Austin Edward those two ideas are practically the same. a circle literally is the collection of points a given distance from another point, on a plane.
@@rH441000 a "side" is a "line segment that joins two vertices in a shape or two-dimensional figure." If a circle is a two dimensional figure that is defined by its vertices, then it MUST have sides.
a circle only practically has infinite sides. mathematically it has zero sides because no matter how closely you look a piece of circle, that piece will always seem an arch and never a "line"
It is just a matter of classification, I don't think it has much difference if you either define it as 0 or infinite. It is a very unique shape after all. Unless you are doing math for the sake of math, called pure mathematics-of which actually ended up having very useful physical implications such as water behavior- utility is much more important than just an issue of classification of a rare case, at least that is what I think
you would have to write taylor series for it (no pi needed). but to use it you would have to convert 180 degrees to radians (pi/2). so the video is pointless
tampicokid i meant you could write the thaylor series for sin(x). By "it" I meant any x. So you can take sines of any angle without knowing pi. The problem here is that we can't find the angle they want us to input.
Thank you for this wonderfully explanatory video. I am not a mathematician, but over the years I have come to love math. Lately I've been wondering how do we know that the ratio of the circumference of a circle to its diameter is Pi, an irrational number that goes on into infinity without repeating? After all, we couldn't hope to measure it more than to a handful of numbers after the decimal point. The video does a great job for fools like me even if it's not the exact method Archimedes used. Thank you.
This method, by using inscribed polygons, guarantees that each result will be < π ; that is, it approaches the desired limit from below. If you instead use circumscribed polygons, each result is > π ; that is, it approaches the desired limit from above. When you do the trigonometry for that, you find that you simply replace the "sin" with "tan": π = lim [n→∞] n tan(180º/n) If you want to get a bit more sophisticated, you can show that the limit of the ratio of errors, tan-to-sin, is 2, so that a weighted average, (n/3) [ 2 sin(180º/n) + tan(180º/n) ] will converge to π faster than either sequence alone.
Not exactly sure what you're asking, but start at 1:50 in the video, and onward to 3:20 or so. The circle has radius r = ½, which is also the hypotenuse, c, of right triangle ADB, with right angle at D. c = ½ BD = a = (½ the side of a regular polygon of n sides) = ½ s θ = ∠BAD = ½ the central angle of the polygon; so θ = ½·2π/n = π/n Now you know that, in right triangle ADB, a = c sinθ = ½ sinθ So: sinθ = 2a = s = the side of the polygon So what things are you asking about, that have "different angles"?
+ C Noone Well, you *can* state that; but you'd be wrong. In particular, Archimedes, who as you say, didn't carry all the way to the end (that is, to infinity!), his procedure for finding π ; he *did*, however, demonstrate that it lies between these lower and upper bounds: 3¹⁰/₇₁ < π < 3¹/₇ ; that is, 3.140845... < π < 3.142857... It follows then, that π < 3.1446...
So The Sumerians And Archimedes (Proposition 1) and 3 x r 2: Are wrong Despite all three being in total agreement as to having 10, 800 square centimetres to the circle's area, I really don't think so and perhaps you would also like to try and disprove my twelve steps to the sphere? Reference Estimating the wealth; Encyclopedia Britannica. A Babylonian cuneiform tablet written some 3,000 years ago treats problems about dams, wells, water clocks, and excavations. It also has an exercise in circular enclosures with an implied value of π pi = 3. The contractor for King Solomon's swimming pool, who made a pond 10 cubits across and 30 cubits around (1 Kings 7:23) used the same value, which would be correct if π is estimated as 3. Quote: *which would be correct if π is estimated as 3* The sheer lack of insight conveyed in this comment is truly astounding. The Sumerians were masters of mathematics, straight linear geometry, and curved differential geometry millennia before the Greeks came along, and it is they who using their differential geometry skills, were the inventors of clocks and time. Fundamentally A circle is a round shape, whose curvature consists of a continuing series of points equidistant from a fixed point (it's centre). The curved edge length of a round shape is a continuing series of points equidistant from a fixed point (it's centre). The circumference of a circle is the width (area) of a "radiated" drawn line, that serves to surround and enclose the radiated shape (area) of the circle. Oxford English Dictionary Circle: a round plane figure whose boundary (the circumference).....consists of points equidistant from a fixed point (the centre). The first part of this dictionary definition is based in/on the Euclidean linear thinking of more than two millennia ago. The second part of this dictionary definition is more on par with modern day physics, whereby it can be understood that the equidistant points to the circles centre are spatial points not linear. Circles shapes and bodies do not have any boundary lines as per Euclidean thinking, all circles have a quantity of area. and all bodies have a quantity of volume. However, i.e. when we look at a dinner plate from a Euclidean perspective we see it outlined against its background, just as we perceive a silhouette being outlined against a lighted background. Whereas, when we look at a dinner plate from the physic's perspective, we are aware that there is no outline as such, but rather that the denser atomic structure and reflecting the colour of the plate, is simply standing out from the less dense surrounding atmosphere. As opposed to nature, in order to make a circle we have to use a drawing compass to radiate a circle, while simultaneously drawing a line around it to enclose it, and so define the limits of its area. However Mother nature has no need, she simply radiates unbounded circles and cycles (continuum's) of energy conversions in motion, ad-infinitude. THE LENGTH OF A CIRCLES EDGE Using a 120-centimetre length of diameter multiply this by 3 1. The circle's edge length is 360 cm long 2. The circle's edge has 360 degrees of subdivision 3. The circle's edge has 360 degrees and each degree is 1 centimetre long SUMERIAN METHOD FOR CALCULATING THE AREA OF A CIRCLE Using a 120-centimetre length of diameter multiply this by 3 1. The Circles Edge is 360 cm long 2. Multiply the 360 centimetres "Edge Length" by itself = 129, 600 square centimetres 3. Divide 129, 600 by 12 = 10, 800 Square Centimetres to the Area of the Circle ARCHIMEDES: PROPOSITION 1. The area of any circle is equal to a right-angled triangle in which one of the sides about the triangle is equal to the radius, and the other to the circumference of the circle. Archimedes Triangle The Circle in question has a 120-centimetre Diameter length 1. The base right-angle is equal to the radius of 60 centimetres 2. The area of the circle is equal to the above right-angle triangle, which has one side that is equal to the 60-centimetre radius, and the other to the 360-centimetre circumference of the circle 3. The 360-centimetre height of the right-angle is equal to 6 x the 60-centimetre radius length 4. (1r) 60 centimetres x (6r) 360 centimetres is 21, 600 square centimetres the area of the rectangle 5. Half of the rectangle is 10, 800 square centimetres 6. The area of the triangle is half of the 1r x 6r rectangle 7. Half of the 1r x 6r rectangle is 1r x 3r 8. (1r) 60 centimeters x (3r) 180 centimeters = 10, 800 square centimeters THREE TIMES THE RADIUS SQUARED 1. The Diameter of the Circle is 120 centimetres 2. The diameter x 120 centimetres gives, 14, 400 square centimetres to the square of the diameter 3. The 60-centimetre radius x 60 centimetres gives, 3, 600 square centimetres to the square of the radius 4. The square of the radius x 3 gives, 10, 800 square centimetres to the area of the Circle SUMERIAN AREA: 10, 800 square centimetres ARCHIMEDEAN AREA 10, 800 square centimetres THREE TIMES THE RADIUS SQUARED: 10, 800 square centimetres Twelve Steps From The Cube, To The Sphere (Volume & Surface Area) Calculating the surface area and volume of a 6-centimetre diameter sphere, obtained from a 6-centimetre cube. 1. Measure the (a) cubes height to obtain its Diameter Line, which in this case is 6 centimetre’. 2. Multiply 6 cm x 6 cm to obtain the square area of one face of the cube; and also add them together to obtain the length of the perimeter to the square face = Length 24 cm, Square area 36 sq cm. 3. Multiply the square area, by the length of diameter line to obtain the cubic capacity = 216 cubic cm. 4. Divide the cubic capacity by 4, to obtain one-quarter of the cubic capacity of the cube = 54 cubic cm. 5. Multiply the one quarter cubic capacity by 3. to obtain the cubic capacity of the Cylinder = 162 cubic cm. 6. Multiply the area of one face of the cube by 6, to obtain the cubes surface area = 216 square cm. 7. Divide the cubes surface area by 4, to obtain one-quarter of the cubes surface area = 54 square cm. 8. Multiply the one quarter surface area of the cube by 3, to obtain the three quarter surface area of the Cylinder = 162 square cm. CYLINDER TO SPHERE 9. Divide the Cylinders cubic capacity by 4, to obtain one-quarter of the cubic capacity of the Cylinder = 40 & a half cubic cm. 10. Multiply the one quarter cubic capacity by 3, to obtain the three quarter cubic capacity of the Sphere = 121 & a half cubic cm, to the volume of the Sphere. 11. Divide the Cylinders surface are by 4, to obtain one-quarter of the surface area of the Cylinder = 40 & a half square cm. 12. Multiply the one quarter surface area by 3 to obtain the three quarter surface area of the Sphere = 121 & a half square cm, to the surface area of the Sphere Confirmation by Weight Given that the 6 Centimeter Diameter Line Sphere was obtained from a Wooden Cube weighing 160 grammes, prior to it being turned on a wood lathe into the shape of a sphere The Cylinder of the Cube would weigh 120 grammes The waste wood shavings would weigh 40 grammes Given that the Cylinder weighed 120 grammes The waste wood shavings would weigh 30 grammes. Note: And ironically you can also obtain this same result by volume, using Archimedes Principle. www.fromthecircletothesphere.net
Not weird. It's basically a pi to 2r ratio, like, "how many times pi would fit in 2r. Pi is basically X here. I don't know how many decimals they had and ended up being correct. (Quite sure they had a better method than calculation by observation by using a thread"). Ambiguous sentence...
mathemental used pi to determine sine values in order to determine the value of pi,stay tuned next week when he and charles darwin solve the chicken and the egg paradox
@@josepeixoto3715 The question here is how, at its simplest would you calc pi with pen and paper. Archy did not use trig function at all, because it is not necessary. Trig functions inherently have pi contained in them. Lets say, with pen/paper you are going to calc the sine of 15 deg. You could do it with the power series x - x^3 + x^5 -x^7......... x is the 15 deg angle expressed in radian. Radian is essentially composed of pi. For instance to express your 15 deg in radian, you first must multiply 15 deg by pi/180 deg. Notice you must already know pi. Maybe this was poorly said, but it demos that using trig function to find pi, is simply employing pi to calc pi
@@ronalddump4061 No I get what you're saying. This exactly what I've been looking into lately. Everything I research is using pi to explain how we get pi like that. It doesn't make sense.
i liked this only for showing some logic. (though i had to think it through a bit) here's it in simplest explanation. 1. split a circle into a number of triangle segments (360/n) 2. half into right triangle (180/n) (to get the opposite length with sin) 3. you would double output for both halves, but sin uses radius 1 (meaning diameter 2) so you would also halve to correct that. 4. you have the segment perimeter! tot them all up for an approximate perimeter! (this cheat method wasn't how we got pi btw)
Many people are saying that this is not correct , I don't know if it's true or not but thanks for making efforts to trying to help us please make a video on the true method this was a nice video.
By using Calculus, you can turn the sin function into an infinite series (Which I believe was the original method). Integrals are great a summing an infinite number of 'zeros' to get a constant.
I actually found this on my own feeling I had an inadiquate understanding of pi so that means I’m as smart as Archimedes the only difference… this guy did it 2300 years ago with a quill and paper no calculators no graphing calculators and literally no teaching using his own number system which most of modern society based off what a legend
This is a circular argument, as one needs to know the value of pi to compute sin(180º/n). This is because sin(180º/n) = sin(pi/n), and any value of sin(x) must be computed with a series (or by direct measure drawing a big circle). So, as n goes to infinity, sin(pi/n) goes to pi/n, and n*pi/n = pi. Yes, I know that this is a introductory video to the topic, but I found important to comunicate this.
At 1:40 line AD actually bisects angle BAC. Therefore angle BAD (which is angle theta) is half of angle BAC. I'm sure there's some way of proving this, but its tedious.
It's because triangle ABC is isosceles (AB = AC = radius = 1/2). In isosceles triangles, the altitude to the base is also the angle bisector. Therefore AD is the angle bisector of BAC, and thus BAD is half of BAC.
i want to ask, HOW did Archimedes know in the first place that a circle's circumference is 2pi*r??? You guys mention as if this equation is something all kids BC knew. Archimedes needs to get PI first before he could write down the equation and you guys mess it all up by mentioning the equation first.
rawan hs 2pi*r = pi*d meaning pi is just the ratio of any circle's circumference divided by its diameter. So just take string, measure its length then make it circular and measure its diameter and divide them.
π is the result of the division between the greek words ΩΚΕΑΝΟΣ (ocean) and and ΝΕΙΛΟΣ ( Nile river) when you transform them into numbers according to the ancient greek numbering system where Ω=800 etc. It is very amazing how you may realize the real world on a different basis. Imagine that this division is older then 5000 years.
Just figured out this formula independently before finding this video. It's amazingly simple, yet all we ever hear about is that ugly and cumbersome "pi=(4/1)-(4/3)+(4/5)..." or "throw a bunch of stick at horizontal lines" or some crap like that.
This was not Archimedes method, he used Pythagoras formulas, there was no angle in the formula. This video is BS! If you are using angle to calculate using computer, then you have to use Pi() to calculate radians. So it is already calculating backwards the value of Pi computer already knows.
DarthHater100 I don't know what I was thinking by my comment now I am reading it again. It doesn't matter if it is radians or degrees, *RPdigital" 's comment is right, this method is circular.
However, note that sines can be calculated without pi. Sine 30 degrees = 0.5 and then you need a half angle formula for sine 15 degrees (or use the sine difference formula since sine 45 degrees is also empirical), sine 7.5 degrees, etc... It's a lot of work but it's doable.
WireMiceOverKingWhea Sure, the 45, 60, 30 deg. etc angles work out to perfectly and you can use some calculus to get half angles and what not, but this is still circular lol. He's not using perfect angles.
Retro Gaming - Clash Of Clans i also got that on my head but on the other hand it wouldn't really make sense because infinity times something is always equal to infinity as for the 180 something devided by infinity will always add up to infinity.
By that logic calculating pi precisely be equivalent to dividing by 0 or a divergent infinite sum. That makes more more intuitive sense of euler’s infinite series calculations ending up with pi
-Take the first three odd integers: 1,3,5 -Double them thusly: 113355 -Divide the last three by the first three thusly: 355/113 There ya go, Pi accurate to 6 decimal places!
Triangle ABC is an isosceles triangle, since AB and AC are the radii of the same circle. For any isosceles triangle, the median from the symmetry vertex is the bisector of that vertex, as well as the it being perpendicular to the opposing side. This is easily demonstrated by using reductio ad absurdum and falsifying that 2 sides are of equal length, which was the initial premise.
After a bit of further looking through this, it's clear to see that n sin(180°/n) as n-> inf It will always result in what the numerator of the sin is. Since sin of 180° is pi, it gives you pi. So if I put a 1 or 2 or 3 up there, as the number n gets larger the result will alway be the sin of the numerator.
A very creative video. It attracted so much interest. Maybe I should inject more creativity into my videos. How can I be more creative with my Maths lessons?
I swear to you that i was literally just thinking about that exact method when i saw the demonstration of why the sum of all angles in a convex poligon is equal to 180*(n-2). I went to youtube just to realize that archimedes already discovered about it two thousand years ago, lol
Alternatively, take an unit circle centered at O. If Pm is a regular polygon of m sides inscribed inside the unit circle and if Am is the area of that inscribed polygon the area A2m of P2m is defined as A2m = (m/2)√(2-2√(1-(2Am/m)²)). Starting with A4 = 2 (the area of a square of sides √2), one can calculate π more accurately by taking higher and higher values of m. Intuitively, since the area of the unit circle is πr²=π1²=π, and since higher and higher values of m gives better approximation of its area as is evident from the video, this must be true. Now to get the first equation, A2m = (m/2)√(2-2√(1-(2Am/m)²)), one has to do a little more work. Taking two consecutive vertices (say A and B) of the inscribed regular polygon and the centre of the circle O defines two half lines OA and OB. These two along with the half line AB defines a triangle OAB. For convenience OB is supposed to lie on the positive x axis and OA could be reached through an anticlockwise rotation of arbitrary degrees or positive radians, determined by the inscribed regular polygon. Draw a perpendicular from A to OB meeting it at C. Then the area of OAB is related to area of OAC by ar(OAB)=(1/2)√((1-√(1-16(ar(OAC))²)/2). (Hint: if x and y are two sides of the triangle OAC then ar(OAC)=xy and x²+y²=1.) Use this to get A2m = (m/2)√(2-2√(1-(2Am/m)²)). There are many more interesting methods.
Try this: The cord length of 1/60° (minute of angle) included angle in inches. At exactly 300 yards (10800) inches. sin 1/120° x 10800 x 2 = PI.....My calculator rounds to 9 digits, excel can round to 100 digits.
You proved that: π= n.sin(π/n), as n goes to infinity which leads to: π/n =sin(π/n), as n goes to infinity which is true for any value of π, and it does not need to be proven, because the opposite side of the angle coincide with the opposite arc as the angle approaces zero. I think the ancients calculated the value of π experimentally.
Question, if you use radians and not degrees, then, because we already know a circle with radius=1/2 has circumference=pi; then the (length of a side of the polygon)=sin(theta) and (theta)=(pi/2n) ---- which is wrong. Theta=(pi/n). If you're thinking of it in degrees you must divide by two, but if you're thinking in arc-lengths there's no need, but then what (theta) are we measuring? What am I missing?
There's no perfect circle in nature. The universe is pixelated at the planck scale, just as minecraft and your computer screen are made of pixels. Geometric circles are 3.14..., and kinematic circles are 4!
Ok so pi is n*sin(pi/n) where n approaches infnity. But now we have a problem what to put as pi here. Fortunately we have that useful video that told us the solution so we can write pi as n*sin(n*sin(pi/n)/n) but now we have a problem what to put as pi here. Fortunately we have that useful video that told us the solution so we can write pi as n*sin(n*sin(n*sin(pi/n)/n)/n) but now we have a problem what to put as pi here. Fortunately we have that useful video that told us the solution so we can write pi as n*sin(n*sin(n*sin(n*sin(pi/n)/n)/n)/n) but now we... oh wait.
Archimedes lived a long time before Aryabhata. Though, what does it matter? Greek or Indian, they were both human. Also, pi, or how many circumferences of a circle for into its diameter, is not a rational number, that is, can't be expressed as a ratio of two whole numbers, which means that you can't actually get an "exact value of pi".
(180/x)*(sin(x)/cos(0.5x)) , x has to be in degrees and the closer you get it to 0, the better you get it. Oh, and you can't simplify it using identities because it won't work then (I think). I think its takes away a solution or something. But yeah, this works.
+ tampicokid: "there are power series for pi, but they['re] all derived from knowing pi in the first place." No they aren't. π = 4(1 - ⅓ + ⅕ - + . . .) = 2(1 + ⅓ + 1·2/3·5 + 1·2·3/3·5·7 + 1·2·3·4/3·5·7·9 + . . .) There's absolutely no reference to π on the right hand side of those. And you can find scads of others on the Wikipedia page on π .
tampicokid No, the method in the video is not "circular" in the logical sense. Rather, it comes from the Taylor series for arctan: tan⁻¹x = x - ⅓x³ + ⅕x⁵ - + ... And then knowing that tan⁻¹1 = ¼π gives you the series used in the vid. Note carefully that that Taylor series is valid only for -1 ≤ x ≤ 1 so x = 1 is right at the edge of validity. On your other remark, I fail to see how φ, the Golden Ratio, is related to π. For one thing, φ = ½(1 + √5) is algebraic, being the positive root of x² - x - 1 = 0 while π is transcendental; i.e., not the root of any polynomial equation with all-integer coefficients. So you're right to say that there's no real connection between π and φ; consequently, there's no finding "exact π in the typically divided golden rectangle." Unless you mean that you can draw a quarter-circle inside any square.
+tampicokid: The construction you give of a "golden spiral" isn't the true logarithmic spiral that passes through the corners of the squares in an infinitely-sectioned golden rectangle; it consists rather, of quarter-circular arcs, each of length ½πs, for each square of side s; the total length of which is a geometric series with a₀ = π , r = 1/φ = φ-1 whose sum is S = a₀/(1-r) = π/(2-φ) = φ²π = (φ+1)π. But it contains "π" only because the quarter-circular arc of radius = 2, has length = π . Which has nothing to do with φ, per se. But as I said, this is not a true logarithmic spiral; its curvature is discontinuous wrt path length, while that of a true logarithmic spiral is continuous. Such a spiral drawn on the infinitely-tesselated golden rectangle, doesn't stay "inside the box(es)," while passing through the required corners. And off hand, I don't have the formula for the length of a logarithmic spiral of given initial radius and given radial aspect angle. Does it involve π?
+tampicokid: Actually, the differences alternate in sign, and the integers are what are known as Lucas ("loo-kah," because their inventor, M. Lucas, was French) numbers, which are sums of pairs of Fibonacci numbers that are consecutive-but-one: Lₐ = Fₐ₊₁ + Fₐ₋₁ ; a = 0, 1, 2, 3, ... Lₐ = 2, 1, 3, 4, 7, 11, 18, 29, ... x = φª = Lₐ + (-φ)⁻ª = Lₐ + (-1)ª/x . . . . . this is your discovery; it can be cast and solved as a quadratic equation: x² - Lₐx - (-1)ª = 0 x = φª = ½Lₐ + √[¼Lₐ² - (-1)ª] And that can be shown from the solution of the recursion for the Fibonacci numbers: Fₐ = [φª - (-φ)⁻ª]/√5 and the reverse formula: φª = Fₐφ + Fₐ₋₁ .
What about the upper bound that archimedes used? I remember learning that archimedes drew a figure inside and outside a circle of radius 1/2 in order get a more accurate answer. By the way, if this is the method he used to calculate pi, then it would be similar to rule within limits of trigonometric functions.
I Have A Doubt In This Question Is That You Say Sin Theta is opposite/hypotenuse i.e a/1/2 But Is Sin Theta Is Not equal To Perpendicular/hypotenuse that Will Be Equal To In This Case Will Be B/1/2 PLZ CORRECT ME
HOLD ON, you said that point A was the radius, and that point B was the first point on the circumference. Then you simply added C, and inbetween D, to be perpindicular to them. After which you said Theta was the equation created from the right angles symmetrical from B and C, from D, all connected to A? How does the curved circumference of a sphere with D being perpindicular to point B & C, create the perfect 90.degree angles? Wouldn't it have to be parallel?
Hi, I was messing around with factorials yesterday. And came up with a formula for pi. I was wondering how many digits this formula is accurate to. (-1/2)!(2)
I know this may be a bit of a silly question but... If he knew that Pi was a constant value that appear in every circle no matter it's small or big, why didn't he take a piece of paper (or anything like), fold it like a cilinder, take the measures and simply divide it?
lim xsin (n/x) =lim sin (n/x)/(1/x)= x->oo x->oo t=1/x x->oo ,t->0 lim sin (nt)/t = n t->0 so yeah any number would work not just pi :) and sin function won't work unless x in radians nice circular tho :)
Him realising that the more sides your polygon has the better the approximation is basically the true discovery of integral calculus.
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A circle is usually defined as having 0 sides when, in fact, it has infinite.
Austin Edward those two ideas are practically the same. a circle literally is the collection of points a given distance from another point, on a plane.
@@rH441000 a "side" is a "line segment that joins two vertices in a shape or two-dimensional figure." If a circle is a two dimensional figure that is defined by its vertices, then it MUST have sides.
a circle only practically has infinite sides. mathematically it has zero sides because no matter how closely you look a piece of circle, that piece will always seem an arch and never a "line"
It is just a matter of classification, I don't think it has much difference if you either define it as 0 or infinite. It is a very unique shape after all. Unless you are doing math for the sake of math, called pure mathematics-of which actually ended up having very useful physical implications such as water behavior- utility is much more important than just an issue of classification of a rare case, at least that is what I think
@@johnjordan3552 it has zero sides. period.
Now the problem is: how do you compute sin(180°/n)?
Presumably, you construct an enormous right-angled-triangle with angle theta and take the ratio of the opposite to the hypotenuse.
That's exactly my question... sin is defined in terms pi :/
you would have to write taylor series for it (no pi needed). but to use it you would have to convert 180 degrees to radians (pi/2). so the video is pointless
tampicokid i meant you could write the thaylor series for sin(x). By "it" I meant any x. So you can take sines of any angle without knowing pi.
The problem here is that we can't find the angle they want us to input.
tampicokid You can calculate sine of any number i give you. sin(1) sin(2) sin(34). where 1 is 1 radian, 2 is 2 radians.
Clearly explained, set by step, with an accurate and clear diagram. Excellent video. Thanks.
😢
3:48 3:50 3:50 3:51 3:51
Thank you for this wonderfully explanatory video. I am not a mathematician, but over the years I have come to love math. Lately I've been wondering how do we know that the ratio of the circumference of a circle to its diameter is Pi, an irrational number that goes on into infinity without repeating? After all, we couldn't hope to measure it more than to a handful of numbers after the decimal point. The video does a great job for fools like me even if it's not the exact method Archimedes used. Thank you.
The most agreeable definition of pie I have heard so far
I've been looking for a good derivation for at least 1hour now and this one is perfect.
mathematiconline, thank you so much for your videos. You're a great teacher, and your videos make learning math enjoyable.
This method, by using inscribed polygons, guarantees that each result will be < π ; that is, it approaches the desired limit from below.
If you instead use circumscribed polygons, each result is > π ; that is, it approaches the desired limit from above.
When you do the trigonometry for that, you find that you simply replace the "sin" with "tan":
π = lim [n→∞] n tan(180º/n)
If you want to get a bit more sophisticated, you can show that the limit of the ratio of errors, tan-to-sin, is 2, so that a weighted average,
(n/3) [ 2 sin(180º/n) + tan(180º/n) ]
will converge to π faster than either sequence alone.
sir, could you please tell me how it,'s possible that
sin theta is equal to the side of the polygon (2a=2a)while both have a different angles
Not exactly sure what you're asking, but start at 1:50 in the video, and onward to 3:20 or so.
The circle has radius r = ½, which is also the hypotenuse, c, of right triangle ADB, with right angle at D.
c = ½
BD = a = (½ the side of a regular polygon of n sides) = ½ s
θ = ∠BAD = ½ the central angle of the polygon; so
θ = ½·2π/n = π/n
Now you know that, in right triangle ADB,
a = c sinθ = ½ sinθ
So:
sinθ = 2a = s = the side of the polygon
So what things are you asking about, that have "different angles"?
+ C Noone
Well, you *can* state that; but you'd be wrong.
In particular, Archimedes, who as you say, didn't carry all the way to the end (that is, to infinity!), his procedure for finding π ; he *did*, however, demonstrate that it lies between these lower and upper bounds:
3¹⁰/₇₁ < π < 3¹/₇ ; that is,
3.140845... < π < 3.142857...
It follows then, that π < 3.1446...
So The Sumerians And Archimedes (Proposition 1) and 3 x r 2: Are wrong Despite all three being in total agreement as to having 10, 800 square centimetres to the circle's area,
I really don't think so and perhaps you would also like to try and disprove my twelve steps to the sphere?
Reference Estimating the wealth; Encyclopedia Britannica.
A Babylonian cuneiform tablet written some 3,000 years ago treats problems about dams, wells, water clocks, and excavations. It also has an exercise in circular enclosures with an implied value of π pi = 3. The contractor for King Solomon's swimming pool, who made a pond 10 cubits across and 30 cubits around (1 Kings 7:23) used the same value, which would be correct if π is estimated as 3.
Quote: *which would be correct if π is estimated as 3*
The sheer lack of insight conveyed in this comment is truly astounding.
The Sumerians were masters of mathematics, straight linear geometry, and curved differential geometry millennia before the Greeks came along, and it is they who using their differential geometry skills, were the inventors of clocks and time.
Fundamentally
A circle is a round shape, whose curvature consists of a continuing series of points equidistant from a fixed point (it's centre).
The curved edge length of a round shape is a continuing series of points equidistant from a fixed point (it's centre).
The circumference of a circle is the width (area) of a "radiated" drawn line, that serves to surround and enclose the radiated shape (area) of the circle.
Oxford English Dictionary
Circle: a round plane figure whose boundary (the circumference).....consists of points equidistant from a fixed point (the centre).
The first part of this dictionary definition is based in/on the Euclidean linear thinking of more than two millennia ago.
The second part of this dictionary definition is more on par with modern day physics, whereby it can be understood that the equidistant points to the circles centre are spatial points not linear.
Circles shapes and bodies do not have any boundary lines as per Euclidean thinking, all circles have a quantity of area. and all bodies have a quantity of volume.
However, i.e. when we look at a dinner plate from a Euclidean perspective we see it outlined against its background, just as we perceive a silhouette being outlined against a lighted background.
Whereas, when we look at a dinner plate from the physic's perspective, we are aware that there is no outline as such, but rather that the denser atomic structure and reflecting the colour of the plate, is simply standing out from the less dense surrounding atmosphere.
As opposed to nature, in order to make a circle we have to use a drawing compass to radiate a circle, while simultaneously drawing a line around it to enclose it, and so define the limits of its area.
However Mother nature has no need, she simply radiates unbounded circles and cycles (continuum's) of energy conversions in motion, ad-infinitude.
THE LENGTH OF A CIRCLES EDGE
Using a 120-centimetre length of diameter multiply this by 3
1. The circle's edge length is 360 cm long
2. The circle's edge has 360 degrees of subdivision
3. The circle's edge has 360 degrees and each degree is 1 centimetre long
SUMERIAN METHOD FOR CALCULATING THE AREA OF A CIRCLE
Using a 120-centimetre length of diameter multiply this by 3
1. The Circles Edge is 360 cm long
2. Multiply the 360 centimetres "Edge Length" by itself = 129, 600 square centimetres
3. Divide 129, 600 by 12 =
10, 800 Square Centimetres to the Area of the Circle
ARCHIMEDES: PROPOSITION 1.
The area of any circle is equal to a right-angled triangle in which one of the sides about the triangle is equal to the radius, and the other to the circumference of the circle.
Archimedes Triangle
The Circle in question has a 120-centimetre Diameter length
1. The base right-angle is equal to the radius of 60 centimetres
2. The area of the circle is equal to the above right-angle triangle, which has one side that is equal to the 60-centimetre radius, and the other to the 360-centimetre circumference of the circle
3. The 360-centimetre height of the right-angle is equal to 6 x the 60-centimetre radius length
4. (1r) 60 centimetres x (6r) 360 centimetres is 21, 600 square centimetres the area of the rectangle
5. Half of the rectangle is 10, 800 square centimetres
6. The area of the triangle is half of the 1r x 6r rectangle
7. Half of the 1r x 6r rectangle is 1r x 3r
8. (1r) 60 centimeters x (3r) 180 centimeters =
10, 800 square centimeters
THREE TIMES THE RADIUS SQUARED
1. The Diameter of the Circle is 120 centimetres
2. The diameter x 120 centimetres gives, 14, 400 square centimetres to the square of the diameter
3. The 60-centimetre radius x 60 centimetres gives, 3, 600 square centimetres to the square of the radius
4. The square of the radius x 3 gives, 10, 800 square centimetres to the area of the Circle
SUMERIAN AREA: 10, 800 square centimetres
ARCHIMEDEAN AREA 10, 800 square centimetres
THREE TIMES THE RADIUS SQUARED: 10, 800 square centimetres
Twelve Steps From The Cube, To The Sphere (Volume & Surface Area)
Calculating the surface area and volume of a 6-centimetre diameter sphere, obtained from a 6-centimetre cube.
1. Measure the (a) cubes height to obtain its Diameter Line, which in this case is 6 centimetre’.
2. Multiply 6 cm x 6 cm to obtain the square area of one face of the cube; and also add them together to obtain the length of the perimeter to the square face = Length 24 cm, Square area 36 sq cm.
3. Multiply the square area, by the length of diameter line to obtain the cubic capacity = 216 cubic cm.
4. Divide the cubic capacity by 4, to obtain one-quarter of the cubic capacity of the cube = 54 cubic cm.
5. Multiply the one quarter cubic capacity by 3. to obtain the cubic capacity of the Cylinder = 162 cubic cm.
6. Multiply the area of one face of the cube by 6, to obtain the cubes surface area = 216 square cm.
7. Divide the cubes surface area by 4, to obtain one-quarter of the cubes surface area = 54 square cm.
8. Multiply the one quarter surface area of the cube by 3, to obtain the three quarter surface area of the Cylinder = 162 square cm.
CYLINDER TO SPHERE
9. Divide the Cylinders cubic capacity by 4, to obtain one-quarter of the cubic capacity of the Cylinder = 40 & a half cubic cm.
10. Multiply the one quarter cubic capacity by 3, to obtain the three quarter cubic capacity of the Sphere = 121 & a half cubic cm, to the volume of the Sphere.
11. Divide the Cylinders surface are by 4, to obtain one-quarter of the surface area of the Cylinder = 40 & a half square cm.
12. Multiply the one quarter surface area by 3 to obtain the three quarter surface area of the Sphere = 121 & a half square cm, to the surface area of the Sphere
Confirmation by Weight
Given that the 6 Centimeter Diameter Line Sphere was obtained from a Wooden Cube weighing 160 grammes, prior to it being turned on a wood lathe into the shape of a sphere
The Cylinder of the Cube would weigh 120 grammes
The waste wood shavings would weigh 40 grammes
Given that the Cylinder weighed 120 grammes
The waste wood shavings would weigh 30 grammes.
Note: And ironically you can also obtain this same result by volume, using Archimedes Principle.
www.fromthecircletothesphere.net
i just doon't know what sin and theta is but i can surely tell he explained extremely well
Thank you for clear and easy explanation, I look for on many channels but your is the best. Really deserves like and subscribe.
"How did we get pi?
Remember that the radius of a circle is equal to 2(pi)r"
Weird flex but ok
It actually makes sense see his previous video
It's high school maths
Pi = 4(1/1-1/3+1/5-1/7+1/9-1/11+1/13-1/15...)
Not weird. It's basically a pi to 2r ratio, like, "how many times pi would fit in 2r. Pi is basically X here. I don't know how many decimals they had and ended up being correct. (Quite sure they had a better method than calculation by observation by using a thread"). Ambiguous sentence...
2πr
mathemental used pi to determine sine values in order to determine the value of pi,stay tuned next week when he and charles darwin solve the chicken and the egg paradox
used pi to find the sine,how is that? thanks
@@josepeixoto3715 The question here is how, at its simplest would you calc pi with pen and paper. Archy did not use trig function at all, because it is not necessary.
Trig functions inherently have pi contained in them. Lets say, with pen/paper you are going to calc the sine of 15 deg. You could do it with the power series x - x^3 + x^5 -x^7......... x is the 15 deg angle expressed in radian. Radian is essentially composed of pi. For instance to express your 15 deg in radian, you first must multiply 15 deg by pi/180 deg. Notice you must already know pi.
Maybe this was poorly said, but it demos that using trig function to find pi, is simply employing pi to calc pi
@@ronalddump4061 No I get what you're saying. This exactly what I've been looking into lately. Everything I research is using pi to explain how we get pi like that. It doesn't make sense.
Your videos are awesome and very informative and are on a different level from most explanations, Thank You.
i liked this only for showing some logic. (though i had to think it through a bit) here's it in simplest explanation.
1. split a circle into a number of triangle segments (360/n) 2. half into right triangle (180/n) (to get the opposite length with sin)
3. you would double output for both halves, but sin uses radius 1 (meaning diameter 2) so you would also halve to correct that.
4. you have the segment perimeter! tot them all up for an approximate perimeter! (this cheat method wasn't how we got pi btw)
The music is Cello Suite No.1 - Prelude, thanks for watching.
Many people are saying that this is not correct , I don't know if it's true or not but thanks for making efforts to trying to help us please make a video on the true method this was a nice video.
It, in fact, is NOT correct! Because, simply put, Archy did not use trig function to calc pi
WOW, Great video with understandable math!! Thanks! I could never have believen that it can be calculated so easily!
whoever came up with this method is really clever, should get the fields medal
best outro for a youtube video I have ever seen in my life
This channel is criminally underrated
Excellent. Wonderfully clear presentation. Thanks.
Excellent explanation
Thank you for this. I have a geometry test tomorrow and our
teacher never teaches us anything
LOL What kind of teacher is that?
By using Calculus, you can turn the sin function into an infinite series (Which I believe was the original method). Integrals are great a summing an infinite number of 'zeros' to get a constant.
Thank you for elaborate the method of π in such a easy manner
Very insightful video. Learned something new. Thanks for sharing
I actually found this on my own feeling I had an inadiquate understanding of pi so that means I’m as smart as Archimedes the only difference…
this guy did it 2300 years ago with a quill and paper no calculators no graphing calculators and literally no teaching using his own number system which most of modern society based off what a legend
This is a circular argument, as one needs to know the value of pi to compute sin(180º/n). This is because sin(180º/n) = sin(pi/n), and any value of sin(x) must be computed with a series (or by direct measure drawing a big circle). So, as n goes to infinity, sin(pi/n) goes to pi/n, and n*pi/n = pi. Yes, I know that this is a introductory video to the topic, but I found important to comunicate this.
great video .it helped me a lot to clear my doubt.the way of illustrating was awesome.thankyou
Great explanation, very intuitive and fast. Good job
At 1:40 line AD actually bisects angle BAC. Therefore angle BAD (which is angle theta) is half of angle BAC. I'm sure there's some way of proving this, but its tedious.
It's because triangle ABC is isosceles (AB = AC = radius = 1/2). In isosceles triangles, the altitude to the base is also the angle bisector. Therefore AD is the angle bisector of BAC, and thus BAD is half of BAC.
This may have been how Archimedes calculated Pi, but I don't think Archimedes had algebra to work with.
This is not the corect solution of pi is equal to 3.14 or 22÷7
This is not the corect solution of pi is equal to 3.14 or 22÷7
Shankar Sanap no thats only approximate
@@shankarsanap4724 dun dun dutsch!!
i want to ask, HOW did Archimedes know in the first place that a circle's circumference is 2pi*r??? You guys mention as if this equation is something all kids BC knew. Archimedes needs to get PI first before he could write down the equation and you guys mess it all up by mentioning the equation first.
Because Archimedes based his stuff of and Indian mathematician called ArryaBath!
rawan hs 2pi*r = pi*d meaning pi is just the ratio of any circle's circumference divided by its diameter. So just take string, measure its length then make it circular and measure its diameter and divide them.
Because thats how pi is DEFINED. The ratio between the circumpherence(C) and the diameter(d). So:
C/d=pi
C=pi×d
d=2r (by definition), so
C=2×pi×r
It was observed that circumference of circle increased linearly with increase in radius/diameter, so that constant was named pi.
B8. RV
it's mind boggling!
best explanation so far
Fantastic !! Thanx a lot for the video... Have a great Life !!
π is the result of the division between the greek words ΩΚΕΑΝΟΣ (ocean) and and ΝΕΙΛΟΣ ( Nile river) when you transform them into numbers according to the ancient greek numbering system where Ω=800 etc. It is very amazing how you may realize the real world on a different basis. Imagine that this division is older then 5000 years.
Tf you on abt??
Just figured out this formula independently before finding this video.
It's amazingly simple, yet all we ever hear about is that ugly and cumbersome "pi=(4/1)-(4/3)+(4/5)..." or "throw a bunch of stick at horizontal lines" or some crap like that.
FINALLY,i get it, Archi was a thinker... thanks
Thanks for your insight!
This was not Archimedes method, he used Pythagoras formulas, there was no angle in the formula. This video is BS! If you are using angle to calculate using computer, then you have to use Pi() to calculate radians. So it is already calculating backwards the value of Pi computer already knows.
But he's not using radians, he's using degrees.
DarthHater100 he could still not calculate sin
DarthHater100
I don't know what I was thinking by my comment now I am reading it again. It doesn't matter if it is radians or degrees, *RPdigital" 's comment is right, this method is circular.
However, note that sines can be calculated without pi. Sine 30 degrees = 0.5 and then you need a half angle formula for sine 15 degrees (or use the sine difference formula since sine 45 degrees is also empirical), sine 7.5 degrees, etc... It's a lot of work but it's doable.
WireMiceOverKingWhea Sure, the 45, 60, 30 deg. etc angles work out to perfectly and you can use some calculus to get half angles and what not, but this is still circular lol. He's not using perfect angles.
I took a modern simple interpretation of Archimedes' Method, but the overall approach to calculating pi is the same.
Ramanujan has calculated the value of pie accurately using a more appropriate method
🙋🙋🙋🙋🙋🙋👌👍👍👍👍👍👍
0:34 Thank you for adding that.
So, the exact value for pi would be: *∞*Sin(180/∞)*
Retro Gaming - Clash Of Clans i also got that on my head but on the other hand it wouldn't really make sense because infinity times something is always equal to infinity as for the 180 something devided by infinity will always add up to infinity.
Infinite is not a number, it's a concept. But I guess you could re-write that as a limit of x approaching infinite.
@@CMAR872 and you will get an (infinite/infinite)*sin(180) which is undefined ..
Ali Alhasan that’s why it’s a limit though you’re not actually plugging infinity in because that’s clearly impossible
Can any one help me?, the numbers at the beginning aren't right, by putting them in calculator the results are different!
By that logic calculating pi precisely be equivalent to dividing by 0 or a divergent infinite sum. That makes more more intuitive sense of euler’s infinite series calculations ending up with pi
Excellent video! I enjoyed this, thank you :)
-Take the first three odd integers: 1,3,5
-Double them thusly: 113355
-Divide the last three by the first three thusly: 355/113
There ya go, Pi accurate to 6 decimal places!
@element5817he said accurate to 6 places
nice! this is a great idea to find Pi.
Triangle ABC is an isosceles triangle, since AB and AC are the radii of the same circle.
For any isosceles triangle, the median from the symmetry vertex is the bisector of that vertex, as well as the it being perpendicular to the opposing side.
This is easily demonstrated by using reductio ad absurdum and falsifying that 2 sides are of equal length, which was the initial premise.
wow really impressive method
So is this why pie has no end in figures? Because it's an approximate estimate the the closer you refine it?
After a bit of further looking through this, it's clear to see that
n sin(180°/n) as n-> inf
It will always result in what the numerator of the sin is. Since sin of 180° is pi, it gives you pi. So if I put a 1 or 2 or 3 up there, as the number n gets larger the result will alway be the sin of the numerator.
Yes. It seems to me that he just used pi to find pi. If you have 180 degrees then you already have pi.
What programs or software you use to make this great video?
A very creative video. It attracted so much interest.
Maybe I should inject more creativity into my videos.
How can I be more creative with my Maths lessons?
thank you, I love you my friend
very informative! thank you.
Very Informative video
I swear to you that i was literally just thinking about that exact method when i saw the demonstration of why the sum of all angles in a convex poligon is equal to 180*(n-2). I went to youtube just to realize that archimedes already discovered about it two thousand years ago, lol
Wow thnx I was wondering how we was calculating pi
Alternatively, take an unit circle centered at O. If Pm is a regular polygon of m sides inscribed inside the unit circle and if Am is the area of that inscribed polygon the area A2m of P2m is defined as A2m = (m/2)√(2-2√(1-(2Am/m)²)). Starting with A4 = 2 (the area of a square of sides √2), one can calculate π more accurately by taking higher and higher values of m.
Intuitively, since the area of the unit circle is πr²=π1²=π, and since higher and higher values of m gives better approximation of its area as is evident from the video, this must be true.
Now to get the first equation, A2m = (m/2)√(2-2√(1-(2Am/m)²)), one has to do a little more work. Taking two consecutive vertices (say A and B) of the inscribed regular polygon and the centre of the circle O defines two half lines OA and OB. These two along with the half line AB defines a triangle OAB. For convenience OB is supposed to lie on the positive x axis and OA could be reached through an anticlockwise rotation of arbitrary degrees or positive radians, determined by the inscribed regular polygon. Draw a perpendicular from A to OB meeting it at C. Then the area of OAB is related to area of OAC by ar(OAB)=(1/2)√((1-√(1-16(ar(OAC))²)/2). (Hint: if x and y are two sides of the triangle OAC then ar(OAC)=xy and x²+y²=1.) Use this to get A2m = (m/2)√(2-2√(1-(2Am/m)²)).
There are many more interesting methods.
Excellent! ... but how to prove it is 22/7 ?
So your using π to caculate π?
Try this: The cord length of 1/60° (minute of angle) included angle in inches. At exactly 300 yards (10800) inches. sin 1/120° x 10800 x 2 = PI.....My calculator rounds to 9 digits, excel can round to 100 digits.
Did Archimedes have trig tables available? I thought he strictly used Pythagoras / square root calculations
You proved that:
π= n.sin(π/n), as n goes to infinity
which leads to:
π/n =sin(π/n), as n goes to infinity
which is true for any value of π,
and it does not need to be proven, because the opposite side of the angle coincide with the opposite arc as the angle approaces zero.
I think the ancients calculated the value of π experimentally.
7 years ago and i have never seen that explanation before in school.
Who tf sits and thinks about this. This is straight genius!
Nice explanation
By far the best demo up to now!
i just thought the same method...im happy that i thought like him
Question, if you use radians and not degrees, then, because we already know a circle with radius=1/2 has circumference=pi; then the (length of a side of the polygon)=sin(theta) and (theta)=(pi/2n) ---- which is wrong. Theta=(pi/n). If you're thinking of it in degrees you must divide by two, but if you're thinking in arc-lengths there's no need, but then what (theta) are we measuring? What am I missing?
Correct value of pie discovered by Ramanajun
Great explanation!
There's no perfect circle in nature. The universe is pixelated at the planck scale, just as minecraft and your computer screen are made of pixels. Geometric circles are 3.14..., and kinematic circles are 4!
Gravock hey, can you please explain more?
you should research plancks, there is too much to explain.
Can you tell me which the sequence that approaches pi from above( the genuinely decline function)
Very interesting Good job i recommend watching this.#
Archimedes also used the EXTERIOR polygons to establish the RANGE of the true value of PI. Not just the inscribed polygons.
Ok so pi is n*sin(pi/n) where n approaches infnity. But now we have a problem what to put as pi here. Fortunately we have that useful video that told us the solution so we can write pi as n*sin(n*sin(pi/n)/n) but now we have a problem what to put as pi here. Fortunately we have that useful video that told us the solution so we can write pi as n*sin(n*sin(n*sin(pi/n)/n)/n) but now we have a problem what to put as pi here. Fortunately we have that useful video that told us the solution so we can write pi as n*sin(n*sin(n*sin(n*sin(pi/n)/n)/n)/n) but now we... oh wait.
Do we have some special triangles for calculating very small values of sin(theta) ?
aryabhatta was the first person to find exact value of pi.
Archimedes lived a long time before Aryabhata. Though, what does it matter? Greek or Indian, they were both human. Also, pi, or how many circumferences of a circle for into its diameter, is not a rational number, that is, can't be expressed as a ratio of two whole numbers, which means that you can't actually get an "exact value of pi".
(180/x)*(sin(x)/cos(0.5x)) , x has to be in degrees and the closer you get it to 0, the better you get it. Oh, and you can't simplify it using identities because it won't work then (I think). I think its takes away a solution or something. But yeah, this works.
Archimedes,didnt make pi,he just thought of infinitely random numbers and after that he ate some pie,then he called it pi
Yeah, he thought up indefinite number of digits in pie whilst eating pie for 24 hours straight.
Good sir keep going
too bad 180 deg is equal to pi, so the equation is lim n--> infinity (n(sin(pi/n))) = pi. kind of recursive :/
This method works pretty good if you are from programming background.
Who watching before the big exam tomorrow
Oh oooooooooooh
Not me though😅
We can't use trig functions because they are based on radians. Circular.
+יהודה שמחה ולדמן circular is the entire point!
+ tampicokid:
"there are power series for pi, but they['re] all derived from knowing pi in the first place."
No they aren't.
π = 4(1 - ⅓ + ⅕ - + . . .)
= 2(1 + ⅓ + 1·2/3·5 + 1·2·3/3·5·7 + 1·2·3·4/3·5·7·9 + . . .)
There's absolutely no reference to π on the right hand side of those.
And you can find scads of others on the Wikipedia page on π .
tampicokid No, the method in the video is not "circular" in the logical sense. Rather, it comes from the Taylor series for arctan:
tan⁻¹x = x - ⅓x³ + ⅕x⁵ - + ...
And then knowing that
tan⁻¹1 = ¼π
gives you the series used in the vid.
Note carefully that that Taylor series is valid only for
-1 ≤ x ≤ 1
so x = 1 is right at the edge of validity.
On your other remark, I fail to see how φ, the Golden Ratio, is related to π.
For one thing, φ = ½(1 + √5) is algebraic, being the positive root of
x² - x - 1 = 0
while π is transcendental; i.e., not the root of any polynomial equation with all-integer coefficients.
So you're right to say that there's no real connection between π and φ; consequently, there's no finding
"exact π in the typically divided golden rectangle."
Unless you mean that you can draw a quarter-circle inside any square.
+tampicokid:
The construction you give of a "golden spiral" isn't the true logarithmic spiral that passes through the corners of the squares in an infinitely-sectioned golden rectangle; it consists rather, of quarter-circular arcs, each of length ½πs, for each square of side s; the total length of which is a geometric series with
a₀ = π , r = 1/φ = φ-1
whose sum is
S = a₀/(1-r) = π/(2-φ) = φ²π = (φ+1)π.
But it contains "π" only because the quarter-circular arc of radius = 2, has length = π .
Which has nothing to do with φ, per se.
But as I said, this is not a true logarithmic spiral; its curvature is discontinuous wrt path length, while that of a true logarithmic spiral is continuous. Such a spiral drawn on the infinitely-tesselated golden rectangle, doesn't stay "inside the box(es)," while passing through the required corners. And off hand, I don't have the formula for the length of a logarithmic spiral of given initial radius and given radial aspect angle.
Does it involve π?
+tampicokid:
Actually, the differences alternate in sign, and the integers are what are known as Lucas ("loo-kah," because their inventor, M. Lucas, was French) numbers, which are sums of pairs of Fibonacci numbers that are consecutive-but-one:
Lₐ = Fₐ₊₁ + Fₐ₋₁ ; a = 0, 1, 2, 3, ...
Lₐ = 2, 1, 3, 4, 7, 11, 18, 29, ...
x = φª = Lₐ + (-φ)⁻ª = Lₐ + (-1)ª/x . . . . . this is your discovery; it can be cast and solved as a quadratic equation:
x² - Lₐx - (-1)ª = 0
x = φª = ½Lₐ + √[¼Lₐ² - (-1)ª]
And that can be shown from the solution of the recursion for the Fibonacci numbers:
Fₐ = [φª - (-φ)⁻ª]/√5
and the reverse formula:
φª = Fₐφ + Fₐ₋₁
.
What about the upper bound that archimedes used? I remember learning that archimedes drew a figure inside and outside a circle of radius 1/2 in order get a more accurate answer.
By the way, if this is the method he used to calculate pi, then it would be similar to rule within limits of trigonometric functions.
Use the tangents, instead of c=1/2 we have b=1/2. So tan∆=2a. This gives π=n×tan(∆/n).
I Have A Doubt In This Question Is That You Say Sin Theta is opposite/hypotenuse i.e a/1/2 But Is Sin Theta Is Not equal To Perpendicular/hypotenuse that Will Be Equal To In This Case Will Be B/1/2
PLZ CORRECT ME
4:48 - That person can not only stand on water, but has a house there, too! There's no island :D
HOLD ON, you said that point A was the radius, and that point B was the first point on the circumference. Then you simply added C, and inbetween D, to be perpindicular to them.
After which you said Theta was the equation created from the right angles symmetrical from B and C, from D, all connected to A?
How does the curved circumference of a sphere with D being perpindicular to point B & C, create the perfect 90.degree angles? Wouldn't it have to be parallel?
Could you represent pi as a limit as n approaches infinity in this case?
More surprising is to notice n approaches infinity and sin of an angle approaching zero! Pi equals to infinity times sin of zero!
Hi, I was messing around with factorials yesterday. And came up with a formula for pi. I was wondering how many digits this formula is accurate to. (-1/2)!(2)
exactly.
Syndee Gaming How did you come up with this?
I was messing about with factorials. as I said
You're using the gama function to find (-1/2)! ?
great job:
Double.MAX_VALUE * java.lang.Math.sin(180D/Double.MAX_VALUE)
exactly what i thought, but maybe use max(unsigned int) x unsigned double (math.sin(180d/max(unisigned int))
I know this may be a bit of a silly question but... If he knew that Pi was a constant value that appear in every circle no matter it's small or big, why didn't he take a piece of paper (or anything like), fold it like a cilinder, take the measures and simply divide it?
Because that would not have been accurate
lim xsin (n/x) =lim sin (n/x)/(1/x)=
x->oo x->oo
t=1/x x->oo ,t->0
lim sin (nt)/t = n
t->0
so yeah any number would work not just pi :) and sin function won't work unless x in radians
nice circular tho :)
I was able to deduce these laws without the help of anyone, I wish to be a scientist like them