series without any recourse to geometry.   

       Similarly, Root 2 could be constructed in a similar way. No geometry required.   

       They'd still both exist, and they'd still be *applicable* to geometric functions, but they'd still be irrational.   

       This may be cheating, but you can define Pi as a repeating (alternating) series of root 2's. Even if that doesn't meet the criteria for differentiating between a "physical" thing and an abstract one, it does further the case that these two are very tightly linked together.   

       Likewise, Euler's formula provides a direct link between Pi and the exponential constant, e. That formula is couched within Complex Numbers, which are (I suppose) inherently geometric in their composition. But again, if you mess with Pi, you also mess with e, and if you alter the value of e, then exponentation goes out the window, and by direct association multiplication, addition and everything else.   

       More strongly, since each of these things (where "things" means *any* number at all) show that each number (rational or otherwise) are just different ways of talking about the same thing.   

       Since you can extend this out to any other number, it follows that they *must* have the values that they have, or otherwise 1 doesn't equal 1 anymore.
zen_tom, Jun 18 2012
  

       How about redefining Pi as 1, and then defining a new number, Ip, with a value of 0.318309886....   

       Then instead of the circumference being = 2 * Pi * radius,   

       We could just say that the radius is = Ip * Circumference / 2.   

       Sorted.
zen_tom, Jun 18 2012
  

       But [bigsleep], why do you need reams of exegesis on a whole load of Iron age mythical texts to lead you to the conclusion "be nice"?
pocmloc, Jun 18 2012
  

       //The bible also claims that insects have four legs.//   

       I don't seem to recall formal scientific taxonomy as having been invented before biblical writings.   

       Let pi remain irrational. po, on the other hand, should stay completely rational. We need a sane po.
RayfordSteele, Jun 18 2012
  

       Leviticus eleven twenty-three says, in the KJV, "But all other flying creeping things, which have four feet, shall be an abomination unto you." It refers to animals which fly and walk on all fours, shortly after mentioning locusts, beetles and grasshoppers, and the Hebrew does use a form derived from its word for "four". To my mind, that's a reference to insects. The Israelites may not have had a modern taxonomy in mind and probably thought of all small invertebrates as similar, but there are no other animals which both fly and have more than two walking limbs. It's sometimes claimed that the hopping legs are not considered legs, but the text also refers to beetles. However, i agree that the idea that the Bible is supposed to be making scientific assertions is anachronistic, and presumably the same applies to mathematics. Concerning the precise nature of the moral code, it's memetic and probably worked better in that set of circumstances than it does here and now, but there were other cultures in similar circumstances which were, for example, less opposed to gender role blurring, for instance in Pre-Columbian North America.   

       Anyway, circles. Pi makes sense in Euclidean geometry as a constant, but geometry is not Euclidean. It's a very important constant but doesn't quite mean what it seems to mean, because in this Universe, any circle, even if perfect, would be slightly off, since the space in which it is drawn is not flat.
nineteenthly, Jun 18 2012
  

       // Pi doesn't change based on the size of the circle you are working with. Pi is simply the appropriate value for the largest circle possible//   

       Ah, OK, so your "pi" is simply a theoretical value for a circle with an arbitrarily-chosen and imprecisely-known size. That's obviously better than a theoretical value which is independent of the size of the circle and is known with arbitrary precision, but can you just remind me exactly why?   

       // The “mathematical abstraction” is wrong// No, the mathematical abstraction is right, as a mathematical abstraction.   

       But, whatever. If you want to define a new type of "bricklayer's pi", that's fine, although it would be churlish to insist on giving it the same Greek letter as the one everyone already uses for the mathematical pi.   

       It would also be worth bearing in mind that you'll need to define new, pragmatic values for the square roots of all the non-square numbers.   

       At the same time, you'd need also to redefine a lot of the of the rational numbers, based on commensurability of the two numbers. I'm thinking of things like "1/3rd", which is really just a mathematical symbol (like pi) for a number which cannot be written out. You'd have to divide the diameter of the universe by 3, and then see how much of a Planck unit you had left over, and then...   

       Moreover, we might find that the diameter of the universe is not an even number of Planck units, in which case the "Halfbakery" will have to be given an inconveniently long name.   

       One last question: there are plenty of irrational mathematical constants which have a less direct relationship to physical objects than pi does. How do you propose to redefine them?
MaxwellBuchanan, Jun 18 2012
  

       Irrational numbers exist wherever one tool for measurement is laid up against another. Your tool for measuring circumference is different from your tool for measuring radius, yielding an irrational number. This fact that different tools of measurement do not perfectly match one another, cannot perfectly match one another, is a perfectly valid observation. It helps, not hinders, the application of mathematics to reality. Hate it all you want, 10 planks is a unit of measure, a 10 plank circle will appear to have an irrational radius. Yes, irrational.
WcW, Jun 18 2012
  

       Also, while tou is useful it is simply 2pi. So I'm not going to get my panties in a twist about it.
WcW, Jun 18 2012
  

       If it's two*pi, shouldn't it have twice the number of legs rather than half?
nineteenthly, Jun 18 2012
  

       Read the manifesto: The legs have a line across the top, so they are in the denominator. Hence half the number of legs. One early paper on this topic proposed a 3 legged pi-like symbol, but adoption seems much less likely if it is a symbols that isn't available in most font packs.
scad mientist, Jun 18 2012
  

       // Trying to prove that there are an infinite number of decimal places to pi is as futile... // I agree completely that it is futile to probe the depths of pi or any of the irrational numbers that we find useful in day-to-day life. I don't think there is anything more of interest to be learned along that avenue, but I completely disagree that we should declare that pi or any of these other values is a rational number. Pi (or Tau) being irrational is not some magical commentary on the ratio between the dimensions of a circle. It just demonstrates one limitation of our very useful system of numerical representation. For practical purposes we overcome this shortcoming by making sure we use enough digits for the task at hand. For theoretical purposes we call it irrational and use a different representation that isn’t as convenient to work with.   

       If there is some confusion caused by the emphasis placed on irrational numbers in junior-high math classes, that ought to be addressed with better teaching, not by attempting to redefine a concept to hide the limitations of our numerical representation system.
scad mientist, Jun 18 2012
  

       hmmm, so, visually if you roll a circle it will roll past three point one four yadda-yadda other circles of the same size to complete one revolution.
Is there any number of times it can roll to come up with a number of revolutions which give an answer that is a whole number?
  

       Is that even a thing?   

       The answer will always be irrational. All non incremental measurements are irrational. That's the point! (Planks be near damned!)
WcW, Jun 18 2012
  

       //Is there any number of times it can roll to come up with a number of revolutions which give an answer that is a whole number?//   

       That is why pi is so beautiful. No matter how large the circle, you can never make a measuring stick which will fit a whole number of times into both the radius and the diameter, no matter how small a measuring stick you try.   

       The same is true of the square root of all integers (except the perfect squares like 1, 4, 9...), and indeed for most numbers.
MaxwellBuchanan, Jun 18 2012
  

       //That's obviously better than a theoretical value which is independent of the size of the circle and is known with arbitrary precision, but can you just remind me exactly why?//   

       Because it's /correct/, that's why. If a theoretical value cannot possibly exist in our universe, then it's pointless to talk about it. You can theorize all you want about God, or the arbitrary precision of pi, but science as we know it is unequipped to discuss such things. Simply put, by insisting pi has an infinitely precise value, or even a more precise value than it could possibly have, you are being unscientific. Far be it from me to deny you your religious beliefs about the nature of pi, though.
ytk, Jun 18 2012
  

       [ytk]:
1: You keep going on about measuring, and not being "abstract". Maths is not about measuring; it is abstract - that's the whole point of maths!
2: As [MaxwellBuchanan] has pointed out, pi shows up all over the place completely unrelated to circles, so your "circle measuring" premise is flawed.
3: In using the Planck length as your "unit", the absolute size of the circle would matter, as a smaller circle would have a (slightly) smaller number of steps. (Eg: if I use the radius of a circle as the step "unit", I get a pi value of 3; 6 steps of the radius around the circle, ie: a hexagonal approximation.)
Conclusion: Use an approximation for measuring, but theoretical maths uses the theoretical version, which is irrational.
neutrinos_shadow, Jun 18 2012
  

       //insisting pi has an infinitely precise value, or even a more precise value than it could possibly have, you are being unscientific.//   

       Pi is a mathematical abstraction. I'm not insisting that there's any circle in the universe with any degree of precision. I am simply observing that the series 4*(1-1/3+1/5-1/7...) converges toward a limit, which most people call pi.   

       I also note that the ratio of a real, physical circle's circumference to its diameter approximates to pi.   

       OK, I'll make you a deal. If you want to use the name 'pi', you can have it. I'll call mine 'Bob'. So, now we can both be happy.   

       Now that we've resolved that one, can we start on phi? I think it's only fair that I get to keep the name "phi", but I think it would be OK if you called the bricklayer's version "fi".   

       Also, can you tell me what value you'd like for a third?
MaxwellBuchanan, Jun 18 2012
  

       //If a theoretical value cannot possibly exist in our universe, then it's pointless to talk about it.//   

       Please Google "computable numbers". [-]
Wrongfellow, Jun 18 2012
  

       A few observations:   

       [Chronicles 4:2] Also he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about.   

       Looks like 3, to me   

       How do we know that pi is constantly associated with what we believe to be circles and spheres throughout the universe? We are beginning to understand that the universe is possibly a toroidal (though not in the sense we know a toroid) shape and may be subject to multi-dimensional distortions we cannot even visualise, let alone describe mathematically.   

       In primary school, through which I passed before the advent of calculators and personal computers, we approximated pi with 22/7 = 3.142857 recurring. Near enough for 10 year old kids to use. Then in high school we used 355/113 = 3.2415929. Again, near as you'll get without picking all of the nits off the monkey for something to do.   

       I'm not sure why people are so fascinated with pi. 1/3 and 2/3 give you infinite decimal progressions *and* they're ones we can all remember to as many places as we have time to recite. Knowing pi past 6 places is largely pointless, though I have a mnemonic to recall it to thirty places. Beyond that it's pretty much academic.
UnaBubba, Jun 18 2012
  

       //How do we know that pi is constantly associated with what we believe to be circles and spheres ... the universe is possibly a toroidal//   

       First of all, me and [ytk] have agreed that pi will hencefroth be known as Bob, leaving "pi" to be used for an approximation of Bob.   

       That said, it doesn't matter what shape the universe is. Pi is defined for circles in flat, two- dimensional space, even though we live in three dimensions. (In fact, mathematicians don't like the circle business to define pi, and it's instead defined in terms of angles and cosines, but that's a hamster of an entirely different lineage.)
MaxwellBuchanan, Jun 18 2012
  

       //Pi is defined for circles in flat, two- dimensional space, even though we live in three dimensions//   

       Exactly. Multiple dimensions and non-Euclidean planes stuff it up. There are always radians (rad), which are so much more predictable that pis.
UnaBubba, Jun 18 2012
  

       Well, I'm glad we can finally agree to disagree, [Max], but I still don't see why this is so hard for you to understand. Finding the actual value of pi is easy. You simply measure the circumference of the entire universe in Planck lengths, divide by its diameter in the same units, and, as they say, (the limit of the series 4*(1-1/3+1/5-1/7...))'s your uncle.
ytk, Jun 18 2012
  

       Are you asserting that the Planck length is somehow indivisible, [ytk]?
UnaBubba, Jun 18 2012
  

       //Are you asserting that the Planck length is somehow indivisible, [ytk]?//   

       I'm not asserting it. I'm stating a widely known scientific fact.
ytk, Jun 18 2012
  

       1.616199 × 10-35 metres is the length of a Planck unit.   

       Therefore 8.080995 x 10-36 metres would be half of a Planck unit, right?
UnaBubba, Jun 18 2012
  

       //Therefore 8.080995 x 10-36 metres would be half of a Planck unit, right?//   

       Wrong.
ytk, Jun 18 2012
  

       OK, please explain why I'm wrong.   

       My understanding is that there may be an apparently insurmountable obstacle in observing processes that occur within a Planck time at this point but there is no way of knowing whether a process that occurs within the elapsement of a single Planck unit is not possibly divisible into process steps using technology still to be forwarded (reversed?) to us from my contacts in the future.
UnaBubba, Jun 18 2012
  

       //Wrong// - Wow. Nice flat statement there, [ytk].   

       Particularly ironic for being in the context of taking a provable fact and popularly replacing it with a falsehood, just because it is more convenient to teach our children lies.   

       (Were you to calculate that one of the bones up there was mine, you'd be correct to a very high - nay, even infinite - accuracy.)
lurch, Jun 18 2012
  

       I have no real place in a math battle--other than the observation bunker, perhaps--but this one gets my goat as well.   

       Pi is pi. It is inviolate.   

       Just because we never calcuate it beyond a dozen decimal places for any practical use does not give us the right to declare it rational. It's not a theory or a formula or even a law, it's a _fundamental property_. We have no more right to do that than we have to declare the Sun the center of the Universe or to declare the proton indivisible.
Alterother, Jun 18 2012
  

       //Wow. Nice flat statement there, [ytk].//   

       I had hoped that this would prompt [UB] to research the Planck length on his own, and discover /why/ his statement was wrong. The information isn't hard to find, and you can find better and more thorough explanations of it via a quick Wikipedia or Google search than I could possibly offer here. However, in retrospect, the statement probably came across as a peremptory dismissal, which is entirely my fault.   

       Accordingly, here is an explanation for you, [UB]: From the Wikipedia article on Planck length: “…the Planck length is the length scale at which the structure of spacetime becomes dominated by quantum effects, and it would become impossible to determine the difference between two locations less than one Planck length apart.”, and “In string theory, the Planck length is the order of magnitude of the oscillating strings that form elementary particles, and shorter lengths do not make physical sense.”   

       In other words, it makes no sense to talk about “half a Planck length”, because if two points were less than a Planck length apart from each other, it is physically impossible to distinguish them from each other. Not really hard, or beyond our technological capability to do so—it's impossible, and therefore they're really the same point. It has no meaning in our universe for two discrete points to be less than a Planck length apart. It's one of the weird things about quantum physics that you just kind of have to accept because it can't possibly be otherwise.   

       So, it's wrong to talk about “8.080995 x 10-36 metres”, because such a distance can't exist. You can't move half a Planck length, nor can you measure it, because it doesn't exist. The only distance less than a single Planck length is 0.   

       //Just because we never calcuate it beyond a dozen decimal places for any practical use does not give us the right to declare it rational.//   

       If you take any given circle that could possibly exist in our universe, and calculate the ratio of its circumference to its diameter, you will get a rational number. The only way to get an irrational number is to use a circle that's not bound by the laws of our universe. Okay, fine, but then we have to say that mathematics is not bound by the laws of our universe. But it is also theoretically possible that mathematics works differently in some other universe—we simply don't know enough to be able to say. It would then follow that mathematics /is/ possibly bound by the laws of the universe it's in. So, there are only two conclusions we can draw from this:   

       1) Mathematics is bound by the laws of our universe, and therefore pi is rational (though we may not know precisely what it is).
2) Pi is not rational, and therefore math is not bound by the laws of our universe, and must be the same in every universe.
  

       Since concluding 2 would lead to a contradiction (because math may be different in other universes), then the only conclusion left is 1, and pi must be rational. Actually, what we can conclude is that, if math can be different in another universe, pi must be rational in this one. However, since our current thinking points to the possibility that math is different in other universes, we have to at least allow for the possibility that pi is rational in this one.   

       //It's not a theory or a formula or even a law, it's a _fundamental property_.//   

       So is the Planck length. It seems what we have is a case of dueling fundamental properties.   

       //We have no more right to do that than we have to declare the Sun the center of the Universe or to declare the proton indivisible.//   

       Well, depending on what you mean by “the center of the universe”, we can declare any point to be its center, including the Sun if we so choose (this gets back to what I mentioned earlier regarding the Sun orbiting the Earth or vice versa). There is no “absolute” frame of reference that can be said to be the true “center” of the universe. From the point of view of the Sun, it is stationary and the entire universe moves about it. This is true for any frame of reference. Einstein demonstrated this with the special theory of relativity. As for the proton, it may or may not be indivisible. The Planck length, however, is—and that's the crux of my argument.
ytk, Jun 18 2012
  

       So are you claiming 2, that math must be the same in every universe?
ytk, Jun 19 2012
  

       There are no perfect circles, perfectly flat surfaces and so forth in physical reality and when we say a real object is spherical or make other statements of a similar kind, we are always slightly wrong. An analogous process takes place when we use numerical notation, in that it takes us away from the actual numbers and we get bogged down in the patterns we can make with them, which are often confused with the numbers themselves. However, those patterns are still mathematically interesting. Both those issues apply to pi because not only are there no real objects whose perimeter is exactly pi times any of their various diameters (since they are not perfect circles anyway, they have a large number of diameters), but also the string of symbols we end up with when we try to calculate that ratio is to some extent an artifact of our system of numerical notation. Having said that, that string of symbols has various interesting features such as being arbitrarily close to a "normal", that is, truly random, sequence. Those features of pi are useful. That particular feature makes it useful as a source of random numbers, for instance, though there are doubtless better ones. That means that pi has a life of its own. Similarly, a rational approximation of pi might have its own interesting mathematical features. Both of them are real, incidentally, in a non- mathematical sense of the word.
nineteenthly, Jun 19 2012
  

       // math may be different in other universes//   

       That statement completely misses the point of mathematics, and in particular of mathematics over the last century or so.   

       Mathematics is a game, like chess. We start by choosing a set of axioms - a minimal set of statements; these are equivalent to the starting positions on the chess board. We also choose a set of rules (formal logic) by which to manipulate statements; these are the equivalent of the rules of chess.   

       From the axioms, by applying the rules, we derive theorems. This is equivalent to saying that from the starting positions on the board, by applying the rules of chess, we arrive at a position.   

       The axioms are to some extent arbitrary (there are other sets of axioms you can choose). The rules of logic are also somewhat arbitrary. However, those axioms and rules seem to give results (games) that we find interesting and which, in some cases, are useful. So, that is how we choose to play the game.   

       So, saying "maths may be different in other universes" is like saying "chess may be different in other universes". As long as I define what I mean by "chess", it's independent of the universe in which I play it.   

       If, in another universe, pawns can move three squares instead of one, then that's fine - it's just not my version of "chess".
MaxwellBuchanan, Jun 19 2012
  

       It might not be fine at all, in fact, because it might make the game unplayable - haven't thought it through, but another variant - a piece with combined Queen's and Knight's moves - ruins the game. In the same way, some versions of maths might ultimately just not work, and we might never know which.
nineteenthly, Jun 19 2012
  

       Well, [Max], I just think that by only considering the entire universe, you're taking a very narrow point of view.   

       Consider that our universe has a fourth dimension—time— that can't be mapped relationally to the three physical dimensions. Pi is useful for computing two dimensional areas and three dimensional volumes, but not for four dimensional volume-seconds (at least, not in the time dimension specifically). So there are some dimensions in which pi has no meaning.   

       Consider a universe in which spatial dimensions don't relate to each other in a linear fashion. In such a universe, pi would vary based on which way you orient your object. So in order to define (mathematical) pi, you have to set up a very narrow set of perfect circumstances based on selectively and arbitrarily choosing physical (and ignoring temporal) properties of our universe that are amenable to your chosen definition. It sounds a little bit like you're cherry picking to arrive at a predetermined conclusion…
ytk, Jun 19 2012
  

       Crooked space in other words.
nineteenthly, Jun 19 2012
  

       //but not for four dimensional volume-seconds// <Pantomime> Oh, yes it does! </Pantomime>   

       Consider it this way, if you want to plot a line where each point on the line is a fixed distance (r)from a point (a,b) you can use a formula in the form
  

       (x-a)²+(y-b)²=r²   

       See all the maths there? The squaring is just multiplying values by themselves, right? Nothing up my sleaves!   

       And yet, having plotted all the values for which x and y fit the above equation, and r = 1, you get a line who's length you don't need to measure.   

       If you arbitrarily decided to fix this value at some random amount (decided by intergalactic committee perhaps) then you wouldn't be able to do the algebra quite so cleanly and errors would creep into your calculations. Errors which repeated iteration (such as that which computers do) would quickly turn into whapping great ones! (e.g. Try to determine the angular velocity of some spinning thing by iterating a starting error factor of 0.0000001% - fine for the first couple of turns, but what if you're talking about a satellite that's been spinning for 20 years, how big is your error compounded by then?)   

       At least if you allow for the transcendental nature of Pi, you don't need to worry about precision until you are finally sure you need an answer that you are going to use for practical purposes, at which you are given the *choice* of deciding at what level of precision you *want* to use.   

       *Forcing* everyone to conform to some arbitrary value is not only wrong on a precision level, it also smacks of authoritarianism, Nanny Statism and frankly suggests that you may be of a Stalinist bent. Sure you can legislate away people's rights, but there will always be people who, in the cause of *freedom* will never bow down to arbitrary authority, even if it is couched in terms of making life "easy". If we're going to do that, why not make life even easier for all by legislating that we all switch to plastic cutlery and wear padded helmets while we're at it, eh? I sometimes find it difficult to understand lobsters (they are so very *hard*) let's wipe them out as well!
zen_tom, Jun 19 2012
  

       And another thing - if we must define everything in purely physical terms (poof! there goes kindness, sociology and logic!) then we need to look more closely at this Plank-length thing.   

       You say that the Plank-length is fundamental, but *how* fundamental is it? The plank-length is itself defined by the Gravitational Constant, the speed of light, and the Plank Constant.   

       l = squareroot ( hG/C³)   

       Where l = Plank Length, h = (Reduced) Plank's Constant, G = Gravitational Constant and C = speed of light.   

       Back to the constants, that's 3 things that we've measured pretty closely here on earth, and which we all accept as being inviolate, static and what have you. But hold on a minute - the speed of light is what, distance over time, right? So our fundamental unit of distance (the Plank Length) is itself defined in circular (pardon the pun) terms. That part aside, fiddling with one of the other constants, and you suddenly get different values for your Plank Length - shakey ground to stand on.   

       Even more shakey is that pesky square root which means what?... that the Plank Length is itself another irrational number!! Oh no! Congratulations, you've abolised one irrational, only to supplant it with another one, Ooops!   

       Still, I suppose it's *much* simpler (note heavy sarcasm here) to define your fundamental values in terms that rely on experimentally determined constants, the meanings of which nobody fully understands (not to mention keeping track of the current estimated size of the universe) rather than the tricky notion of the distance that a simplified (read abstracted) wheel will travel in the course of a full revolution.   

       [edit] See link above - It turns out that the Reduced Plank Constant, one of the constituents of the Plank Length, is itself defined by the value of Pi. So if we are going to redefine Pi in terms of the Plank Length, that means redefining the Plank Length, whcih means redefining the value of Pi, which means...ad inf because The Plank Length relies fundamentally on the value of Pi. Ouch.
zen_tom, Jun 19 2012
  

       //Consider a universe in which......   

       ...by only considering the entire universe, you're taking a very narrow point of view. //   

       You're still missing the point. We have maths for any number of dimensions, for pretty much any shape of space. Pi (I mean Bob) happens to apply to circles in flat two-dimensional space (as well as to spheres and whatever in higher dimensions), which is nice. It doesn't apply (in quite the same way) to circles drawn on a sphere or on a saddle, but we have different maths for those.   

       Maths is a puzzle, a game. It can be played with various rules (that is, various axioms and various formal logics). By very good fortune, one set of axioms and rules seem to give results that are analogous to our own physical universe - so much so that we can predict the motions of planets or the breaking stress on a bridge. So, we sometimes use those results for practical porpoises.   

       Now, as regards declaring Pi to be some fixed rational number, I still have two questions for you, [ytk].   

       (1) What's the point? What I mean is, suppose you convince everyone that Pi is 22/7 (or whatever), and that they should use the name "Bob" for the irrational number that we used to call Pi - what happens next? How does it make my day any better?   

       (2) I ask again - what's your chosen value for 1/3rd (which is rational but happens to be an infinite decimal), or the square root of 2 (which is irrational), or e (which is, if I remember correctly, transcendental like Pi)?   

       (3) Finally, if you are only going to allow numbers that can be represented by physical values in our own universe, how do you answer the question "what would happen if the universe grew by one planck unit"? By your reasoning, the answer has no meaning, because there is no way to represent the size of a universe bigger than its current size.   

       (OK, that was three.)   

       Joking aside, I honestly don't see your point or aim in all of this.   

       [Edit - having re-read the idea, I see that your point is to do away with the 'meaningless' precision imposed by those naughty mathematicians. Well, OK, as long as the point is silly, the argument is irrelevant. You can have your Pi, me and the mathematicians will make do with Bob, and everyone's happy.]
MaxwellBuchanan, Jun 19 2012
  

       I can see a point to using an even lower precision value of pi to save computing time when accuracy is less important. For instance, pi needn't have even five decimal places of accuracy to enable a circle to be drawn on a monitor. I'm not sure it would save much time though, because i imagine all that's worked out in advance or the algorithms for dealing with trig or whatever don't even use pi or would deal just as fast with a higher-precision version.
nineteenthly, Jun 19 2012
  

       Actually, Parmenides tried to convey that what we live in and what we are a part of is not a physical reality as such but rather more akin to a single thought. In my own folly I tried to convince others and defend my master with some sophistry of my own.   

       So please, in discussing this, fall back on the teachings of Parmenides and do not blame him for my shortcommings (sp?).   

       Also it is often forgotten that my example of the turtoise and the hare is not the only one.   

       Now, even though I will not claim to understand all the science that came after me, it seems to me that one cannot move any faster(or slower) than one planck length in one planck time unit and therefore different speeds are not at all possible.   

       I state that that none of the above explains accelleration and that nobody has ever explained or calculated it to a point beyond discussion.   

       Clearly people attribute some magical propensities to what happens within the planck measure of time and space that enables movement of matter, the existence of which depends on the discovery of some definite particle at cern.   

       So, and here you see I did not veer of topic, it is clear that Bob is what(among other things) defines what the universe is and those of us that live in the real world, in between philosophical musings, are great fans of Pi.
zeno, Jun 19 2012
  

       Pi? Sure, pi not?
daseva, Jun 19 2012
  

       //At least if you allow for the transcendental nature of Pi, you don't need to worry about precision until you are finally sure you need an answer that you are going to use for practical purposes, at which you are given the *choice* of deciding at what level of precision you *want* to use.//   

       I think you misunderstand. I'm not proposing that anyone use some arbitrary, but inaccurate value of pi. I'm saying that there is clearly some value pi that, in our universe, is accurate enough to calculate any possible value you might need it for within the margin of error of a Planck length, which is to say no error at all. Beyond that level of precision, it is impossible to determine a difference in any calculation, and therefore that level of precision /is/ the exact, rational value of pi in our universe.   

       //*Forcing* everyone to conform to some arbitrary value is not only wrong on a precision level, it also smacks of authoritarianism, Nanny Statism and frankly suggests that you may be of a Stalinist bent.//   

       I ask you, which of us is the communist, the authoritarian? Is it I, who declares that we have no right to fiddle with the natural order of things, no right to tell Nature that she is incorrect when she clearly teaches us that pi is rational? Or is it you, who says “No! Nature is wrong! We must correct her according to our established theory!” If anything, my outlook is far closer to a sort of mathematical Objectivism, where only hard evidence survives. You, on the other hand, seem to prefer a centrally planned system of mathematics based on your preconceived notions of what must be "right" for the people. Well, Atlas has finally shrugged, “comrade”! No longer will the great be constrained by the small!   

       Well, I mean, I guess it will be, if the “great” is the circle the size of the universe and the “small” is the Planck length, but you get my point.   

       //the Reduced Plank Constant, one of the constituents of the Plank Length, is itself defined by the value of Pi//   

       You have it backwards. The reduced Planck constant is just the Planck constant divided by 2*pi, which is the standard method of converting between cycles per second and angular frequency (i.e. revolutions per second). The Planck constant is not itself defined by pi.   

       //What's the point? What I mean is, suppose you convince everyone that Pi is 22/7 (or whatever), and that they should use the name "Bob" for the irrational number that we used to call Pi - what happens next? How does it make my day any better?//   

       I think the truth is purpose enough.   

       //I ask again - what's your chosen value for 1/3rd (which is rational but happens to be an infinite decimal), or the square root of 2 (which is irrational), or e (which is, if I remember correctly, transcendental like Pi)//   

       Only pi is defined in terms of a real world measurement. As I've said before, irrational numbers in general are just fine.   

       //Finally, if you are only going to allow numbers that can be represented by physical values in our own universe//   

       I'm not doing that. I'm just saying that, like it's meaningless to talk about a distance of half a Planck length, it's meaningless to talk about the precision of pi past what is usable in our universe. It cannot be observed or tested, nor can it be applied in any manner. So, if something cannot be seen, and cannot be felt, and cannot possibly affect anything in any way, in what manner does it exist? That sounds to me like a question for a theologian or a philosopher, not a scientist. Science is not equipped to deal with things that cannot be tested experientially, and something with no possible form or presence that has no possible effect on anything cannot be tested, and, as far as science is concerned, does not exist.   

       I did a little bit of research on this, and believe it or not I'm not the first person to come up with this theory that pi is rational. As I understand the argument, there are really two values for pi: the calculated value, and the measured value as I've described it. In our universe, pi has the measured value, which is rational but constantly changing with the size of the universe. The measured value of pi is always slightly smaller than the calculated value. As the size of the universe approaches infinity, the measured value approaches the calculated value. Thus, the calculated value is really a theoretical value describing what pi /should/ be, and the measured value describes what it actually is. I don't really think the reasoning is the sticking point here, and we seem to agree (at least somewhat) on that.   

       So what's the idea then? It's really more a point of principle, since we have no way of knowing precisely what the value of measured pi is at any given time. Everyone would just have to use an approximation as they do now. But by teaching that even though pi is theoretically irrational, in our universe it must be rational (and here's a quick explanation why), we tie mathematics and physics together, make a point about the inadequacy of mathematics alone to describe our universe, and simultaneously introduce the intriguing concept that what we think of as an analog world is, in fact, really more digital in nature than we realize. It's a subtle point, I guess, and one that will probably be lost on most students. But I wish the point had been made to me when I was being taught about pi. To me, the concept of our universe being divided into discrete quanta is fascinating, and it might have sparked an interest in further exploration of the sciences that could have changed the entire trajectory of my academic career.   

       Well, probably not, but I have to blame my miserable grades on /something/, and the Newtonian view of the universe seems as good a scapegoat as any.
ytk, Jun 19 2012
  

       //Only pi is defined in terms of a real world measurement.//   

       Actually, modern definitions of Pi are in terms of cosines, or as the limit of a summed series, rather than in terms of a circle.   

       Also, sqrt(2) is definable as the length of the diagonal of the unit square.   

       In other words, Pi is no more "real world" than sqrt(2) or, for that matter, e.
MaxwellBuchanan, Jun 20 2012
  

       Am I the only one whose goat is thoroughly got by e not being rational?
Voice, Jun 20 2012
  

       Well, of course e is irrational. You'd have to be some sort of lunatic to argue otherwise.
ytk, Jun 20 2012
  

       It just seems like if e is irrational our math must be wrong. I can't explain it any better than that
Voice, Jun 20 2012
  

       Don't worry about it, [Voice]. Someone recently told me I'm not allowed to imagine half of an imaginary distance, where both of them are far too small to ever see.   

       Apparently we're allowed imagine something
0.000 000 000 000 000 000 000 000 000 000 000 001 616 199
metres long, but not something
0.000 000 000 000 000 000 000 000 000 000 000 000 808 099 500
metres long.
  

       Irrationality is all around us, it seems.
UnaBubba, Jun 20 2012
  

       //it's meaningless to talk about a distance of half a Planck length// It is no less meaningless to talk about a distance of one Planck length. That you can have two points or particles exactly one Planck length apart, but no closer, is utter nonsense. The Planck length is not a cosmic Snap To Grid.
spidermother, Jun 20 2012
  

       So, when objects move through space, do they do it by Plank length stairsteps, or is that only a limitation when applied to measuring matter? Forgive me, I'm far out of my architect's ruler comfort zone.
RayfordSteele, Jun 20 2012
  

       All of which reminds me of another point. If distance is quantized into Planck lengths, and if I make a square ten Planks on a side, what's its diagonal?
MaxwellBuchanan, Jun 20 2012
  

       A cross brace?
pocmloc, Jun 20 2012
  

       Sometimes you meet a maths nut who's as thick as two Plancks.
UnaBubba, Jun 20 2012
  

       <considers entering into a round of physicist puns>   

         

         

       <decides that he'd rutherford a river of molten lava than sink so low>
MaxwellBuchanan, Jun 20 2012
  

       I hadron, I lepton him. He certainly has a fermion him. Meson a new drug, now I wear a pion my head.
UnaBubba, Jun 20 2012
  

       //if I make a square ten Planks on a side, what's its diagonal?//   

       14 plancks and a "titch"(C)
AusCan531, Jun 20 2012
  

       Unless of course, you really meant 'planks' and not 'plancks' in which case I withdraw the above anno.
AusCan531, Jun 20 2012
  

       //Meson a new drug// what happened to your old trino? And is it true you can no longer get an electron?
MaxwellBuchanan, Jun 20 2012
  

       Physics gives me a hadron.
mitxela, Jun 20 2012
  

       Dyslexia does the same, to me.
UnaBubba, Jun 20 2012
  

       //Physics gives me a hadron.// - ha! [marked-for-tagline]
zen_tom, Jun 21 2012
  

       //So, when objects move through space, do they do it by Plank length stairsteps, or is that only a limitation when applied to measuring matter? // Closer to the second one. Stuff does not happen in Planck lengths or integer multiples thereof.   

       I think a close analogy is the lower wavelength limit for water surface waves, where viscosity overwhelmes inertia. It becomes increasingly difficult to measure or produce waves as you approach that limit, and impossible *at or below* that limit, but it would be silly to speak of water waves as propagating in steps of that size, and the same holds for the Planck length.   

       Similarly, the speed of light could loosely be described as the upper limit for the velocity of massive particles, but it is mistake to think of them as travelling at that velocity.   

       (Disclaimer: I'm not a proper physicist, and will gladly be proved wrong if anyone really knows better)   

       //If distance is quantized into Planck lengths, and if I make a square ten Planks on a side, what's its diagonal?// AFAIK, distance is not quantised into Planck lengths. It is not possible to construct or measure such a square, so the question is meaningless. Or, to put it the other way, your question is a reductio ad absurdum argument that length is not so quantised.
spidermother, Jun 23 2012
  

       It has been thought of doing this before. [link]
AusCan531, Jul 21 2012
  

       I say we round pi to the nearest 10.
phundug, Jul 23 2012
  

       You know what I don't get, how can pi not be a rational number? A rational number is a number that can be expressed as a ratio of two integers.   

       And pi is the ratio of the circumference to the diameter. So it is inherently a ratio. It's just not a ratio that can be expressed in terms of integers... So wait, does that put it outside the realm of rational numbers?   

       Maybe there is a pair of integers out there that represent pi exactly.
EdwinBakery, Jul 23 2012
  

       If we are going to define Pi as the ratio of the circumfrence of a universe sized circle and the diameter of the universe, then we run into a problem.   

       If the planck length is the smallest possible unit, then the universe is not circular, but an n-gon figure with planck length edges. As such, it's circumfrence is n*Lp. Therefore under rational calculations (per YTK) Pi drops out completely, so there is no need to rationalize it.
MechE, Jul 23 2012
  

       Stuff and nonsense. 1/7 can indeed be represented exactly on a computer. To pick just one example, the (superb) calculator program Pari stores and represents rational numbers, including 1/7, exactly. Pi, on the other hand, cannot be represented exactly by any finite means - the definition enables arbitrarily close approximations to be found, but not an exact value.
spidermother, Jul 25 2012
  

       Again, my readiness to assent is incomplete. For instance, the exact, irrational value (1/7)^(1/2) can be correctly represented and manipulated with no loss of precision (until, as you say, you choose to display a decimal or other approximation). But Pi cannot be exactly represented, in a computer or otherwise, by a finite set of arithmetic operations on integers. Hence Pari, etc. simply store Pi in symbolic form, and calculate an approximate representation when asked to.   

       (Many mainstream programs, such as Microsoft Excel, insist on representing everything as floating point numbers, but they're simply mathematically naive)
spidermother, Jul 25 2012
  

       Regardless of how fractions and pi are represented by software, there is a fundamental difference.   

       One seventh can be precisely represented by "1/7". In base 7 it would be "0.1". Conversely, one tenth cannot be conveniently represented in base 7, but can be represented as "1/10", or "0.1" which is the same thing.   

       Pi, on the other hand, cannot be represented by any ratio of two numbers. Nor can it be represented by a decimal in any base (except in base pi, obviously).   

       Pi, to my mind, is wonderful and mysterious - a very simple number which cannot be represented by any ratio in any base - it's sort of kind of like the numeric equivalent of a fractal.   

       And Pi is transcendental, not just irrational. In other words, it cannot be represented by any equation. (Root 2, as a counterexample, is irrational but can be represented as the solution to the equation X^2=2.)
MaxwellBuchanan, Jul 25 2012
  

       //And Pi is transcendental, not just irrational. In other words, it cannot be represented by any equation.//   

       e^(x*i) = -1, for 0<x<5   

       2*asin(1) = x   

       x = lim n->inf (2^(4n+1)*n!^4) / ((2n+1)(2n)!^2)
ytk, Jul 25 2012
  

       My apologies - it is not the solution to any polynomial equation.
MaxwellBuchanan, Jul 25 2012
  

       Boys! Boys! ;-)
gnomethang, Jul 25 2012
  

       Thanks for the heads-up, [gnome], but I'm not interested.
MaxwellBuchanan, Jul 25 2012
  

       Cool. I hadn't seen this before.   

       ...So who's to say that a circle the size of the universe is the most precision that Pi might be needed for? It's easy to come up with more complex geometry that would need further precision, ie what about if you tried to calculate the strand length of a helix of pitch and diameter = 1 metre, who's axis is concentric around said universe-sized cirlce. There's another couple of orders of magnitude of accuracy you'll need to get it right within one planck length. What about if you wanted to calculate the volume of the sphere created by rotating this universe-circle, down to cubic planck-lengths. More precision required.   

       It's not really about the size of the universe, it's about the most complex problem you're trying to solve, which isn't even arbitrary - it's unknowable.   

       It could be argued that there is a "fundamental" valule for Pi. That is by using up the entire mass of the universe and storing the value of pi in every bit of information available.   

       ...But then it depends on how you write down the number, and if you use any compression algorithms.   

       ....So this whole debate reduces to the absurd.   

       Interesting read, though.
Custardguts, Aug 22 2013
  

       Is anyone who is obsessed with Pi known as a Pivert?
xenzag, Aug 22 2013
  

       No, they're usually just called "fatty" ...
8th of 7, Aug 23 2013
  

       Piophile.
Alterother, Aug 23 2013
  

       Given that the universe is turning out to be made of smaller and smaller bits, the precision of pi needs to get better and better.   

       Strings are meant to wibble on scales far, far below Planck lengths, shirley?   

       And then there are all those other dimensions which are wrapped up even smaller, and will need their own pis, which will need to be related to our pi.
MaxwellBuchanan, Aug 24 2013
  

       Insteady of being called tiny, they'll be called piny.
xenzag, Aug 24 2013
  

       //Strings are meant to wibble on scales far, far below Planck lengths, shirley?//   

       I acknowledge that I understand very little about string theory (although most scientists who study it would probably say the same thing). Nevertheless, it seems you have touched on a paradox here. I would posit that there are only three possibilities:

1) If the Planck length is indeed the smallest possible unit of measurement, the requirement for strings to vibrate on scales smaller than that is a reductio ad absurdum of string theory.
2) Or vice versa.
3) Our understanding of both the Planck length and string theory is incomplete, such that, like the “circles within circles” model of Ptolemaic astronomy, both phenomena are accurate representations of the Universe when examined out of context, but taken together can only be explained by some other phenomenon (yet unrealized by us) that neatly accounts for both. Like the Ptolemaic model, neither one is “wrong”—just not the easiest depiction to comprehend.

//Given that the universe is turning out to be made of smaller and smaller bits, the precision of pi needs to get better and better.//
  

       So what about this: Rather than trying to find a fixed, universal value of pi, we can theorize a “local pi” value that represents the greatest possible precision that can be attained for the current calculation, at a given point in the Universe, under a specified set of circumstances. We needn't know exactly what the value of “local pi” is at any given time (that's an engineering problem), but knowing that there is some such value gives us a precisely defined target to reach in terms of accuracy. We may not know exactly what that target is, but we do know that we can, at least theoretically, reach perfection.
ytk, Aug 25 2013
  

       Thing is, all of these arguments seem to revolve around the idea that pi is the ratio of some circle's circumference to its diameter.   

       It isn't. Pi is many things, including the limit value of any of several different series which involve only integers (and powers thereof). What is remarkable and beautiful is that elementary operations on integers can give you such a beautiful number.   

       So, once again, I fail to see the point of this proposal to rationalize pi. To begin with, if you do that, then a whole bunch of other, much more elementary things come unglued.   

       For example, if pi can be expressed as a ratio, you can prove that all the integers are equal to one another. Yet integers are useful things used by real people to do helpful things with.   

       So, help me out here, and explain to me the point of this idea.
MaxwellBuchanan, Aug 25 2013
  

       //Thing is, all of these arguments seem to revolve around the idea that pi is the ratio of some circle's circumference to its diameter.//   

       Quite the opposite. In fact, your argument seems to be based on precisely that—that there is some circle whose ratio divided by its diameter requires an infinite degree of precision of pi to precisely express. I am saying that there is not, and cannot be such a circle in our universe.   

       //Pi is many things, including the limit value of any of several different series which involve only integers (and powers thereof). What is remarkable and beautiful is that elementary operations on integers can give you such a beautiful number.//   

       Or perhaps those things all happen to sum up to an irrational number that happens to also coincide with the theoretical limit of what pi in our universe would be if not for the existence of quantum phenomena. The fact that A -> B, and C -> B, does not establish that A = C, no matter how similar A and C may outwardly seem.   

       //For example, if pi can be expressed as a ratio, you can prove that all the integers are equal to one another.//   

       On a quantum scale, all integers *are* equal to one another, in the sense that an equation can be considered where the solution is the superposition of all integers. Isn't this simply an expression of the many-worlds interpretation? Set up some experiment where the outcome is dependent on a quantum phenomenon, and the range is the set of all integers (or even all real numbers). Before the system is observed and a result determined, there exists a single Universe where the solution consists of every number superimposed; after it is observed, the Universe splits off into an infinite number of equally probable Universes, wherein the solution is some different number.   

       Can any of those Universes be said to be the “correct” one? None more so than any other. Therefore, since all solutions are equally correct, all numbers (and thus all integers) must be equal to each other. Q.E.D.   

       Of course, it would be a fallacy of logic to use this to prove that pi must indeed be rational, unless there exists a biconditional relationship between the rationality of pi and the equivalence of all integers (i.e., pi is rational if and only if all integers are equal to each other). But the equivalence of all integers would nevertheless would tend to strongly suggest that it is indeed the case that pi is rational, or at least that it's /not/ impossible for it to be so.
ytk, Aug 26 2013
  

       // In fact, your argument seems to be based on precisely that—that there is some circle whose ratio divided by its diameter requires an infinite degree of precision of pi to precisely express. //   

       No, I'm not. I don't care whether pi represents a circle or not.   

       //On a quantum scale, all integers *are* equal to one another// Well, that's a silly thing to say. Still, at least your grasp of quantum mechanics is on a par with that of mathematics.   

       However, setting such points aside, answer me this. Let us imagine that pi is declared to be only 59 (or only 127, or only 8) digits long. How is my life improved?
MaxwellBuchanan, Aug 26 2013
  

       You're of course asked to memorize all of them in primary school, which improves your memory.   

       Me, I've memorized them to an indeterminate number of digits. I just don't bother putting them into the correct order or worrying about repeats.
RayfordSteele, Aug 27 2013
  

       Praying Mantis anybody?   

       I know this is from 2012 spilling over to 2013 and now is 3.1415 years later, but just wanted to note a few interesting things:   

       1. Rabbi Eliyahu of Vilnius noticed 250 years ago something interesting:   

       The verse reads (my literal translation): And he made the 'sea' by casting it ten arm from his lip to his lip around, five in arm his standing, and his line thirty in arm will round it around.   

       Hebrew is written without vowels.   

       Rabbi Eliyahu noticed that the word QaWo' meaning 'and- his- line' has a written version QF read today as Kav and in ancient times as Kow and there's a read version QFE read today as Kavvoe and in ancient times: Kaawoe.   

       The numeric values of these Hebrew letters are 5 for the E, 6 for the Wow and 100 for the Quf.   

       (The first 9 letters are with values of 1-9,
the next ten are with values of 10-90,
and finally the last four are with values of 100-400).
  

       So (QF / QFE) x 30/10 = 106/111 x 3 = 3.141509   

       2. Insects explicitly mentioned in the bible and which we can be sure we understand the terms correctly are ants, locust, bees, wasps, lice and grasshoppers.   

       But then there are terms that are NOT clear at all: There are three similar sounding words (in ancient Hebrew pronunciation, which can be deduced in many ways) sometimes translated as dealing with insects: Remmess Shekqess and Sherress. (These are pronouced in modern Hebrew a bit differently).   

       And as a side remark: Just to mention that almost all nouns in Hebrew can be used as verbs. So people can ride a horse by horsing it, and wear a shoe by shoing it. In the same way the Remmess can be Romssed (whatever those terms mean).   

       Shekqess undoubtedly means disgust (notice how the English and Hebrew words sound alike, and in modern Hebrew even more so: Shecketz). This is deduced from the many mentions of the word in that context, and its continued similar use in other Semitic languages like Arabic.   

       The Torah tells us to be disgusted by these Shequess animals (whatever they are) by using the verb Shaquess. Something like: "Be disgusted of the dis-guests."   

       Sherress means either to spawn - to give birth to many live descendants or to lie in one place (as crocodiles do).   

       The exact translation of Remmess - is not clear. It could mean 'slithery things'. The similar noun Romess means sliding around, or it could also mean 'trampling' things, squatting down, hiding things, or squashing them. It is mentioned many times as a term for all those that living beings inside the ground or close to it.   

       b. The verse reads: "All Sherress of flight that walks on four: Shekqess he for (plural) you." the next verses discuss the locust, so the claim that Leviticus 11:20 is translated "All winged insects that walk upon all fours are detestable to you" is probably correct (although in any case there's no "all" in the verse).   

       The Jewish traditional (Talmudic) explanation prohibits any flying animal with four OR MORE legs... (so that includes bats on the one hand, and ants on the other)   

       But it doesn't mean that all winged insects walk on four. It is a list of types of insects that are allowed or denied.   

       The next verse reads (my own literal translation):   

       But this you will eat of all the Sherress of flight that walk on four:
Those that (written version: Loa - meaning: no, read version: Loe - meaning: him) bending-knees above the legs to jump upon with them on the faces of earth, these of them (you) eat: The locust to its kind, the Salaam to its kind, the Hargoll to its kind, the Haggav to its kind.
  

       Haggav is probably a grasshopper since it looks like people ("There was the land of giants, and we felt like Haggavs in our own eyes, and so we were in their eyes").   

       Hargol has something to do with a large foot. Is it the praying mantis? In modern Hebrew the mantis is a Gamal Shlomo - King Solomon's camel. - sounds similar...   

       Salaam may have something to do with a rock Sella or with a basket Sall.   

       During Passover in 2013 Adam Matan took pictures and videos of the locust infestation in southern Israel including close ups of individual locusts on the ground. They were ready to leap, and stood and walked on their four legs...   

       So: The verses may be allowing insects that walk on six such as ants and dragonflies, and denying permission to eat only those that "walk on four", such as most Arcididae (locust grasshoppers and crickets) which have four walking legs and two jumping legs, and which on the other hand usually stand on the four hind legs holding the two front legs out for climbing and grabbing, bees which have four long back legs and two seemingly "hands", and denying the eating of the praying mantis which walks on four. Or it may be a list of animals that usually stand on four (although Holech means walk not stand). It may be that the large skipping legs were not considered "walking legs" - although all giant grasshoppers clearly walk with all six.   

       Enough apologetics for today.
pashute, Jan 21 2016
  

       // rationalize pi   

       circumference / diameter. That's a ratio.   

       Fields metal, please.
Cuit_au_Four, Jan 21 2016
  

       Here you go: [link]
notexactly, Jan 21 2016
  

       I'm not convinced dragonflies can walk, although presumably their nymphs can. What I came here to say, though, is that in practical terms pi may still need to be considered irrational because there is a lot of time and I can imagine, for example, two black holes orbiting a common centre getting out of kilter as the kalpas pass because people use pi to calculate their period, so basically you never know when it might come in handy even if it takes a Graham's Number of years to do so. Therefore even physically it has to remain irrational to ensure its utility.
nineteenthly, Jan 22 2016
  

       Looks like the visible universe is something like 10^70 Planck lengths across. So does that suggest that this idea proposes standardising pi(approx) for daily use, to 10^70 decimal places?   

       Given 10ˆ6 characters in a normal printed book, and 10ˆ6 books in a modest-sized private library, and 10ˆ9 households in the world, and 10ˆ3 exoplanets so far discovered, that takes us to 10ˆ24 digits. About one third of the way there!   

       I guess we need to think about using extra-thin paper and very small print. Shouldn't be too much of a problem.
pocmloc, Jan 22 2016
  

       Looks like current world data strage is about 10ˆ21 bytes. So we will be stuck with paper calculations for a few years yet while we build the tech to handle this number electronically.
pocmloc, Jan 22 2016
  

       //So does that suggest that this idea proposes standardising pi(approx) for daily use, to 10^70 decimal places?//   

       Uh, no. That would be 70 decimal places.   

       You would only need 10^70 decimal places if the universe were 10^(10^70) Planck units across.
MaxwellBuchanan, Jan 22 2016
  

       Margin of error, old boy. Better too big than too small.
pocmloc, Jan 22 2016
  

       The Planck time needs to be taken into consideration too though. Supposedly the stelliferous era will last a total of 10^14 years. Given that the Planck time is of the order of 10^43 seconds and the length of the stelliferous era alone is something like 3 x 10^21 seconds, that makes sixty-four decimal places advisable in terms of the Universe as we know it, in order to take account of, for example, two identical iron lumps of equal mass orbiting a metre from each other. After the stars have gone out, and after stars created by collisions between red dwarfs (sp?) have also gone out, there will be occasional leptons circling at distances of gigaparsecs from each other very slowly. These will be interacting very slightly electromagnetically and gravitationally, and consequently will be describing conic section trajectories related to the value of pi. If a pair is orbiting in a near- circle, after 10^70 orbits or so, a discrepancy will occur between the relationship between the velocity of the particles and the radius of the orbits predicted by a rational approximation of pi and that predicted by pi itself, given a value of seventy decimal places.   

       Even in the case of something happening in the stelliferous era, the Planck length exponent and the Planck time exponent would need to be added together to ensure that such a discrepancy could not take place in the observable Universe during this time, so that's already well over a hundred decimal places required.   

       Then there's the Nyquist-Shannon sampling theorem to be taken into consideration, which I feel sure is relevant but can't quite work out how, but it would probably only add a few places - sorry to be vague but I can't really grasp what it means. I imagine some error would accumulate if this were not taken into consideration.   

       Regarding the Bible, it surely just comes down to the fact that Biblical Hebrew lacks a method of expressing that degree of mathematical precision, doesn't it?
nineteenthly, Jan 23 2016
  

       Also, let's suppose I want to fill the universe with string, the string being one Planck unit in diameter. I do this by starting with an atom in the middle of the universe, and wrapping the string around it repeatedly to build up a ball, until I've filled the universe. I'm going to need all the digits I can handle, to work out the length of string I need.
MaxwellBuchanan, Jan 23 2016
  

       I disagree, in theory at 1p scale everything is minecraft rules. Your string is cubic, and filling space is cubic and going diagonally is impossible. Your ball of string will have to fill the space in a cubic fashion; L x W x H. Or put better, it will have to be as long as the total resolution of the universe.
WcW, Jan 23 2016
  

       So pi=4
pocmloc, Jan 24 2016
  

       I'm glad that's settled. I'll have a slice.
whatrock, Jan 24 2016
  

       This discussion is irrational
pashute, Feb 12 2018
  

       // The Bible also claims insects have four legs//   

       Are you sure you're not thinking of Aristotle, [xixly]?
pertinax, Feb 13 2018
  

       Maybe the front two are arms.
Ian Tindale, Feb 13 2018
  

       I don't think anyone has ever claimed that Aristotle had four legs. Even Aristotle himself made no such claim, and he was notoriously unreliable.
MaxwellBuchanan, Feb 13 2018
  

       Here: No Aristotle has 3 legs. All Aristotles have 2 legs more than no Aristotle. Therefore all Aristotles have 5 legs. And if they have 5, then they also have 4.
RayfordSteele, Feb 13 2018
  

       What about foreheads?
Ian Tindale, Feb 13 2018
  

       Better than two heads, and quite useful for smacking when someone thinks that pi is only for measuring things.
RayfordSteele, Feb 13 2018
  

       //at 1p scale everything is minecraft rules. Your string is cubic//   

       Couldn't everything at that scale be packed into a stack of offset hex grids, like certain kinds of atomic lattice? On the one hand you would have some trouble with orthogonality, but on the other hand you would have room for slightly more quanta of everything, by wasting less space in the quantum corners.
pertinax, Feb 13 2018
  

       // Better than two heads //   

       "Foreheads good, two heads better" ?   

       George Orwell would be so proud ...
8th of 7, Feb 13 2018
  
      
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