Archimedes, Inventing Calculus, and the Value of Pi

 

Let’s get one thing straight upfront; I blame @SaintAugustine. If you’ll recall he wrote a post about a month ago about Leibniz, who is smarter than I. I offered up a joking reply to his question naming someone else who’s smarter than I, the Greek mathematician Archimedes. He’s probably most famous for discovering the principle of displacement one day in the bath. He announced it by shouting “Eureka!” and (depending on the account) running through the streets to announce it without so much as a towel. Aside from the bathtub thing Archimedes is the mad scientist of antiquity, having devised weapons of war (including the first known death ray) to face off Roman invaders. That part of the legend has probably grown in the telling.

A Boy and His Circles

Matter of fact, Archimedes didn’t survive the Second Punic war. Rome sacked Syracuse after a long siege (having thus far been stymied by Mad Science). Archimedes was drawing diagrams in the dirt when a Roman soldier found him. “Stop disturbing my circles!” yells Archimedes. The Roman, not about to be lectured by some weird old guy, runs him through with his sword. Probably for the best; the mind shudders at what might have resulted had that Roman, equally interested in circles, stopped his looting in order to contemplate conic sections.

Archimedes was the first person to calculate the value of π as far as 3.14. That’s harder than it sounds. Before him, the usual way to do it was to take a wheel and string a rope around it to see what you could get. Depending on the roundness of the wheel, the stretchiness of the rope, and how much fiddling you’re willing to do the technique isn’t all that accurate. The Egyptians had narrowed the value down as far as 3.1 that way. Archimedes not only one-upped their accuracy, he also was the first to devise a method of reckoning the value of π that didn’t measure physical circles. He worked out a geometric relationship that allows you to approximate the value. Here, take a look at this:

Diagrams by means of AutoCAD. No other graphics program I’ve worked with allows you to draw measured distances; it’s all freehand. Do artists not train in geometry anymore? 

What we’ve got is a square inscribed in a circle inscribed in a square. The area of the circle has to be less than that of the big square (or it wouldn’t fit) but more than the area of the small square (or that square wouldn’t fit.) We know how to calculate the area of squares; it’s the length of a side, err, squared. We know how to calculate the area of a circle, it has to do with the constant π. To figure out π we figure out the area of the circle, which is between that of the large square and the small square.

How Big of a Pi are We Talking?

We know how to calculate the area of the circle from the radius:

Area = π*radius²

Let’s be tricky, and assume we’re dealing with an idealized circle with a radius of one. Therefore the area of the circle is going to be precisely π. It then remains to figure out how big our squares are.

First, the large square. Since we know the radius of the circle is 1, we know its diameter is 2. Since the circle touches the square on either side, we know the diameter of the circle is exactly as wide as the width of the square. And since our square is a square we know that its area is 2×2 = 4.

Second, the smaller square. Hold on, let me draw some diagonals in there:

The small square doesn’t have to be tilted that way for the geometry to work, it just makes it easier to draw. 

Again, we know the diameter is two, which also happens to be the diagonal of our smaller square. Using the Pythagorean theorem and the fact that the sides of the square are exactly the same length, we know that

diagonal² = side² + side²

Since the diagonal is 2 and our sides are precisely as long as each other, then

4 = 2*side²

2 = side²

But the square of the side length is also the area of the square, so we know the area of the smaller square is 2.

Based on that calculation we know the area of the circle (and hence the value of π) is somewhere between 2 and 4. Not exactly ground-breaking, I know. Archimedes upon making that discovery hardly shouted “Eureka”. But he can make it more accurate. You get better results when you use more sides to your polygon; the more sides they have the closer they get to a circle. I did the math once with a hexagon:

Hexagons are superior to squares in the quality of roundiness.

End result, I can tell you that π is between 2.598 and 3.468. Archimedes though isn’t satisfied. He’s not going to impress the Egyptian-in-the-streets with that discovery. When he derived his famous result he doubled his hexagon, then kept doing it until he got a 96 sided polygon. Here, let me draw one for you:

AutoCAD has a command for drawing regular polygons. Yes, that’s got straight lines and angled corners, no matter how much it looks like it’s actually a circle.

Calculus and People Smarter than Me

This is where I got into trouble. You see, Archimedes didn’t have to stop with a 96-gon. The only thing standing between him and even more digits of π is that the arithmetic gets harder. What if you use a polygon with more sides? Five hundred, or fifty thousand? You get an even more accurate representation of the number. Why stop there? Why not take it to the limit? As n gets arbitrarily large your polygons get arbitrarily close to the size of a circle. This whole calculating the area under a curve thing sounds a great deal like calculus. That brings me back to Leibniz. [1]

Could you use the Archimedes method to calculate π and generalize it into calculus? That was the question I was working on one night as I was trying to fall asleep. To figure out if the Archimedes method could be generalized I’d have to know the Archimedes method. Nothing for it but to re-derive the method using what I already know. (Mamas don’t let your boys grow up to be physicists. They’re always trying to pull that re-deriving nonsense.)

Here’s Looking at Euclid

First, a couple of assumptions. These ought to be provable by the geometry of Euclid, but not having access to a copy of the Elements lying there with my eyes closed, I took ’em on faith barring further difficulties.

  • A regular polygon inscribed inside a circle will touch only on the corners of the polygon.
  • A circle inscribed inside a regular polygon will touch only on the midpoint of the polygon’s sides.
  • A regular polygon with an even number of sides will have straight diagonals which go from one corner, through the mid point, and hit the other corner.

That one I could check against squares, hexagons, and octagons, being the only even-sided regular polygons I could picture in my head. Oh, and while I’m listing assumptions I suppose I should add this one in too:

  • The sum of the angles of a regular polygon is 180° * (sides – 2)

Alright? Let’s go. We’re going to work out our general case with an arbitrary regular n-gon, which for the sake of picturing I’m going to draw as an octagon.

Why an octagon? Because a hexagon was too easy. 

Because our n-gon is a regular polygon all the sides are the same size and all the angles are the same size. Draw a diagonal between each pair of opposite corners.

Also because an octagon gets an extra bonus diagonal over those wimpy hexagons.

Your polygon is now divided into n pie slices. Each slice is a triangle bordered by an outside edge and two half-diagonals. Each pie slice is congruent to each other pie slice. It’s an isosceles triangle, with the peak at the midpoint of your n-gon. Next, we’re going to zoom in on one pie slice, and we’re going to split it in half. Drop a vertical from the midpoint of your n-gon to the opposing side.

You ever try to be polite and only take half the last piece of pizza, then get into a war where you’re splitting the slice ever finer and finer? No? Just me? 

Since our slice is isosceles we know the vertical is going to hit the midpoint there and make a right angle.

Everyone’s Got to Have an Angle

The sum of the angles in an n-gon is going to be 180° * (n-2). Since it’s a regular n-gon we know they’re all the same size (and that there are also exactly n angles), hence

n * individual angle size = (180° * n) – 360°

individual angle size = 180° – 360°/n

Now back to our sliced and diced n-gon. In terms of the diagram, I’m looking at the outer octagon right now. We bisected an angle to draw our diagonal. (I can feel Euclid glaring at me that I haven’t properly proven that a diagonal bisects the angle. Euclid can get stuffed; I’m rolling.) Therefore the angle between any diagonal and the edge segment it connects to is half our individual angle size, [90°-180°/n]. And we dropped a vertical, so we also have a 90° angle between the vertical and the side. That gives us two out of our three angles in a triangle. A triangle has 180 degrees in it. That lets us calculate the angle between the diagonal and the vertical (the angle with the midpoint as its vertex).

180° = 90° + [90°-180°/n] + (our midpoint angle)°.

Hey, that works out neatly; it’s going to be exactly [180°/n]. Now we know all the angles of our triangle. Here, lemme neatly label our diagram:

Captions added in HyperSnap because manually drawing them in is easier than convincing AutoCAD to do it for me. 

Caution: Trigonometry Ahead

With three angles we know most everything about our triangle except for its area. To get the size we need the length of one of the edges. The good news here is that we can tell that from the nature of our problem. Remember we’re trying to calculate π here, right? That means we’ve got a circle with a radius of 1. Remember how we’re calculating both an upper and a lower bound for the area of the circle? That means we’re going to have to calculate the area twice, once for the inside n-gon and once for the outside one. Luckily, our first two bullet-pointed properties up above there let us do that; one for each. First we’ll look at the outside n-gon which conveniently has a vertical with length 1

The height is easy. The base is going to cost extra. 

The area of a triangle is half the base times the height. We’ve got the height, that’s easy. We need the base. Working with the midpoint angle (which was [180°/n]), we know from basic trigonometry (say it with me! Soh-Cah-Toa!) the tangent of the angle corresponds to the ratio of the opposite side to the adjacent, or

tan[180°/n] = (triangle base) / (length of our vertical)

For our outside polygon we know the length of the vertical is 1, so

triangle base = tan[180°/n] 

area of our triangles = 1/2 * tan[180°/n]

Cool? Now we can calculate the area of our n-gon. Remember the n-gon was divided into n slices by the diagonals, and since we dropped a vertical we’ve split one of those in half, so we’ve got 2*n of these triangles.

area of the Polygon = 2*n*(area of our triangles)

area of the Polygon = 2*n*(1/2 * tan[180°/n] )

The area of our outside n-gon is officially

n*tan[180°/n]

That’s our upper bound for π.

Calculating the Lower Bound

Our lower bound is done by taking a similar calculation on a similar (in the technical sense too!) triangle for the smaller n-gon. Lemme caption one of those for ya:

A brief look at the mini map to remember where we are.

On our tilted triangle, the radius is now our hypotenuse. The argument is similar except we need to calculate the vertical as well as the base. The good news is we know our hypotenuse is 1, so our Soh and Cah are easy to calculate:

sin[180°/n] = (opposite side) / (hypotenuse)

cos[180°/n] = (adjacent side) / (hypotenuse)

Since both those hypotenuses are 1, we have formulas for our base (the opposite side) and our height (the adjacent side). Skipping ahead to the full n-gon, we have our lower bound for π:

n*sin[180°/n]*cos[180°/n] < π < n*tan[180°/n]

for any n that’s both an even number and greater than 2.

Enough Calculation, Get Back to the Story

As a matter of fact, I’ve sort of lost the thread myself.  A bare minimum of looking on the internet tells me that Archimedes actually used a different method. Oh, he still approximated π by looking at a 96-gon (sort of), but he did so via square roots rather than resorting to trigonometry. In retrospect that makes sense; calculating a sine exactly is not a trivial task. It does you a limited amount of good to know precisely how one thing you can’t calculate explicitly relates to another thing you can’t calculate explicitly. [2]

Square roots though are straightforward to calculate. In principle. Archimedes had to do this by hand. In Roman numerals. Uphill both ways. You ever try to do something as simple as multiplication in Roman numerals? I have. You have to make a table and then count your X’s so you’ve got VII of them. Taking a square root? I shudder at the thought. [3] I took the easy route for this and plugged the numbers into a spreadsheet.

That isn’t to say everything is copacetic with our modern calculators. My spreadsheet promptly assumed I wanted my angles calculated in radians and gave me wrong answers. I can work in radians, but working with π to calculate π sort of begs the question. I plugged it into Wolfram Alpha, and it helpfully assumed I wanted to convert into radians. No, I wanted a graph of the function. But never mind that. Here’s what it gives me for Archimedes 96-gon:

3.139 < π < 3.143

There you go, two decimal places, clean as day. Or, as the old boy himself put it: “The ratio of the circumference of any circle to its diameter is less than III and I by VII, but greater than III and X by LXXI, now stick that in your papyrus and smoke it you Egyptian ignoramus!” I, uh, may have edited that quote slightly.

I’ve got more power on hand than Archimedes did. Let’s let Wolfram Alpha loose on a closer approximation. When I try a 50,000-gon I get this:

3.141592545 < π < 3.141592657

Which is correct to eight decimal places. It took Wolfram Alpha a couple seconds to think about that last tangent calculation so I think I’ll leave off on asking it questions for fear of accidentally Spocking it.

What About the Calculus?

Could Archimedes have actually done it? I don’t know. Between his day and the invention of calculus, roughly eighteen hundred years passed. In that time mathematics advanced a great deal. The new techniques aren’t just theorems that were subsequently proven; they involve new ways of thinking about numbers. What’s the square root of three? I say it’s 1.732, three decimals being enough approximation for what you’re doing. Archimedes would have given you a ratio; it’s very close to 265/153. That’s accurate to four decimals. He understood fractions better than anyone I’ve ever met. Decimals would have to wait on the invention of a number system with place notation.

Okay, but if you allowed Archimedes decimals could he have worked it out? Should we assume he understood the Cartesian plane too? What other necessary conceptual frameworks were invented in the millennia since his death? If we dropped him in the year 1650 with all the knowledge of math available at the time I expect he’d be neck-and-neck with Newton and Leibniz in discovering the calculus. Possibly not with his circles. Or possibly with.

The way integral calculus is usually taught these days you start with Riemann sums. That is, you take an arbitrary squiggle and you look to find the area under that curve. You approximate it by drawing a whole bunch of rectangles that top out at your curve. To get a better approximation you draw more rectangles, thinner ones. The more rectangles you use the closer your approximation converges on the real area under that curve. Calculus is taking an infinite sum of those rectangles which have grown infinitely thin to fit ’em all in there.

What Archimedes does to calculate π is set up an infinite sum. You can approximate any curve with a series of circles of various radii; it might be possible to generate a way of doing calculus from that idea. You approximate a squiggly line with a series of curves generated from an infinite series of circles, which are in turn approximated by infinite-sided polygons. It’s not obvious to me whether that technique is actually possible or not. The only thing to do is to work it out myself. Truth be told, it’s been so long since I worked with the underpinnings of calculus that I view the prospect of working out the details with some trepidation.

Lying there that night with my eyes closed, sketching polygons and dropping verticals in my mind’s eye, I got all the way to the trig functions up above (with one error not reproduced here.) At that point, I had gotten myself so excited by the problem that I had to wrench my mind onto other things or I’d never get to sleep at all. Should I find myself musing on the subject on another sleepless night I’ll be sure and let you know what I figure out.


[1] As a matter of fact, not Leibniz but the other guy who invented calculus, Sir Isaac Newton, also invented the next great way to approximate π. Around roughly the same time a continental scholar had calculated — using Archimedes method and enormous amounts of labor — π to about eight places. Newton doodled out more digits than that during the plague year because he was bored. In case you weren’t already feeling like you hadn’t gotten enough done in quarantine.

[2] Another classic example of two unknowns, Sir Isaac Newton’s universal gravitation. Any two objects pull on each other, right? How hard? It’s easy enough to measure how quickly something falls, but how much does the Earth weigh? If we knew how much the Earth weighed we could work out the constant of proportionality. If we had the constant we could work out the mass of the Earth. If we could measure two other objects attracting each other gravitationally we’d have an answer, but the force is stupendously weak. This was resolved by a feller — name of Cavendish — who invented an extremely sensitive balance. Sensitive enough to detect the gravity between two objects of known mass. That let him work out the Gravitational constant and thereby weigh the Earth

[3] You know who developed a great way to calculate square roots? Sir Isaac Newton. He — alright, I’ll shut up about Newton for now.

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  1. Saint Augustine Member
    Saint Augustine
    @SaintAugustine

    I never took calculus, so I think I’m out about halfway through here.

    But I can tell you this for sure: The value of pie is relative. It depends on the pie and on how hungry you are.

    The value of a chicken pot pie right now is rather high.

    • #1
  2. HankRhody Freelance Philosopher Contributor
    HankRhody Freelance Philosopher
    @HankRhody

    Saint Augustine (View Comment):

    The value of a chicken pot pie right now is rather high.

    No argument here.

    • #2
  3. Gary McVey Contributor
    Gary McVey
    @GaryMcVey

    Another @hankrhody mindbender! Thing is, like all of Hank’s mathematical posts, they’re all worth going over until they become perfectly clear. Hank’s explanations always convey the reader to a point of understanding.

    The problem is, like the brain-building mental tests of the long-vanished Krell in the 1956 movie Forbidden Planet, you may be driven insane by implacable logic before you reach total genius. A small price to pay. 

    • #3
  4. kedavis Coolidge
    kedavis
    @kedavis

    Wow.

    But I’m a little puzzled about one thing.  Were you using the trig functions just to show how it works, or are you claiming that Archimedes used them as part of his process?  Last I heard, those functions/calculations weren’t discovered until several centuries after Archimedes was around.

    • #4
  5. HankRhody Freelance Philosopher Contributor
    HankRhody Freelance Philosopher
    @HankRhody

    kedavis (View Comment):

    Wow.

    But I’m a little puzzled about one thing.  Were you using the trig functions just to show how it works, or are you claiming that Archimedes used them as part of his process?  Last I heard, those functions/calculations weren’t discovered until several centuries after Archimedes was around.

    I used the trig functions because I was working out a method as I went along. Archimedes, well, if you want to look at it it’s here. He’s computing the same boundaries (the size of the n-gons) but he’s going about it with different tools, working out the proportionality between the diagonals on the polygons and the radius of the circle. 

     

    • #5
  6. Hang On Member
    Hang On
    @HangOn

    You missed a pun. When the Roman ran him through with a sword, Archie’s was screwed.

    <I hate autocorrect. >

    • #6
  7. WiesbadenJake Coolidge
    WiesbadenJake
    @WiesbadenJake

    Totally awesome!

    • #7
  8. Adios Muchacho Member
    Adios Muchacho
    @OldDanRhody

    Hang On (View Comment):

    You missed a pun. When the Roman ran him through with a sword, Archie’s was screwed.

    <I hate autocorrect. >

    For those of you who were dozing in class, here’s an illustration of Archimedes’  famous screw pump:

    What is a Archimedean screw? | World of interesting facts ...

    • #8
  9. Ekosj Member
    Ekosj
    @Ekosj

    This is the appropriate spot for my Ricochet meets Calculus story…

    I learned the basics of Calculus in college; and a great deal more in grad school.   I even taught a calculus-based Economics 101 class.   But while I was fascicle manipulating the equations, I never really had a visceral understanding of just what it all meant.

    Then I came across a book titled A Tour of the Calculus by Dr David Berlinski.  (Yes – that Berlinski … Claire’s dad.)   It was an epiphany.   The proverbial scales fell from my eyes and for the first time I really felt like I knew … like I genuinely understood what was going on here.   I was so excited, in fact, that I tried to phone the good Dr B at Stanford to tell him what a stupendous thing he had done.   He was not at his office.   But whomever I was blathering on to was so taken with my story that they gave me his home phone number.   I called.   I think it was a young Claire who answered.   I continued blathering to her.   The teenaged eye-roll was audible from across the country.   She called for her Dad.   Her tone added the unspoken “There’s some nut on the phone.”   He was very gracious accepting my compliments and we had a nice chat.  But I’m sure his next call was to his office demanding to know who gave out his number.

    When I first joined Ricochet and figured out that Ricochet’s Claire Berlinski was Dr Berlinski’s daughter I almost quit the site out of embarrassment.

     

    • #9
  10. EHerring Coolidge
    EHerring
    @EHerring

    Love this.it is great for those of us ready to progress to something more advanced than cubes and cube roots.

    • #10
  11. Henry Castaigne Member
    Henry Castaigne
    @HenryCastaigne

    Whenever you deny evolution. You are stabbing Archimedes in the gut and preventing the advancement of humanity by about 2,000 years.

    • #11
  12. Hang On Member
    Hang On
    @HangOn

    Henry Castaigne (View Comment):

    Whenever you deny evolution. You are stabbing Archimedes in the gut and preventing the advancement of humanity by about 2,000 years.

    Especially when evolution is nothing more than chemical thermodynamics. 

    • #12
  13. Mark Alexander Inactive
    Mark Alexander
    @MarkAlexander

    • #13
  14. kedavis Coolidge
    kedavis
    @kedavis

    Mark Alexander (View Comment):

     

    There’s something similar in this episode, close to the end:

     

    • #14
  15. kedavis Coolidge
    kedavis
    @kedavis

    Adios Muchacho (View Comment):

    Hang On (View Comment):

    You missed a pun. When the Roman ran him through with a sword, Archie’s was screwed.

    <I hate autocorrect. >

    For those of you who were dozing in class, here’s an illustration of Archimedes’ famous screw pump:

    What is a Archimedean screw? | World of interesting facts ...

     

     

    • #15
  16. DaveSchmidt Coolidge
    DaveSchmidt
    @DaveSchmidt

    Saint Augustine (View Comment):

    I never took calculus, so I think I’m out about halfway through here.

    But I can tell you this for sure: The value of pie is relative. It depends on the pie and on how hungry you are.

    The value of a chicken pot pie right now is rather high.

    Some of the Libertarians here might be tempted to leave out the chicken. 

    • #16
  17. WillowSpring Member
    WillowSpring
    @WillowSpring

    I missed if you pointed it out, but yesterday (3/14)  was “Pi day”. 

    I sent my retired engineer friend a coffee mug with about 2,000 digits of Pi printed on it as a Pi day gift (I couldn’t get the “My password is the last 8 digits of Pi” cup delivered in time)

    True to form, he called me at 1:59 (that is 3/14:1:59) to thank me for it.

    I have a feeling that Engineer humor is a rarely appreciated trait.

     

    • #17
  18. kedavis Coolidge
    kedavis
    @kedavis

    WillowSpring (View Comment):

    I missed if you pointed it out, but yesterday (3/14) was “Pi day”.

    I sent my retired engineer friend a coffee mug with about 2,000 digits of Pi printed on it as a Pi day gift (I couldn’t get the “My password is the last 8 digits of Pi” cup delivered in time)

    True to form, he called me at 1:59 (that is 3/14:1:59) to thank me for it.

    I have a feeling that Engineer humor is a rarely appreciated trait.

     

    But if it wasn’t at 1:59:26 (or rounded up to 27) it doesn’t count.

    • #18
  19. Adios Muchacho Member
    Adios Muchacho
    @OldDanRhody

    WillowSpring (View Comment):
    I have a feeling that Engineer humor is a rarely appreciated trait

    Fact check: True

    • #19
  20. kedavis Coolidge
    kedavis
    @kedavis

    Adios Muchacho (View Comment):

    WillowSpring (View Comment):
    I have a feeling that Engineer humor is a rarely appreciated trait

    Fact check: True

    Actually there’s a lot of Engineer humor, but it’s about other people making fun OF engineers.

    • #20
  21. Percival Thatcher
    Percival
    @Percival

    WillowSpring (View Comment):
    I have a feeling that Engineer humor is a rarely appreciated trait.

    A lot of INTPs in engineering.

     

    • #21
  22. HankRhody Freelance Philosopher Contributor
    HankRhody Freelance Philosopher
    @HankRhody

    WillowSpring (View Comment):

    I missed if you pointed it out, but yesterday (3/14)  was “Pi day”. 

    I did struggle a bit to get the post polished and published under the wire, but didn’t call the date out specifically since by the time I hit “post” it was well past the hour that an editor would promote it, and hence it wouldn’t hit the main feed in time for π day.

    I memorized the alt code to type that symbol editing this post and darn it if I ain’t gonna use it. alt 227 gets you π, alt 0176 gets you °, alt 0133 gets you … as a single character — and while I’m here — alt 0151 gets you the — symbol. I’ve already forgotten the one that gives you the squared symbol.

    • #22
  23. Percival Thatcher
    Percival
    @Percival

    HankRhody Freelance Philosopher (View Comment):

    WillowSpring (View Comment):

    I missed if you pointed it out, but yesterday (3/14) was “Pi day”.

    I did struggle a bit to get the post polished and published under the wire, but didn’t call the date out specifically since by the time I hit “post” it was well past the hour that an editor would promote it, and hence it wouldn’t hit the main feed in time for π day.

    I memorized the alt code to type that symbol editing this post and darn it if I ain’t gonna use it. alt 227 gets you π, alt 0176 gets you °, alt 0133 gets you … as a single character — and while I’m here — alt 0151 gets you the — symbol. I’ve already forgotten the one that gives you the squared symbol.

    Alt 253 = ²

    Today is semicolon day, by the way. Alt 315 0r alt 0315. It doesn’t matter.

    • #23
  24. The Reticulator Member
    The Reticulator
    @TheReticulator

    Ekosj (View Comment):
    But while I was fascicle

    Facile? (I never thought of you as a fascist.)

    • #24
  25. HankRhody Freelance Philosopher Contributor
    HankRhody Freelance Philosopher
    @HankRhody

    The Reticulator (View Comment):

    Ekosj (View Comment):
    But while I was fascicle

    Facile? (I never thought of you as a fascist.)

    Not just a facist; a Fascicle is a Nazi soldier frozen on the Eastern Front. Possibly with a stick involved; I don’t know.

    • #25
  26. HankRhody Freelance Philosopher Contributor
    HankRhody Freelance Philosopher
    @HankRhody

    HankRhody Freelance Philosopher: I got all the way to the trig functions up above (with one error not reproduced here.)

    A friend of mine was kind enough to link me to the below video (which uses a similar method to mine, but despite it’s claims is also not the exact method Archimedes used.) What really interests me about the video is that it gives a different lower bound than I derived:

    n * sin[180°/n]

    Which brings me back to that error I mentioned. When I first got  my results I plugged ’em into a spreadsheet to see how accurate they were. Not at all; the series diverged. So I checked my math, and I figured out I had missed a term on the lower bound. I hadn’t accounted for

    HankRhody Freelance Philosopher:

    The argument is similar except we need to calculate the vertical as well as the base. The good news is we know our hypotenuse is 1, so our Soh and Cah are easy to calculate:

    sin[180°/n] = (opposite side) / (hypotenuse)

    cos[180°/n] = (adjacent side) / (hypotenuse)

    I had missed that the adjacent side wasn’t of length 1, so I had neglected the cosine term. Which means the erroneous result I got was actually the lower bound that the guy making that video derived.

    You know what? He’s still right about it, and I was still wrong. The perimeter of his n-gon is different from the area of my n-gon, and his calculation actually leads to a correct answer.

    Visually, (lemme pull up that last diagram again) we’re looking at the length of the left diagonal. The one with the marked right angle with our inner n-gon (I’m not going to redraw the diagram for this footnote.) The difference between his numbers and my numbers is the length of that line. If the line is length one (pushed all the way to the circle’s edge) then our equations would be exactly the same. And, if you imagine the n-gon getting tighter and tighter to the circle, the closer and closer that is to true.

    In symbols I’m bringing up one of my favorite physics kung-fu techniques; the Small Angle Approximation. Here we take the approximation that, for very small angles, cos(x) = 1. If both he and I use an extremely large n, then 180°/n is so close to zero as to not really matter, and my n*cos[180°/n]*sin[180°/n] reduces to n*1*sin[180°/n], which is his result.

    Now here’s the worst part. Because n*cos[180°/n]*sin[180°/n] is always going to be slightly smaller than n*sin[180°/n] (assuming n is more than 2), my lower bound is going to always be a little looser than his lower bound, and hence he gets the prize for being the better mathematician. Curse him!

    • #26
  27. EHerring Coolidge
    EHerring
    @EHerring

    kedavis (View Comment):

    Adios Muchacho (View Comment):

    Hang On (View Comment):

    You missed a pun. When the Roman ran him through with a sword, Archie’s was screwed.

    <I hate autocorrect. >

    For those of you who were dozing in class, here’s an illustration of Archimedes’ famous screw pump:

    What is a Archimedean screw? | World of interesting facts ...

     

     

    I’m a physicist, not a hippy😂😂😂😂😂😂😂😂😂😂😂

    • #27
  28. EHerring Coolidge
    EHerring
    @EHerring

    Percival (View Comment):

    HankRhody Freelance Philosopher (View Comment):

    WillowSpring (View Comment):

    I missed if you pointed it out, but yesterday (3/14) was “Pi day”.

    I did struggle a bit to get the post polished and published under the wire, but didn’t call the date out specifically since by the time I hit “post” it was well past the hour that an editor would promote it, and hence it wouldn’t hit the main feed in time for π day.

    I memorized the alt code to type that symbol editing this post and darn it if I ain’t gonna use it. alt 227 gets you π, alt 0176 gets you °, alt 0133 gets you … as a single character — and while I’m here — alt 0151 gets you the — symbol. I’ve already forgotten the one that gives you the squared symbol.

    Alt 253 = ²

    Today is semicolon day, by the way. Alt 315 0r alt 0315. It doesn’t matter.

    Dang, now I want to go play on my computer.

    • #28
  29. Adios Muchacho Member
    Adios Muchacho
    @OldDanRhody

    HankRhody Freelance Philosopher (View Comment):
    What really interests me about the video

    is that it leaves me wondering whether or not it was narrated by a robot.

    • #29
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