[HN Gopher] How Pi Almost Wasn't
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How Pi Almost Wasn't
Author : jamespropp
Score : 79 points
Date : 2025-03-13 22:54 UTC (1 days ago)
(HTM) web link (mathenchant.wordpress.com)
(TXT) w3m dump (mathenchant.wordpress.com)
| gerdesj wrote:
| Looking at the list of 10 equations.
|
| For me 2,5,8,9,10 being simplified with Pi=3.141 is the best
| outcome, especially 5.
|
| I'm definitely: e^{i \pi} + 1 = 0
| tonyarkles wrote:
| I agree with you on the absolute beauty of e^{i\pi}+1=0, but I
| think I would also see a different beauty in e^{ik\pi}=1 for
| all integer k (if we had \pi=6.28... instead)
| orangecat wrote:
| I would argue that #2 is better with tau; yes there's a factor
| of 1/2, but it should be there for the same reason that kinetic
| energy is 1/2 _mv*2 and distance traveled under constant
| acceleration is 1 /2_at*2. All of those are the result of
| integrating a linear function.
| yen223 wrote:
| For all integers k, e^(i k\tau) = 1
|
| This is way more beautiful to me than the pi version e^(i
| 2k\pi) = 1
| jmclnx wrote:
| Most of the article was over my head, but it hit HN because
| tomorrow is PI Day. It is a nice article though. A Quote:
|
| >we celebrate Good-Enough Pi Day in honor of the approximation
| 3.1, which is close enough for most purposes. It could be
| celebrated on either the 3rd of the 1st month ... or on the 1st
| day of the 3rd month...
|
| at the end it went to the fact in other countries there is no
| month 14 :)
| ahazred8ta wrote:
| Approximate Pi day is in July: 21/7.
|
| 1 Kings 7:23 describes a basin in Solomon's Temple as ten
| cubits across and a cord of thirty cubits encircles it. The
| word for 'cord' has two spellings with two numerical values:
| qvh (111) and qv (106). Now, 3 x 111/106 = 333/106 = 3.1415
| leereeves wrote:
| Shouldn't that be 22/7?
| dekhn wrote:
| I'm still surprised that Pi isn't considered a universal physical
| constant, given Buffon's needle depends on (IIUC) the physical
| curvature of nonrelativistic space.
| IHLayman wrote:
| Is it because pi isn't measured, but calculated? The wikipedia
| article (https://en.m.wikipedia.org/wiki/Physical_constant)
| makes a distinction between a mathematical constant and a
| physical constant, stating that the latter cannot be calculated
| but instead needed to be measured experimentally... Pi could be
| measured experimentally, but it has an exact definition and can
| be calculated outside of any experiment.
| illini1 wrote:
| Buffon's needle assumes a flat space, while a non-Euclidean
| space or geometry would affect the probability leading to a
| different value other than pi. You can treat the space around
| us as Euclidean, but that isn't true for every part of the
| universe.
| not2b wrote:
| Even in a non-Euclidean space with positive or negative
| curvature, the limit of the ratio of the circumference to the
| diameter of a circle as the diameter goes to zero is pi.
| mitthrowaway2 wrote:
| Up until 2019, it sort of was, via the magnetic constant mu_0 =
| 4 pi x 10^-7 H/m exactly. Unfortunately the 2019 revision of SI
| redefined it.
| ants_everywhere wrote:
| If you run Buffon's needle a lot you come to the conclusion
| that we live in a _locally Euclidean_ space. That 's fine and
| good. We also live in what you might consider a "locally
| Newtonian" world, but when things get very big or very small,
| the Newtonian approximation breaks down.
|
| The ratio of a circle's circumference to its diameter (or
| equivalently the sum of triangle angles) has the same problem.
| If you want general relativity to work, then we need to live in
| a curved spacetime. Depending on whether that spacetime is
| positively or negatively curved, the angles of a very large
| triangle may add up to more than or less than pi radians.
| readthenotes1 wrote:
| I usually hold that the only sensible date ordering is
|
| yyyy-mm-dd
|
| So that files sort properly.
|
| I'm happy to see that this enabled pi day, whereas the Europeans
| would prefer it not exist.
| Etheryte wrote:
| ISO 8601 is the standard date format in a number of European
| countries, but you do you.
| roelschroeven wrote:
| It is? I've only ever seen dd/mm/yyyy or dd/mm/yy as standard
| date format, with different separators instead of / depending
| on the country.
| dspillett wrote:
| It isn't used on any official forms and such that I'm aware
| of, at least outside technical circles, but people don't
| tend to have a problem with YMD ordered formats when they
| are used.
| Etheryte wrote:
| Sweden used YYYY-MM-DD before ISO 8601 was even written,
| and many other countries use it too.
| dspillett wrote:
| I know at least some countries further East used it as a
| matter of course, but I wasn't aware any countries
| in/near Europe did.
| johnisgood wrote:
| YYYY-MM-DD+Z is pretty much the standard.
| scbrg wrote:
| It's used in Sweden, at least.
|
| Source: Me, having lived there most of my life.
|
| Second source: https://en.wikipedia.org/wiki/Date_and_time_
| notation_in_Swed...
| layer8 wrote:
| File sorting normally isn't determined by the date format used.
| You're probably thinking of other contexts.
| behringer wrote:
| Dates in the file names can be sorted by that date if you use
| the correct date format.
|
| This is common to do.
| layer8 wrote:
| Right, I thought this was about locale formats and the
| equivalent of `ls -t`. It didn't occur to me that the
| sensicality of using a lexicographic date format in file
| names might be under contention.
| scythe wrote:
| The Europeans have 22/7, as long as they write dates with the
| slash.
| drsopp wrote:
| It is common in Norway to also write 22.7.
| dspillett wrote:
| _> yyyy-mm-dd_
|
| _> whereas the Europeans would prefer it not exist._
|
| I (a UKian) use YMD ordered formats everywhere possible and
| have done for many years, a few decades in fact, perhaps even
| before I know of ISO8601. I've seen more pushback about it from
| USians than us EUians, for no reason other than it isn't
| mm/dd/yyyy. In my experience most people outside technical
| circles on both sides of the big wet are unaware and just use
| their local legacy format, when EUians see yyyy-mm-dd (or the
| undecorated yyyymmdd) they tend to get it and accept it (though
| often keep using what they have always used), where some very
| vocal USians take a different format existing as a direct
| comment that their usual format is wrong and feel that as a
| personal slight rather than technical matter.
| thaumasiotes wrote:
| > William Oughtred, in his 1631 work Clavis Mathematicae (The Key
| of Mathematics), used the notation "p/d" where p is the
| circumference of a circle and d its diameter
|
| This notation would make more sense if you described it as
| referring to the "perimeter" of a circle, which it seems almost
| certain is what William Oughtred had in mind. (Though maybe he
| was thinking "periphery", which is a more exact match to
| "circumference". I don't know where "circumference" came from;
| there is no conceptual difference to explain the difference in
| word use, and it's strange for a Latin word to be used in
| geometry anyway.)
| jszymborski wrote:
| The far superior constant, Tau, mentioned. I am glad Mr. Propp
| has heard the good word.
| djmips wrote:
| He's got it wrong. People don't support tau like he supported
| the Red Sox, the support tau because it's actually a handy
| constant. Pi still exists!
|
| Tau is not an inconsequential lost cause. I think it will
| slowly win over everyone.
| abnry wrote:
| The real nerds celebrate octal pi day on 3/11. The crowds are too
| big on 3/14 anyways.
|
| https://imgur.com/a/gczeqkz
| twiceaday wrote:
| The real nerds celebrate Tau day on 6/28.
| abnry wrote:
| That's just controlled opposition.
| hgomersall wrote:
| Only in America though.
| lern_too_spel wrote:
| To be clear, the real nerds celebrate octal pi day on the octal
| date 3/11, which is 3/9 to the decimal date partisans.
| nokun7 wrote:
| Aryabhata, an exceptional Indian mathematician born in 476 CE,
| left a lasting mark on mathematics and astronomy. At the age of
| 23, in 499 CE, he calculated pi to be approximately 3.1416 and
| suggested its irrational nature, relying on insights from Vedic
| traditions. He's also widely recognized for introducing zero as a
| numeral and developing various mathematical and astronomical
| concepts. That said, the value of pi had been explored even
| earlier by another Indian mathematician, Baudhayana, around the
| 6th century BCE. Baudhayana not only worked out pi but also laid
| out what we now call the Pythagorean Theorem, long before it
| reached European scholars.
| theamk wrote:
| I've read Tau Manifesto [0], and I am now convinced that 3.14...
| is not the best circle constant, "turn" (tau, t) with value
| 6.28... would be a much better choice.
|
| How many radians in full circle? 1 tau (turn). What about 1/3
| circle (120 degree)? 1/3 tau. What about euler's constant?
| e^i*tau=1, or in other words rotate vector by 1 turn and end up
| at start position.
|
| So beautiful. So unreachable - pi has so much legacy, there is
| zero chance of changing it.
|
| [0] https://tauday.com/tau-manifesto
| fractallyte wrote:
| Tau replacing pi? Sure, just start using it. Publish with it.
| If we want better, we have to make it happen!
| djmips wrote:
| I use it in my code at least.
| brookst wrote:
| Area of a circle?
| deathanatos wrote:
| Yes, even that one gets more beautiful, too.
|
| Look at the usual equation: A = pr2. Why _is_ there no "2"
| there?
|
| Let's derive it, and in particular, let's derive it from the
| onion proof, which is that a circle's area is composed of
| many small circles, arranged concentrically, like a 2D onion:
|
| A = [?]_0^r 2pt dt
|
| There's that blasted 2 again. The tau form is more beautiful:
|
| A = [?]_0^r tt dt
|
| Integrate it, and you'll get A = tr2/2, the constant being a
| result of the integral.
|
| That is, to me, the usual equation is more properly A =
| 2pr2/2, the two 2s being different in their origins, and we
| just usually use & memorize the simplified form.
| hinkley wrote:
| Unfortunately the ancients didn't invent calculus. Pi had
| been in use a long time when Liebniz and Newton came along.
| deathanatos wrote:
| Another way to look at that would have been visible to
| non-calculus bearing ancients: A = pr2
| C = pD
|
| ... why do we arbitrarily use r in one equation, and D in
| the other? (... because we're using the wrong constant,
| and it bugs us, and we're sweeping that under the
| mathematical rug.)
| adrian_b wrote:
| There are some advantages also for taking Pi/2 as the circle
| constant.
|
| The choice between 2xPi and Pi/2 is equivalent with the choice
| between the cycle and the right angle as the unit of plane
| angle (the right angle corresponds with the "imaginary" unit,
| i.e. any point on the unit circle corresponds to i^x, with x in
| right angles). When extended to 3 dimensions, the corresponding
| constants for the solid angle become 4xPi and Pi/2, so Pi/2 is
| more consistent (it corresponds to taking the orthogonal
| trihedron as the unit of solid angle). The consistency of Pi/2
| remains true for higher-dimensional spaces, but that has little
| practical importance.
|
| "Pi" is by far the worst choice, in all computational
| applications either 2xPi or Pi/2 are needed, not Pi.
|
| The only case where Pi appears naturally is in measurements,
| because both the circumference and the diameter are much easier
| to measure than the radius, and Pi is the relationship between
| these 2 practical measurements, allowing the conversion of one
| to the other.
|
| Nowadays, the computational uses of Pi are many orders of
| magnitude more frequent than the conversions between the
| practical measurements of diameter and circumference, so the
| use of Pi is undesirable inside computer programs or in
| symbolic processing of mathematical formulae.
|
| In general, it is much better to implement trigonometric
| functions where the argument is either x*Pi/2 or x*2*Pi,
| instead of traditional trigonometric functions, because the
| argument reductions are fast and exact. Sadly, the floating-
| point arithmetic standard defines useless functions of Pi*x,
| like sinPi, cosPi etc., instead of using any of the 2 better
| choices.
|
| The only possible advantage of traditional trigonometric
| functions is at integration or differentiation, where they
| could save a multiplication, but in almost all applications of
| this kind the argument of the function is not x, but an
| expression with at least one multiplication, and the additional
| multiplication with a constant caused by using x*Pi/2 or x*2*Pi
| can frequently be done at compile time, or at run time, but
| only once, not at every computation.
| t-3 wrote:
| What I don't understand is why pi is two taus stuck together
| but tau is 2*pi rather than the other way around.
| smallstepforman wrote:
| This is the first time I encountered the quarter pi formula:
|
| 1 - 1/3 + 1/5 - 1/7 + ...
|
| Amazing ...
| rottc0dd wrote:
| There is a playlist by 3b1b for different interesting
| derivations of Pi:
| https://www.youtube.com/watch?v=8GPy_UMV-08&list=PLZHQObOWTQ...
| adrian_b wrote:
| This kind of sloppy writing is too frequently encountered and it
| is annoying: "Archimedes didn't think of 22/7, 223/71, or pi as
| numbers; to him, they were ratios of magnitudes".
|
| This has nothing to do with "thinking". This is just about
| (incorrect) language translation and mathematical terminology.
|
| In Ancient Greek and Latin and in any other old languages, the
| word "number" designated the result of a counting operation, i.e.
| it corresponds to what in modern mathematical terminology is
| named "natural number", so it _must_ be translated as such in a
| mathematical context.
|
| In Ancient Greek and Latin, the word "magnitude" (or "measure")
| designated the result of a measurement operation, like the
| measurement of a length with a standard ruler or the measurement
| of a weight with weighing scales, i.e. it corresponds to what in
| modern mathematical terminology is named "real number", so it
| _must_ be translated as such in a mathematical context.
|
| There is no difference between Archimedes and a modern
| mathematician, both distinguish natural numbers and real numbers,
| but since they speak different languages, they use different
| words for these 2 concepts. Whenever you read an ancient
| mathematical text, "number" must be understood as "natural
| number", while "magnitude" or "measure" must be understood as
| "real number".
|
| The names do not matter, this is just a problem of language
| translation (which is done usually incorrectly for texts with
| mathematical terms, like also for texts with specialized terms
| from physics, chemistry, biology or mineralogy, because the
| translators have little knowledge about those domains).
|
| The important difference between ancient mathematics and modern
| mathematics is that the ancients did not have a method of
| construction of the real numbers from natural numbers, which has
| been conceived only in the 19th century, by the time of Cantor,
| Dedekind etc. The ancients also did not have the method of
| abstract algebra, where you prove something for a set of axioms
| of a given form (e.g. of group, of ring, of field etc.) and then
| the proof is valid for any set where that set of axioms is
| satisfied. Abstract algebra is also a creation of the 19th
| century.
|
| Because of that, the ancients had 2 independent sets of axioms
| for natural numbers and for real numbers, even if many of them
| were identical in form. The consequence of this was that many
| theorems and demonstrations had to be duplicated for natural
| numbers and for real numbers, because something proven for one of
| them could not be applied directly to the other.
| johnisgood wrote:
| Right, I read Aristotle's metaphysics, the terms used refer to
| concepts for what we use a different word, today.
|
| So what you said is very important to take into consideration.
| whiteandnerdy wrote:
| I think there's a sense in which moderns feel that the reals
| (or at least the rationals) are a natural category: that 5 and
| 0.3 are the same "kind of thing". Mathematicians talk about the
| distinction between different classes of numbers, but to most
| people they're all numbers.
|
| Do you think that sense was shared by the ancients, or do you
| think that the linguistic distinction mirrored a stronger
| mental separation between the two? It sounds like it might have
| done if they had to do the work of duplicating proofs from one
| to the other. Did they have a single word to describe the
| shared concept?
| adrian_b wrote:
| To the ancients, as already clearly expressed by Aristotle,
| natural numbers and real numbers were both _quantities_.
| Quantities were classified in discrete quantities, of which
| natural numbers are an example and continuous quantities, of
| which real numbers are an example. This classification
| remains completely valid today.
|
| The ancient mathematicians and philosophers were using the
| word "quantity" in most contexts where modern people use the
| word "number", i.e. when the word is applied to different
| kinds of "numbers", not just to natural numbers.
|
| There is no difference in thinking between ancients and
| moderns, it is just a difference in the words that happen to
| be used.
|
| Both the similarities and the differences between natural
| numbers and real numbers are well entrenched in most natural
| human languages since many millennia ago, before any
| scientific theory of quantities, numbers and magnitudes, as
| exemplified by the similarities and differences between
| questions like "How many ... do you have?" and "How much ...
| do you have?".
|
| Actually I consider the ancient usage of the words as more
| sound than the modern usage. There appears little
| justification for the modern usage of the word "number"
| instead of the previous usage of "quantity", except that
| "number" is a shorter word than "quantity", so the change in
| terminology is just due to laziness, not to any theoretical
| reason. However what has been gained by saying "number"
| instead of "quantity" when the wider sense is intended, has
| been lost due to the requirement for qualifying "number" as
| "natural", "real", "integer" etc., when the narrower meaning
| is intended.
|
| Etymologically, "number" is the result of counting, which
| real numbers and many other kinds of "numbers" that
| correspond to continuous quantities are not.
| areyousure wrote:
| Suppose we designate a unit line segment, and from that we
| pick out a line segment of (what we now call) length 2 and
| separately a rectangle of area 3. To "an ancient" (like
| Aristotle), does it make sense to add the length and the
| area to get 5? If you then drew a line segment of length 5,
| would they say it has the same magnitude?
|
| (My understanding is "no" to both questions.)
| adrian_b wrote:
| Today, as always, it is a serious mistake to add a length
| with an area.
|
| Nowadays, everyone who attempts to do such things should
| understand that the value of a physical quantity is the
| product between a scalar (i.e. real number) with a unit
| of measurement.
|
| The addition of distinct physical quantities is
| impossible, because the addition between distinct units
| of measurement is undefined (because there is no useful
| definition).
|
| However, if you detach the scalars from the complete
| values of some physical quantities, those are elements of
| the field of the real numbers, so you can do any
| operations with them, not only addition, but you can
| compute arbitrary functions, like transcendental
| functions, e.g. logarithms (which you cannot compute
| using as argument a complete physical quantity, including
| a unit of measurement).
|
| In the ancient Greek mathematics, the notions of physical
| quantity and measurement were not well formalized, even
| if they had some intuitive understanding, so they would
| not normally try to perform invalid operations with
| physical quantities. The concepts of physical quantity
| and measurement became well understood only in the 19th
| century, e.g. in the works of people like Weber and
| Maxwell.
|
| Nevertheless, in the ancient Greek texts there are also
| examples of numbers that were detached from some
| geometric quantities and then used in unrelated
| operations where their origin was no longer taken into
| account. A simple example is the use in various
| arithmetic problems of several kinds of "geometric"
| numbers, e.g. square numbers, triangular numbers, cubic
| numbers or pyramidal numbers, i.e. numbers that were
| computed using formulae for areas or volumes, but which
| were used in applications were there were no areas or
| volumes involved.
|
| Another example is the use of mechanical devices for the
| computation of some irrational or transcendental
| functions, which could be used for solving some famous
| problems like the trisection of an angle, the quadrature
| of a circle or the doubling of a cube. In such mechanical
| devices there were e.g. some lengths that were
| numerically equal with areas or volumes in the related
| problems. (While such mechanical devices, which were
| examples of analog computers, have been built and they
| have solved the problems, such solutions were considered
| cheating, because those problems had been formulated with
| the restriction of using only a straightedge and a
| compass for their solution, which is now known to be
| impossible.)
|
| The essence of analog computing, whence the name "analog"
| comes, is that you have in the analog computer some
| physical quantities that are numerically equal (using
| some arbitrary units) with some physical quantities in
| the problem that must be solved by the computer, even if
| the nature of the physical quantities is very different.
| Analog computing is an important example of detaching the
| real number value from the complete physical quantity,
| and as I have said, there are examples of simple analog
| computers, dedicated to the computation of some
| irrational or transcendental functions, already since the
| Greek Antiquity.
| kazinator wrote:
| Very Eurocentric article. This dude
|
| https://en.wikipedia.org/wiki/Zu_Chongzhi
|
| found in the fifth century AD that p lands between 3.1415926 and
| 3.1415927, using Liu Hui's algorithm on a 12288-gon:
|
| https://en.wikipedia.org/wiki/Liu_Hui%27s_%CF%80_algorithm#L...
|
| He also discovered the 355/113 approximation over a millennium
| before the Europeans:
|
| "[H]owever milu p = 355/113 could not be found in any Greek,
| Indian or Arabian manuscripts, not until 1585 Dutch mathematician
| Adriaan Anthoniszoon obtained this fraction; the Chinese
| possessed this most extraordinary fraction over a whole
| millennium earlier than Europe." (First article above quoting
| scholar Yoshio Mikami)
|
| Also:
|
| https://en.wikipedia.org/wiki/Liu_Hui%27s_%CF%80_algorithm#S...
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(page generated 2025-03-14 23:02 UTC)