How π Emerges From a Forgotten Curve

welcome under another roof where there is always another proof they call me the witch of anazi don't stare like that I'm not crazy but I am no ordinary magician just a humble mathematician and with your permission in this addition a proposition regarding a simple addition what concoction shall I brew come closer I'll share a clue of all my recipes this one shines for this mixture surely defines a most irrational solution despite its rational Constitution to uncover the secrets within with some odd numbers is How We Begin 13579 Etc their reciprocals this bruis progenitor including them all would be an error adding every second is far more clever for now we remove those left out and what remains there can be no doubt this glorious formula I cannot lie yields exactly a quarter of Pi a question Burns I hear you cry keep watching to explore [Music] why so the liet formula for pi somehow if you take the alternating sum of the reciprocals of odd numbers Pi where there is pi there is a circle but what do these reciprocal of odd numbers have to do with circles allow me to show you a curve that stems from a circle which on one hand will produce pi and on the other hand will produce these reciprocals of odd numbers this curve has a special name the witch seriously that's what we call it the witch of ani and there's a fun story behind that name that led me into the deepest research Rabbit Hole I've ever tumbled into and we'll get into that later no Halloween video would be complete without some horrifying mathematics we'll soon be looking at a bit of ukian geometry convergent geometric series and calculus but don't let that scare you I'll explain everything so if you have knowledge gaps in those areas you'll still be able to follow along if you're still uncertain about anything we've covered don't worry you're not alone I receive a lot of emails from viewers saying that they never got a chance to study mathematics at a higher level so what's the best way to learn now luckily there's an answer and that's this video sponsor brilliant by providing a free and easy to ouse platform whatever your current level brilliant has courses for you later on we'll be looking at some ukian geometry and I was shocked to find a gold mine of interactive courses teaching everything from the fundamentals to Advanced results I love how everything is clearly presented these Hands-On modules are key to developing a strong intuition all of their courses are delivered in fun bite-sized chunks motivating you to learn a little bit each day with brilliant customizing your learning experience so that you can learn at your own pace my degrees were in mathematics so personally I'm eager to dive in and and learn some computer science but whatever your learning Ambitions brilliant is the best way to learn try everything brilliant offers today at brilliant.org another roof your first 30 days are absolutely free and the first 200 people to sign up with my link get 20% off their annual premium subscription that's brilliant.org roof or click the link below huge thank you to brilliant for supporting the channel and now back to the video so what the heck is this witch curve to show you we must first journey to a different plane of reality the plan dominated by your former partner and life's most fundamental questions that's right the X and the Y I'm speaking of course of the cartisian plane we will draw a circle such that the x-axis is tangent to the circle and its Center is somewhere on the positive y AIS we'll also draw another tangent to the circle parallel to the x-axis I'll call this line the ceiling since all the action is going to take place in this band now we draw a ray from the origin through the circle until it hits the ceiling and I'll call this L now here's where the witch comes from we plot the magic point with y coordinate equal to where L hits the circle and x coordinate to where L meets the ceiling in effect we're projecting lines down and across and where they meet is our magic point now let's pick another Ray and follow the same steps to plot another magic point and again and again and again doing the same thing on the other side and as our aray varies we Trace out a curve that looks like a broad hump this looks kind of familiar it looks a lot like the normal distribution but it isn't this curve is the witch now the witch pops up in all sorts of places a scaled version of it depicts the Koshi distribution in probability Theory and it approximates a Solon a solitary wave for us it can shed some insight into this alternating sum for us it's the area beneath the witch that hides a lot of surprising mathematics so that's what we'll want to calculate first let's derive a formula for the curve now so far I have haven't actually stated the radius of this circle here you can make this whatever you want so really there isn't a single witch but a whole bunch of witches or whatever the collective noun is a gagle of witches we could leave the radius as R and derive a general formula for a general witch and everything I talk about for the rest of the video does hold in general but to make the algebra smoother I'm just going to proceed with the diameter one case so r equal a half so let's pick a general point on the Witch and call it X notot Y not what we want is y not in terms of xnot that's what a formula is it's a relationship between the coordinates so let's think about how this point came to be extending upwards and across we can recover the raid that spawned the point pause here if you want to work it out because it's a fun problem and it doesn't involve anything beyond School level maths now there are a few different ways to proceed but today we'll do it like this let's look at this point it has some x coordinates say V and it has y-coordinate why not and now we'll observe two relationships that link X X not and Y not the first is that this point here v y lies on the circle so it has to satisfy the circle equation generally Circles of radius R centered at AB have this equation our Circle has Center 0 half with radius half so our Circle has this equation substituting x = v and y = y we derive this so that's something but we also know something else this triangle down here is similar to this triangle here because all of their angles are the same that means that the ratios of their sides are equal now we don't care about V so we'll make V the subject of both of these relationships and then we can equate them and I'm actually going to make v^ squ the subject of each because squaring this one is easier than rooting this one usually avoiding Roots is a good idea so both of these expressions are equal to v^2 so we can equate them now we'll expand this bracket the quarters die and now because y not is strictly greater than zero we can divide through by y not and now we just collect the Y kns factor out divide and boom and because this holds regardless of the choice of coordinates we can just write this as a general formula y = 1 / 1 + x^2 so that's the formula for the witch in The R equals a half case if you did it for the general case of radius R you'll get a formula that looks like this you can see now why I chose Ral a half because it's a heck of a lot simpler now that we have a formula we can calculate and explore the area under the wit this has a super satisfying answer that's fundamentally tied to this circle here before we do however we're going to need a couple of lemas stepping stone results related to this expression up here we essentially want two different ways to write this down one will yield the alternating sum of the odd reciprocals and the other will yield Pi so let's do [Music] that first let's talk about geometric series take a look at these sequences what do they all have in common they all have a rule where you pick a number and that's what you multiply by each time to get from term to term that number can be greater than one giving an increasing sequence less than one giving a decreasing sequence or negative giving an alternating sequence these are called geometric sequences and in general imagine we have a sequence that starts from one and we multiply by some number R each time this will give us a sequence that looks like 1 r r 2 R cub and so on what if we keep going and stop at the R the^ of N and then add them all together thereby forming a geometric series what does this equal there is a formula for this and again there are many ways to derive it but my favorite is to consider what happens when we multiply the whole thing by 1 - R after expanding we can see that every Power of R is going to die apart from R the N +1 on the end so we're left with r subtract R the N +1 and after dividing by 1 - R we obtain an expression for this sum here but now here's some magic What If instead of stopping at R to the^ of n we just keep adding terms forever in other words if we let N approach Infinity well then so long as R is less than 1 this R the^ of n + 1 will approach zero as n approaches infinity and thus we obtain a well-known formula for the infinite sum of a geometric series for example take this famous 1 plus a half plus a qu plus an e and so on we can get a visual intuition for this by taking a unit square of Area 1 adding a angle of area half then area quarter then area eth and so on and we can see that if we keep going forever we Converge on a total area of two and we can verify this with our formula in this case R equals a half so substituting R equals a half into here we get an answer of two as we expect so this is cool and everything but you're probably wondering who cares well this formula here Bears a little bit of resemblance to the which formula can we marry these together well sure these two things say the same thing as long as as R is equal tox2 so this is the formula for an infinite geometric Series where R is equal tox 2 what does that sequence look like well we multiply by x^2 each time so that's 1 subtract x^2 + x 4th take x to the 6 and so on in other words for a given x coordinate calculating this infinite series is an alternative way of generating the y-coordinate of the witch note however that this doesn't necessarily converge but we'll see later that we don't really need to worry about that the punch line though is that this is an alternating series which we know is what we're looking for put a pin in that because now we're going to look at something totally different and that involves a bit of calculus all we want to do is to differentiate yals Aran of X why do we want to differentiate this don't question the wi you'll see why very soon now this is a pretty standard result that I'm sure many of you know already and it can be accomplished with school level calculus again lots of ways to do this but parental advisory there is implicit content here we have X is tan Y and now differentiating with respect to X implicitly we have that X is SEC s of y * Dy by DX tan differentiating to SEC squ is a pretty standard result but if you don't like that you can set tan equal to sin / cos and use the quotient Rule now SEC s y is equal to 1 + tan^ 2 of Y and instead of writing tan^ s of Y I'm going to write tan of Y all 2 I've done that because now remember that Y is equal to R tan of X so I can replace y with r tan of X and we'll see what happens tan of R tan of X must just be X so this whole thing is just 1 + x^2 and if we divide by that we get the Dy by DX is equal to 1/ 1 + x^2 and oh look it's a witch I told you there was a reason we were doing this so with these two lemur let's now put all of the pieces together and find the area under the [Music] witch we'll integrate to calculate the area beneath the curve now remember the curve has this formula which by our first L is equal to this but this actually only converges for X is less than one so we're actually going to find the area of this strip here between 0 and 1 so we're going to integrate between 0 and 1 and because these are equal integrating both must yield the same result ready let's start with the infinite series cuz calculus doesn't really get much easier than this we just add one to each of the powers and divide by the new power and we're going to do this between 0 and 1 obviously this whole thing is zero when x equals 0 so substituting xal 1 we get the alternating series that we've been looking for and this must be equal to this integral up here I think you might see where this is going but let's do it by our second Lemma differentiating AR of X gives us this 1 / 1 + x^2 so integrating 1 over 1 plus x s will give us AR tan of X so now we evaluate this between 1 and0 AR tan of 0 is0 and AR tan of 1 is < / 4 and there we have it by the magic of the witch these expressions are equal both represent the area beneath This Witch between 0 and 1 but and it's a big but I cannot lie this might in some way feel unsatisfying why does the area under the curve have something to do with pi sure the curve stems from a circle but it isn't a circle well don't go away because I found a really inuitive explanation for why that might be the case there is in some sense a clear relationship between the area of the circle and the area under the Witch and what's more we can see it using an elementary calculus free method first I'm sure you're still wondering I'm discussing a curve called The Witch as if it's totally normal why such a Sinister name well indulge me in a short break in the mathematics for an eological tangent the curve is named after The Tragically overlooked mathematician Maria Ani one of the first women to be offered a mathematics professorship the curve was studied earlier by our old friend ferma and a guy named Guido Grandy and yai wrote about the curve in her 1748 treaties on calculus her institut analytic which was eventually translated into English by Cambridge professor John Coulson there are a few conflicting stories about the exact wording in these texts and about whether which came about due to a Mis transation so I felt I had to view these texts with my own eyes so I've gone to my old University Library where they actually have an original copy of the text so let's see what it has to say and now we're actually at the John ryland's library because it turns out this is where the books actually are stupid they kindly let me examine a copy of Ani's institutions I was very nervous about handling a book that's over 250 years old but an enormous thank you to the team at John ryland's library for letting me view these works and there it is La verra which apparently means witch there it is so cool here we clearly see that she uses la verier for the curve and also just look at these old diagrams how awesome are they then we looked at an original copy of Carlson's translation where he writes that the curve is vulgarly called the witch lo and behold the witch of anazi so where did the name verser come from and does it actually mean which well while Guido grandi wasn't the first to study the curve he was the first to name it in 178 Trea he writes which I usually call verser or in Latin versia versia meant a rope used to turn a sail deriving from the Latin Verte which means to turn or change and sinus meaning S curve or among other things a bulgan a ship's sale while Grandy says nothing for his motivation for the name a common explanation for Grandy's name is that verser means turning curve as it derives from this line turning about the origin now at the same time the Italian word a Sierra meant a female adversary or the wife of the devil this would come to mean more generally a female fiend or demon the equivalent of which as we saw verser persisted into Maria and Ya's institutions and this work was much more prominent considered the greatest introduction to calculus at the time so the curve came to be named after her John coulson's translation was published posthumously in 1801 where vers era was translated into wit many writers argue that this was John Carlson's mistake Clifford trusel berates Coulson accusing him of mistaking La verier for lav verier however it was common at the time to refer to a verier as just verser through Aeris where the initial sounds of words are omitted kind of like how we say cause instead of because or how loone derived from alone we see evidence of this in Petro Fan's vocabulary of Tuscany where he writes that because it was taboo to speak of the devil or the wife of the devil people invented different names like verier he cites several writers using verier in this context long before Grandy so we can assume that verier was in common use verser essentially became a homonym it has two different meanings so translating verser to which might have been a misr transation but it wasn't totally without Foundation others like Steven Stigler claim that Grandy was aware of this double meaning and deliberately named The Curve verser as a pun however it happened verier stuck and which persists into English nonetheless and as a final Point TF milron argues that witch Vani is inappropriate as a name for the curve as the words evil connotations do little to honor Maria and yai who lived at a time when women in maths and science struggled for acceptance Maria and yai by the way is reported as being prodigiously intelligent mastering seven languages before turning 11 debating philosophy with her father's circle of academics in her late teens all while acting as tutor to her 20 younger siblings yes 20 after writing one of the era's greatest textbooks she spent the second half of her life devoted to helping the sick and homeless eventually founding her own home for the elderly she donated so much that despite being born into one of the wealthiest families in Milan she died poor at the age of 80 and was buried in a mass grave for the impoverished my use of the term Witch is just a bit of fun it's directed to the curve and none of its Sinister connotations are directed at this remarkable person who I deeply respect anyway let's get back to the mathematics and explore the area under the whole curve because this turns out to have a surprising answer and it reveals the relationship the witch has to the [Music] circle to do this we need to integrate from negative Infinity to positive Infinity but by symmetry I'm just going to integrate from zero to infinity and double the answer at the end we handle this by integrating from 0 to H and then we'll let H tend to Infinity as we know this this just means evaluating Aran of X between 0 and H which is just Aran of H and the limit of this as H TS to Infinity is Pi / 2 and since we have to double the answer at the end we get that the area is pi now that's pretty cool but what's the area of that Circle well using p piun r s remembering that the radius is a half we obtain pi/ 4 that means that the area under the whole Witch is exactly the area of four of these circles and by the way that's true regardless of the size of circle you use to draw the witch how awesome is that and do you know what else has an area of four times this circle a sphere of the same diameter that's right the area under the whole curve defined by a circle of a given diameter has exactly the same area as a sphere of the same diameter I'll try to demonstrate this by starting with this sphere and plotting a big witch cutting it out and into pieces we can see that the curve almost perfectly covers the sphere I say almost I can't actually draw an infinite curve and the flat area cannot perfectly wrap around a curved sphere but hopefully this gives you the idea now my mind was blown by this but all I kept thinking was why why exactly do four of these circles fit Under The Curve is there an intuitive reason I couldn't find one anywhere but I did make a satisfying discovery that sort of puts things into perspective consider only the half wi for now defined by this semicircle we're going to pick two points and consider this band we Trace across finding two points on the circle and now consider this sector I want to compare the area of this sector to the area of its corresponding band and what more we can make this comparison in entirely without calculus we'll start by giving a different description of the curve because finding the area of a sector demands knowledge of this angle in here so we'll redefine the witch based on the angle the Ray makes with the Y AIS so call this angle in here Alpha and we want to describe the coordinates of this magic point in terms of alpha again Paul here if you want to work it out the x coordinate represents the base of this right angle triangle with a height of one and this angle in here is Alpha so that means that X is tan Alpha as for the y-coordinate draw in this line here by sales's theorem an angle subtended from the diameter of the circle is a right angle so here we have another right angle triangle and this time the hypotenuse is one which makes this opposite side sin Alpha and now we're going to project a line across because what we actually want is this length here this is why Y and we can get y by doing one subtract this length and that length is part of this right angle triangle and notice we can work out this angle here which is also Alpha so what is this red length let's call it a and looking at this top right angle triangle we have that sin Alpha equals opposite over hypotenuse so that's a over sin Alpha multiply by sin Alpha and we get that a is equal to sin^2 Alpha but y equals one subtract a in other words one subtract sin s Alpha in other words cos s Alpha so putting it all together we have that if the Ray makes an angle of alpha with the y- axis the magic point it defines is given by the coordinates tan Alpha cos squ Alpha this is called a parametric description of the curve as it depends on this extra parameter Alpha now back to our two points that Define this band we're going to choose them to be very close together the first will be defined by array of angle Alpha the second by array of angle Alpha plus five where five will be taken to be very small my diagram is absolutely not to scale but I'm just drawing it nice and big so you can see what's going on we'll deal with the smallness of fi and its implications in a second but just a reminder we want the area of this sector and the area of this band we'll start with the sector and to find its area we need this angle in here so allow me to draw yet another diagram so that we can find that we'll use the fact that this blue line and both of those red lines are radi so we have two isoceles triangles that means that this angle here is Alpha and this angle here is also Alpha + 5 and because angles in a triangle summ to Pi we know that this one is pi subtract 2 Alpha and this one is pi subtract 2 Alpha + 5 and so that angle between the two red radi is this one subtract this one we'll see that the pi is cancel the two Alphas will cancel and we'll be left with just two F now because fi is very small we're going to make three assumptions first we'll assume that this sector is actually basically a triangle then we'll assume that this band is actually basically a trapezium or trapezoid if you're in North America third we'll allow ourselves to use these well-known small angle approximations we can use them because these functions are incredibly close to those values when we take fi to be very small obviously the smaller we make fi the smaller the errors created by each of these assumptions making F arbitrarily small then summing up all of these areas is basically what it means to do calculus let's continue and find the area of this sector then which we're idealizing as a triangle using this formula for the area of a triangle we have half * half * half * s of 2 pi using our small angle approximation we can say that sin 2i is essentially 2 pi and we're dividing by 8 so all in all we have that this is equal to 5 over 4 now onto the trapezium which is significantly more complicated so let me clear away some of this board using the formula for the area of a trapezium we need to take half of this perpendicular height and then Times by the sum of the parallel sides start by thinking about this perpendicular height H now this comes from the fact that it's the X distance of this band so in other words it's going to be this x coordinate subtract this x coordinate that's going to give that length here so in other words it's tan Alpha + 5 subtract tan Alpha now these red distances here basically just come from the y-coordinates so this will be cos s Alpha + 5 + cos s Alpha and I almost forgot we also need to times this by a half let me get rid of that trapezium now cuz I don't think we need it and now we proceed with the headache of trying to simplify this damn thing now this is a great exercise if you're currently learning about trigonometric identities and that sort of thing so pause now if you want to work it out for yourself otherwise let's continue take a deep breath we're going to start with the COS 2 Alpha + 5 + cos s Alpha now this is equal to cos 2 Alpha + 2 5 + 1 all over 2 plus cos 2 Al + 1 / 2 and that's using the fact that cos s of theta is equal to cos 2 + 1 now we'll factor out that half collect those ones together and now use the double angle formula for cos just about fit they done and now we'll invoke our small angle approximations saying cos 2i is basically 1 and sin 2i is essentially 2 pi now we can see we've got two cos 2 Alphas two of these two of these so we can now re-expand that half into there to simplify things now we don't like that two alpha usually if we take the cosine or S of something in problems like this we want to just do it of Alpha not two alpha so let's use our double angle formula and now finally one subtract sin s Alpha is another cos squ Alpha so we have two cos squ alphas and two 5 sin Alpha cos Alphas so we can bring the two out and we're done although it might not look like it you might be wondering what on Earth are you doing but it'll come clear in a second don't question the which so we can rewrite that whole expression in red with this expression down here and now the next step is to simplify this expression in blue another deep breath and let's start by replacing tan Alpha + 5 with our sum of angles formula and now we'll immediately invoke the small angle approximation to write tan 5 as five and now we want to subtract this tan Alpha think of it as tan Alpha over 1 and now multiply numerator and denominator by 1 subtract 5 tan Alpha now there's a common denominator and we can subtract okay now we can subtract these numerators so we have tan 5 here subtract tan Alpha * 1 So Tan Alpha subtract tan Alpha that's going to die so we'll be left with a 5 and then we have - tan Alpha * - 5 tan Alpha so that's POS 5 tan^ 2 Alpha and now there's a common 5 up here so let me factor that out and now we use the identity that 1 + tan^ 2 Alpha is SEC squ Alpha but I'm actually going to write it as 1 / cos s Alpha and then this original denominator of 1 - 5 tan Alpha I'm going to rewrite that tan Alpha as Sin Alpha over cos Alpha why are we doing that because tan is the worst always get rid of it now let's just simplify this whole thing the numerator will be F and the denominator will be cos s Alpha subtract 5 sin Alpha cos Alpha and we're done so let's replace that blue expression up there with this yellow expression here and we'll watch some fireworks so as grueling as this whole process has been look what happens now the half cancels with the two but that's not all the cos^2 alpha minus 5 sin Alpha cos Alpha cancels as well and is there anything left yes there is a f so after all of that the the area of this idealized band is just five and that is four times the area of this idealized sector cuz we said that that was five over four oh it's so good and that'll be true regardless of the two points we choose as long as we choose them to be very close together so if we split the whole curve up into bands thereby splitting the whole circle up into sectors each band will have area of four times its corresponding sector therefore the area of this whole half which is four times the area of this semicircle and therefore the area of the whole wi is the area of four of those circles oh I just love it this reveals a fundamental connection between the area of the curve and the area of this circle which is why Pi pops up in the area under the witch because the formula for the witch can be represented as this infinite series Pi sneaks in as a result of this infinite Series so to conclude while there might not be a good reason for the name which the curve certainly hides a lot of mathematical Magic thank you kindly for the view I'm glad you watched all the way through to my patrons as always the biggest thank you please consider supporting me too if not don't worry but do be so kind before you leave be sure to subscribe I hope our study of the witch for now did scratch your mathematical itch this has been another proof as always under another roof

Channel: Another Roof

Views: 29,079

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Id: ZcbWjD6HA_s

Length: 29min 37sec (1777 seconds)

Published: Tue Oct 31 2023