Prehistoric Calculus: Discovering Pi

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Pi is mysterious. Sure, you “know” it’s about 3.14159 because you read it in some book. But what if you had no textbooks, no computers, and no calculus (egads!) — just your brain and a piece of paper. Could you find pi?

Archimedes found pi to 99.9% accuracy 2000 years ago — without decimal points or even the number zero! Even better, he devised techniques that became the foundations of calculus. I wish I learned his discovery of pi in school — it helps us understand what makes calculus tick.

How do we find pi?

Pi is the circumference of a circle with diameter 1. How do we get that number?

  • Say pi = 3 and call it a day.
  • Draw a circle with a steady hand, wrap it with string, and measure with your finest ruler.
  • Use door #3

What’s behind door #3? Math!

How did Archimedes do it?

Archimedes didn’t know the circumference of a circle. But he didn’t fret, and started with what he did know: the perimeter of a square. (He actually used hexagons, but squares are easier to work with and draw, so let’s go with that, ok?).

We don’t know a circle’s circumference, but for kicks let’s draw it between two squares:

Pi square estimate

Neat — it’s like a racetrack with inner and outer edges. Whatever the circumference is, it’s somewhere between the perimeters of the squares: more than the inside, less than the outside.

And since squares are, well, square, we find their perimeters easily:

  • Outside square (easy): side = 1, therefore perimeter = 4
  • Inside square (not so easy): The diagonal is 1 (top-to-bottom). Using the Pythagorean theorem, side2 + side2 = 1, therefore side = sqrt(1/2) or side = .7. The perimeter is then .7 * 4 = 2.8.

We may not know where pi is, but that critter is scurrying between 2.8 and 4. Let’s say it’s halfway between, or pi = 3.4.

Squares drool, octagons rule

We estimated pi = 3.4, but honestly we’d be better off with the ruler and string. What makes our guess so bad?

Squares are clunky. They don’t match the circle well, and the gaps make for a loose, error-filled calculation. But, increasing the sides (using the mythical octagon, perhaps) might give us a tighter fit and a better guess (image credit):

Pi Polygon Estimate

Cool! As we yank up the sides, we get closer to the shape of a circle.

So, what’s the perimeter of an octagon? I’m not sure if I learned that formula. While we’re at it, we could use a 16-side-a-gon and a 32-do-decker for better guesses. What are their perimeters again?

Crickey, those are tough questions. Luckily, Archimedes used creative trigonometry to devise formulas for the perimeter of shape when you double the number of sides:

Inside perimeter: One segment of the inside (such as the side of a square) is sin(x/2), where x is the angle spanning a side. For example, one side of the inside square is sin(90/2) = sin(45) ~ .7. The full perimeter is then 4 * .7 = 2.8, as we had before.

Outside perimeter: One segment of the outside is tan(x/2), where x is the angle spanning one side. So, one segment of the outside perimeter is tan(45) = 1, for a total perimeter of 4.

Neat — we have a simple formula! Adding more sides makes the angle smaller:

  • Squares have an inside perimeter of 4 * sin(90/2).
  • Octogons have eight 45-degree angles, for an inside perimeter of 8 * sin(45/2).

Try it out — a square (sides=4) has 91% accuracy, and with an octagon (sides=8) we jump to 98%!

But there’s a problem: Archimedes didn’t have a calculator with a “sin” button! Instead, he used trig identities to rewrite sin and tan in terms of their previous values:

  • New outside perimeter \displaystyle{newOut = \frac{2}{\frac{1}{Inside} + \frac{1}{Outside} }} [harmonic mean]

  • New inside perimeter \displaystyle{newIn = \sqrt{Inside \cdot newOut}} [geometric mean]

These formulas just use arithmetic — no trig required. Since we started with known numbers like sqrt(2) and 1, we can repeatedly apply this formula to increase the number of sides and get a better guess for pi.

By the way, those special means show up in strange places, don’t they? I don’t have a nice intuitive grasp of the trig identities involved, so we’ll save that battle for another day.

Cranking the formula

Starting with 4 sides (a square), we make our way to a better pi (download the spreadsheet):

Every round, we double the sides (4, 8, 16, 32, 64) and shrink the range where pi could be hiding. Let’s assume pi is halfway between the inside and outside boundaries.

After 3 steps (32 sides) we already have 99.9% accuracy. After 7 steps (512 sides) we have the lauded “five nines”. And after 17 steps, or half a million sides, our guess for pi reaches Excel’s accuracy limit. Not a bad technique, Archimedes!

Unfortunately, decimals hadn’t been invented in 250 BC, let alone spreadsheets. So Archimedes had to slave away with these formulas using fractions. He began with hexagons (6 sides) and continued 12, 24, 48, 96 until he’d had enough (ever try to take a square root using fractions alone?). His final estimate for pi, using a shape with 96 sides, was:

\displaystyle{3 \frac{10}{71} < \pi < 3 \frac{1}{7}}

The midpoint puts pi at 3.14185, which is over 99.9% accurate. Not too shabby!

If you enjoy fractions, the mysteriously symmetrical 355/113 is an extremely accurate (99.99999%) estimate of pi and was the best humanity had for nearly a millennium.

Some people use 22/7 for pi, but now you can chuckle “Good grief, 22/7 is merely the upper bound found by Archimedes 2000 years ago!” while adjusting your monocle. There’s even better formulas out there too.

Where’s the Calculus?

Archimedes wasn’t “doing calculus” but he laid the groundwork for its development: start with a crude model (square mimicking a circle) and refine it.

Calculus revolves around these themes:

  • We don’t know the answer, but we’ve got a guess. We had a guess for pi: somewhere between 2.8 and 4. Calculus has many concepts such as Taylor Series to build a guess with varying degrees of accuracy.
  • Let’s make our guess better. Archimedes discovered that adding sides made a better estimate. There are numerical methods to refine a formula again and again. For example, computers can start with a rough guess for the square root and make it better (faster than finding the closest answer from the outset).
  • You can run but not hide. We didn’t know exactly where pi was, but trapped it between two boundaries. As we tightened up the outside limits (pun intended), we knew pi was hiding somewhere inside. This is formally known as the Squeeze Theorem.
  • Pi is an unreachable ideal. Finding pi is a process that never ends. When we see π it really means “You want perfection? That’s nice — everyone wants something. Just start cranking away and stop when pi is good enough.”.

I’ll say it again: Good enough is good enough. A shape with 96 sides was accurate enough for anything Archimedes needed to build.

The idea that “close counts” is weird — shouldn’t math be precise? Math is a model to describe the world. Our equations don’t need to be razor-sharp if the universe and our instruments are fuzzy.

Life Lessons

Even math can have life lessons hidden inside. Sometimes the best is the enemy of the good. Perfectionism (“I need the exact value of pi!”) can impede finding good, usable results.

Whether making estimates or writing software, perhaps you can start with a rough version and improve it over time, without fretting about the perfect model (it worked for Archimedes!). Most of the accuracy may come from the initial stages, and future refinements may be a lot of work for little gain (the Pareto Principle in action).

Ironically, the “crude” techniques seen here led to calculus, which in turn led to better formulas for pi.

Math Lessons

Calculus often lacks an intuitive grounding — we can count apples to test arithmetic, but it’s hard to think about abstract equations that are repeatedly refined.

Archimedes’ discovery of pi is a vivid, concrete example for our toolbox. Just like geometry refines our intuition about lines and angles, calculus defines the rules about equations that get better over time. Examples like this help use intuition as a starting point, instead of learning new ideas in a vacuum.

Later, we’ll discuss what it means for numbers to be “close enough”. Just remember that 96 sides was good enough for Archimedes, and half a million sides is good enough for Excel. We’ve all got our limits.

Other Posts In This Series

  1. A Gentle Introduction To Learning Calculus
  2. How To Understand Derivatives: The Product, Power & Chain Rules
  3. How To Understand Derivatives: The Quotient Rule, Exponents, and Logarithms
  4. An Intuitive Introduction To Limits
  5. Why Do We Need Limits and Infinitesimals?
  6. Learning Calculus: Overcoming Our Artificial Need for Precision
  7. Prehistoric Calculus: Discovering Pi
  8. A Calculus Analogy: Integrals as Multiplication
  9. Calculus: Building Intuition for the Derivative
  10. Understanding Calculus With A Bank Account Metaphor
  11. A Friendly Chat About Whether 0.999... = 1
Kalid Azad loves sharing Aha! moments. BetterExplained is dedicated to learning with intuition, not memorization, and is honored to serve 250k readers monthly.

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74 Comments

  1. Great post. I really liked the formatting too, and the calculator at the end. Worthwhile talk about how you might estimate pi in other ways, such as estimating the number co-prime numbers, or the Buffon Needle problem.

    Just some ideas.

    Again, great post.

    Matt

  2. I like the article, and will definitely use the ideas in my classes, but first there is a minor problem to solve. You have the inside perimeter as the geometric mean of the previous estimates, and the outside perimeter as the harmonic mean. The problem is, the harmonic mean is always /smaller/ than the geometric mean. Maybe they just switch, but I don’t see how yet.

    The page to which you link contains the same mistake. I have no idea where that guy got the trig identities he cites, but I’ll keep working on it.

  3. Update: Okay, I now see I was reading the formulae incorrectly (one refers to newIn rather than Inside), but the formulae themselves are wrong as well (on your page; the page you link to gets them right). They should be

    newOut = harmonicMean(Inside, Outside)
    and
    newIn = geometric(Inside, newOut)

    which is what you use in the spreadsheet.

  4. In my email to you about writing a guest article, I had one that this article just destroys. I talked (a lot) about Archimedes’ discovery that 223/71

  5. The comment form just ate my last comment; I have no idea why that is.

    I continued on from there:

    223/71

  6. Okay, I figured it out. It misinterpreted what I typed as html.

    In my email to you about writing a guest article, I had one that this article just destroys. I talked (a lot) about Archimedes’ discovery that 223/71 is less than pi which is less than 22/7, though I focused more on the concept than the mathematics behind it.

    Although, your square root comment made me think: I have a better explanation for Newton’s method than you had in the Quake Square Root article, so maybe I should write about that…

  7. @Matt: Thanks, glad you enjoyed it! Those are great suggestions, I think it’d be great for a follow-up. I didn’t want to distract from the calculus roots too much in this post, but the needle approach is a fun way to look at probability.

    @Chad: Sorry about the confusion there! Yes, I made a major flub and miswrote the equations (just corrected it), the spreadsheet should have the correct ones.

    @Zac: No worries — I should probably install a live preview plugin so people will know when their comment is getting eaten / mistaken for HTML. Sure, if you have ideas for the square root method feel free to write them down — once the contribution wiki is up I’m sure it’ll be a nice addition :).

  8. hey nice one there for a quick look , although we know the value of pi after all those yrs of forced insertion of the value into our heads, but this gives a better insight to the derivation in a way,
    appreciate the effort!

  9. Thanks Brijesh! Yep, we know pi because we’ve seen it before, but it’s nice to see how we came to that result.

  10. Please explain this:

    >Whatever the circumference is, it’s somewhere between the perimeters of the squares: more than the inside, less than the outside.

    Why is it obvious that the outside square and futhermore 512-sided-thingy has bigger perimeter?

  11. Reading a little more into pi and the ways of calculuating it seem to always lead me to Taylor Series. It would be nice to really understand what’s going on there.

    Pi is a fun number. For some reason, I decided to memorize it to 50 decimal places. The fact that it’s impossible to calculate exactly just makes it even more fun to try and find more.

  12. Well,I’m from Lima,Perú.And I never going to understand the way americans do math.For us “PI”is=3.1416.And if I’m not wrong 22/7 is not a correct anwser.not even that 223/71.Also I found and america you guys solve math problems outside down.My favorite subject is MATH,and II want tobe a math teacher.
    I wish I dont make mad noone with my comments.

  13. Dude the chinese made a much better version than Archimedes sooner. They made ones with roughly 30-100 sides according to my research.

  14. @tekumse: That’s an interesting question, sometimes it’s good to break down these assumptions. The formal name for the inside shape is “inscribed” and the formal name for the outside shape is “circumscribed”.

    The area of the inscribed shape is less than or equal to the area of the circle, since all points are inside the boundary.

    The area of the circumscribed shape is greater than or equal to the area of the circle, since all points are outside the boundary. Therefore, the area of the inscribed polygon is less than or equal to the area of the circumscribed shape.

    For similar shapes, the greater area corresponds to a greater side length (see the Pythagorean theorem for more details). Since we are using similar shapes (squares, octagons, 16-gons, etc.) the circumscribed shape will have a larger side length (and perimeter) than the inscribed one. Hope this helps.

    @Zac: Yep, the Taylor series will be fun. I want to think about it more to see if I can find some insights that link it to everyday analogies :). And 50 digits of pi is pretty precise, enough to estimate the size of the universe to 1 atom’s precision, I think.

    @Miguel: Thanks for the comment, glad you like math. I think most students know 22/7 (or 3.1416) is just an estimate for pi, not an exact value.

    @Anonymous: Don’t believe everything you read!

    @Cheeseburger: That’s interesting, Archimedes made this technique famous but others may have used it as well.

  15. I have a question at the very first assumption in the two squares case:

    How did you make the assumption (assuming precalculus and calculator days and all that) that

    2.8

  16. Hi Ashwin, the comment form may have eaten your comment.

    1/sqrt(2) comes from the Pythagorean theorem — it’s actually sqrt(2)/2 (which is the same thing), and sqrt(2) can be approximated using various algorithms: it’s more than 1, less than 2. It’s more than 5/4 (5/4 squared = 25/16 which is less than 2), and less than 6/4 (6/4 squared is 36/16 which is more than 2).

    Not sure if that was the question but feel free to ask again, sorry about the form.

  17. Hi phyu, 22/7 is an approximation for pi, but it isn’t as accurate as 355/113. Check out the “Cranking the Formula” section for more details.

  18. @ phyu

    355/113 actually simplifies to 22/~7.0028169014084507042253521126761

    the higher up in the fractions you go the farther from 7 the bottom number becomes, which falls right in line with the numeration given in Kalid’s chart.

    Also @ Kalid very nice job in the compilation, always nice to see some interesting math facts!

  19. such a great article! i was really happy that you included that little bit of life lesson at the end there. i have a tattoo of pi to remind myself that life doesn’t always make perfect sense :]

  20. @Holy: Thanks for the comment and additional details!

    @Anna: Glad you enjoyed it! Don’t think I’ve ever met anyone with a pi tattoo but that’s pretty intriguing :). Yep, I think math (or any subject) should enhance your outlook, not just teach facts.

  21. Hi there. First of all, thanks for the article. I think I’ve got a silly question., but it’s driving me nuts!

    My intuition keeps telling me that the inside perimeter (sin(x/2) above) and outside perimeter (tan(x/2) above) should be the same equation – it’s the same shape, just bigger, so the formula should be the same with larger values for x.

    Can you tell me what I’m missing?

  22. Hi Karl, that’s a great question! I had to think about it a bit.

    You’re right, the two shapes (large and small square, large and small octagon) should have the same formula, scaled by some amount. The tricky thing is to realize that x/2 (the angle) should be *the same* in both cases; the angles don’t change no matter what size square you have.

    You want to start with a formula (call it f(x) ) and scale it by some amount, called C: f(x) and C * f(x).

    Looking closer, this is what’s happening: sin(x/2) is the basic formula, and tan(x/2) is really sin(x/2) / cos(x/2).

    Since cosine is between 0-1, the division will actually be a multiplication or scaling. So tan(x/2) is always larger than sin(x/2), giving us the scaling factor we need.

    Again, great question — sin(x/2) and tan(x/2) are really the same formula, but scaled by 1/cos(x/2). Phew :).

  23. Great articles Kalid, any similar insights or an intuitive approach you could share on eulers identity ? This explanation for pi is the one we were actually thought in school, and we were thought about e through continuous growth. (as in the article on e on this site). Both make perfect sense to me but I’m still blown away by eulers identity ( e(i.pi)+1 = 0 ). What is the meaning of this relation between e, i and pi .. is there an intuitive way to look at this you are aware of ?

  24. @enki: Whoops, sorry about the late response, think I missed this. There is an intuitive way to approach Euler’s identity that I’d like to write about (the book Visual Complex Analysis has a take on it, which I highly recommend). Basically, you can view it as a linkage between growth and rotation — but I’ll be writing about this topic in the future.

  25. Great article. This is one of the reasons why I have heard the circle referred to as an “infinigon.”

  26. i have derived a formula for pi which approximates the value of pi.

    pi = lim n*cos((180/n) – 90)
    n->infinity

    higher the value of n, more accurate the value of pi.
    i have derived it based on inscribing a polygon in circle. variable n represents number of sides of polygon.

  27. I don’t understand why pi is an irrational number. Can’t you just measure the circumference accurately and then divide by the diameter – there you have a rational fraction. I can’t see how a constant derived by real division can be an irrational number.

  28. @rishi: Great question. The problem with drawing and measuring a circle is that there’s no such thing as a perfect circle. Anything you draw is just a collection of points (each drop of ink, or each molecule of ink!) and is therefore a very large polygon, maybe with billions of sides.

    We can measure the circumference of this polygon, but it won’t be “pi”, just a very close guess. After all, we could have added more sides and got a better guess.

    One way to see the irrational, neverending decimal is to consider pi the result of an infinite process (adding more and more sides to a polygon to approximate a circle), one we can approximate but never write out completely. Hope this helps!

  29. I like the write-up very much, but find the title a little misleading.

    You give a good description of applying Archimedes method of calculating the numerical value of pi. In fact, this type of successive approximation is useful for computing many other interesting values as well.

    To many, however, the “discovery of pi” is the realization that the ratio of circumference to diameter is the same for ALL circles. Without that, we wouldn’t be talking about the circumference of a unit circle, nor would that value have a special name (pi).

    Adding an intuitive description of that discovery to your write-up would really make it shine.

  30. The idea that the newOutside is the harmonic mean and the newInside is the geometric mean is not very intuitive. Why is newOut being derived from the perimeter of the previous inside and previous outside, similarly with geometric mean?

  31. Thanks for the great write up. I’m returning to calculus after 20 years and your article is helping me finally internalize something I’ve never grasped before.

  32. @Simon: Awesome, glad it was helpful for you!

    @Sapan: Yes, I struggle with that too — I don’t have an intuitive understanding of why it would be the geometric and harmonic mean to figure out those ratios. Right now my understanding is at the level of “the math works” :).

  33. I actually came up with Archimede’s method on my own but I started with a triangle and kept going with more polygons (basically each side of the triangle got another triangle, and so on). Basic geometry got me from the perimeter of one poly to the next. Using my PC i was able to calculate pi to a million decimal places rather quickly (i did a text-compare with one i found online and it was right). I thought i may have stumbled on something new but later i found out it was not so.
    The only interesting thing was that it was recursive and used only basic geometry (right triangles).

  34. @Dedic: That’s a cool story — there’s always something to be said for the joy of discovery, even if you weren’t the first to do so :).

  35. Hi Kalid, Wonderful post…………

    I’m actually a young guy and new to complex stuff but u make it look easy…..

    A question :

    Is a straight line a part of a large circle ???

  36. @Shankar: Glad you liked it! Hrm, I’m not sure what you mean — i.e., is a circle made up of straight line segments? A perfect circle seems never has two points on a perfect line (i.e. if you rotate the circle only one of the points will be “rightmost”, you can’t have both vertically above each other) but reality is quite different :).

  37. Hi Kalid…….What i meant was that a road seems perfectly straight to us……..however its just a part of a large circle called earth……

    So if we keep on extending a straight line on both sides infinitely, will we get a large circle ????

  38. And one more thing………

    How can we be sure that pi is an irrational number.??.

    Maybe after the 100 billionth number after the decimal point, it may repeat itself, thus making it a rational number……….

  39. Dear Khalid,

    Grate article and will look out for other article by you.

    Regarding: PI ~= 335/113

    On PI day (3.14) a french lady emailed me a gift that further to my Quran and Prime Numbers reseach, the 355 days in a Hijri leap year divided by the chapters of The Message (113 chapters) is a very close approximation to PI.

    PI ~= days in a year cycle (circumference) divide by the number of chapters of the message (stright path, diameter)

    Here is a summary for all your readers about the prime numbers in the Quran.

    Quran = Key + Message
    114 chapters = 1 (Al-Fatiha) + 113 (Remaining chapters)
    6236 verses = 7 of The Key + 6229 of The Message

    The Key has 7 verses, 29 words, 139 letters) all are primes, with prime digit sums (7=7, 2+9=11, 1+3+9=13) and amazingly concatnating them left-to-right (729139) and right-to-left (139297) also primes with primes digit sums (7+2+9+1+3+9=31)

    The rest can be found at http://www.heliwave.com pr http://www.primalogy.com.

    Make sure not to miss the 355 days of chapter The Merciful that map to the leap Hijri year 1433AH = 2012 :)

    I suspect the Hijir year becomes leap evey PI years :) a “PI in the Sky” if you like :)

    Ali Adams
    God > infinity

  40. Great article very informative and helpful, the only think I could see needing some furthre explainging this quote

    “faster than finding the closest answer from the outset”

    What is outset?

  41. @GW: Ah, I just meant “rather than finding the closest answer immediately, from the very beginning”.

  42. Great article. I think one of my face-palming moments was when I realized that pi was the result of infinitely improving the number. (This also helped me to understand transcendental numbers, since you need an infinite series of algebraic formulas to reach it.)

    What I think is particularly interesting is how something infinitely complex can make formulas so simple. Instead of picking an approximation (since, a lot of the time, we don’t know ahead of time what this should be), we use the pure number “pi” to allow somebody else to approximate later. Not only that, but it makes the formula easier to read as well by encapsulating the complexity in a single constant. Truly beautiful.

    (Side note: working with image processing and other forms of computer graphics, I sometimes wish “pi” was initially measured with the radius instead of the diameter. That way, we could use the constant itself instead of writing “2*pi” everywhere. The constant really only represents half of the shape of a circle.)

  43. I really enjoyed this article, and it makes complete sense why Archimedes used this method, although i would have never thought of it on my own. I liked the style of the writing too, very easy to understand. The one thing is didn’t understand was the formula for perimeter of the inside and outside shapes. I don’t understand why we use sin. Other than that great article

  44. @Joe: Thanks for the comment! Yeah, one of the weird things about pi is that it’s never “done” — i.e., when does a shape with “infinite” sides become a circle? It raises all sorts of interesting philosophical questions too — i.e., we use pi for calculations but will never encounter a perfect circle in the real world. But the beauty, as you say, is that we encapsulate this whole concept into a symbol which is “use the best approximation of the perfect circle that you can..”.

    I agree on the pi vs. 2*pi thing — have you seen http://tauday.com/?

    @Matthew: Thanks! Great question on the formula — there’s an explanation on why sine is used here:

    http://personal.bgsu.edu/~carother/pi/Pi3b.html

    but I’d like to cover it in more depth myself. Thanks again for the note!

  45. this is a wonderful article. thank you.

    there is only one place where i disagree. I would say that pi can ‘hide but not run’ instead of the other way around.

    cheers

  46. Wow!great information on pi this has really widen my view abt maths.maths is becoming interesting to me.thanks 4 making it interesting.i’ll love 2b a mathematician.

  47. funny, i just only understood the point of taylor series while reading your article on intro to calculus. and its right on this page! thanks for giving me a wonderful “Aha!” moment. love your site.

    if you ever watched the movie 3 idiots, you remind me of one character, Rancho. I hope that turned out as a compliment. :) more success on this and other ideas of yours!

  48. @mel: Thanks for the kind words! Really happy the site is helping with those ahas. I haven’t seen the movie but have heard much about it!

  49. Sides = 4037146 is the first here to show up as 100% Accuracy

    Just so you don’t keep trying
    LIKE I DID.

  50. Is there another article after this? I’m probably not looking properly, but I want to read on!!

  51. I appologize at start if my english is not perfect – it is not my native language, so I express myself on english not as I want, but as I can. I will try to express my opinion on the best possible way. It is a true that our science becomes materialistic and I studied electrical engineering on that way, never thinking about spiritual way of what I learned untill one day, when I figured out something about the numbers and I find the open book in front of myself that never ends.

    Squering the circle means to know how to generate number pi, how to generate its next digit, and not only that. Squering the circle means also to be able to measure how long some curve line is. To measure curve line – it means you have to compare it with some streight line, but it is hard to fit them, isn’t it?
    So, squering the circle is the same as comparing curve line and streight line and finding their corelation… as comparing soul and body and find the corelation… as comparing man and woman and find the corelation…as comparering good and evel and find the corelation.
    Please watch the numbers – 012345678910 – but only as a forms.curve- 0, streight line – 1, combination – 2, only curves – 3, only streight lines – 4, combination – 5, combination – 6, streight lines – 7, curves – 8, combination – 9, streight line -1, curve – 0.Some of them are extremes (only curve or only streight lines) some of them are balance (combination). Each number has its own pair 01, 25, 34, 69, 78, 10 complicated as much as it is, but on some way opposite.
    Now place them in the number pi…as you go more and more discovering number pi as a multiform you actualy know more about each its digit, about extremes and about balance.
    As long you go from digit to digit of the number pi you will find out – at start was circle, at the end is circle – conection between these two is long, narrow and unsecure path very few people are following on the right way these days, holding faith in their souls as a small light of the candel.

  52. the equation above is wrong dipshit, the inner squares side doesnt equal .7, it equals the square root of .5 . pythagoras states in the above equation that a²+b^2=c^2 〖.5〗^2 〖+ .5〗^2=c^2 .5=c^2 c= √(.5) not .7 as stated above. just thought id let you know

  53. Dear Kalid,

    I am now in my 60s, but majored in Mathematics and the History and Philosophy of Science at Melbourne University (Australia) some 30 years ago. The latter major – a fascinating study – focused on the development of several different sciences, including mathematics, physics, chemistry, biology, etc. from times BC to the 20th century.

    Motivations underlying early mathematical development in different areas BC (Eastern Europe/Middle East, India, China) included fascination in number theory and algebra, astronomy (“understanding the movements of the planets or heavens”), religion or belief in divinities (predicting or setting auspicious dates for appeasing the gods or for festivities), agriculture (predicting the seasons), and taxation (calculating approximate land areas under plot, based on shapes and dimensions).

    It therefore should come as no surprise that I appreciated your paper on estimating a “best value” for PI (Discovering PI, 2008). However, to avoid confusion and unhelpful feedback, it may be worthwhile clarifying the dimensions of the inner square, especially for busy teachers and students.

    The inner square has a diameter or hypotenuse of 1, as is obvious from the diagram of the inner square within the circle and within the outer square. Based on elementary geometry for right-angle triangles in a square, we can calculate a^2 + a^2 =1, where “a” is the length of a side of the inner square.

    Simplifying and transposing, we get:
    x) 2a^2=1,
    y)a^2 =1/2 =0.5, whence
    z)a=1/SQRT(2) or a=SQRT(0.5)

    It may be stating the obvious to record these arithmetic identities, but even chess Grand Masters can make blunders, let alone hasty readers!

    PS: My 3 sons have all been strong in Mathematics, and my youngest (now aged 16) is excelling in the subject and is currently doing early-entry University study. He combines Mathematical study with Musical study (playing piano and percussion, and conducting his school orchestra).

  54. Hi Geoff, thanks for the note! Completely agree, that computation is not obvious, I’ve clarified the post. Appreciate the feedback!

  55. Hi!

    Great site! You have really put a lot of effort into it and I am loving it!

    Quick question: I understand how you arrive at 1/sqrt(2) for inner square and 1 for outer square(by diamater=1) and I know that sin(45) = 1/sqrt(2) and tan(45) = 1 but how do you take the leap where for any inside shape one can claim side = sin(x/2) and for outside shape side = tan(x/2)?

    Thanks!

  56. Hi Ali, no problem! Glad you enjoyed the article. I might like to do a follow-up to really understand how those formulas came about.

  57. I am the author that prove the circle is a binary polygon with 2^55 sides.
    You have done great work and I am letting you know to inquire wikipedia.org
    User: Stagno sand box and You will have more information on the subject.
    Thanks
    Giuseppe

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LaTeX: $$e=mc^2$$