Update

pi.md

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 # Pi
 
-Pi (normally written with a Greek alphabet symbol with [Unicode](unicode.md) value U+03C0) is one of the most important and famous [numbers](number.md), equal to approximately 3.14, most popularly defined as the ratio of a circle's circumference to its diameter (but also definable in other ways). It is one of the most fundamental mathematical constants of our universe and appears extremely commonly in [mathematics](math.md), nature and, of course, [programming](programming.md). When written down in traditional decimal system, its digits go on and on without end and show no repetition or simple pattern, appearing "random" and [chaotic](chaos.md) -- as of 2021 pi has been evaluated by [computers](computer.md) to 62831853071796 digits. In significance and properties pi is similar to another famous number: [e](e.md). Pi day is celebrated on March 14.
+Pi (normally written with a Greek alphabet symbol with [Unicode](unicode.md) value U+03C0) is one of the most important and famous [numbers](number.md), equal to approximately 3.14, most popularly defined as the ratio of a circle's circumference to its diameter (but also definable in other ways). It is one of the most fundamental mathematical constants of our universe and appears extremely commonly in [mathematics](math.md), nature and, of course, [programming](programming.md). When written down in traditional decimal system, its digits go on and on without end and show no repetition or simple pattern, appearing "random" and [chaotic](chaos.md) -- as of 2021 pi has been evaluated by [computers](computer.md) to 62831853071796 digits, although approximate values have been known from very early times (e.g. the value (16/9)^2 ~= 3.16 has been known as early as around 1800 BC). In significance and properties pi is similar to another famous number: [e](e.md). Pi day is celebrated on March 14.
 
 { Very nice site about pi: http://www.pi314.net. ~drummyfish }
 
@@ -24,12 +24,16 @@ Additionally contrary to what's sometimes claimed **it is also unproven (though
 
 What makes pi special then? Well, mostly its significance as one of the most fundamental constants that seems to appear extremely commonly in math and nature, it seems to stand very close to the root of description of our universe -- not only does pi show that circles are embedded everywhere in nature, even in very abstract ways, but we find it in [Euler's identity](eulers_identity.md), one of the most important equations, it is related to [complex exponential](complex_exponential.md) and so to [Fourier transform](fourier_transform.md), waves, oscillation, trigonometry ([sin](sin.md), [cos](cos.md), ...) and angles ([radians](radian.md) use pi), it even starts appearing in [number theory](number_theory.md), e.g. the probability of two numbers being relative primes is 6/(pi^2), and so on.
 
-## Approximations And Programming
+## Approximations, Estimations, Measuring And Programming
 
 Evaluating many digits of pi is mathematically [interesting](interesting.md), programs for computing pi are sometimes used as [CPU](cpu.md) [benchmarks](benchmark.md). There are programs that can search for a position of arbitrary string encoded in pi's digits. However in practical computations we can easily get away with pi approximated to just a few decimal digits, **you will NEVER need more than 20 decimal digits**, not even for space flights (NASA said they use 15 places).
 
 One way to judge the quality of pi approximation can be to take the number of pi digits it accurately represents versus how many digits there are in the approximation formula -- this says kind of the approximation's [compression](compression.md) ratio. But other factors may be important too, e.g. simplicity of evaluation, functions used etc.
 
+Also remember, **you can measure pi in real life** by many methods: you can draw a big circle, measure its radius and circumference and then make the division, you can also manually perform the Monte Carlo algorithm (see below) by drawing a circle and then throwing objects around, counting how many fall inside and outside (just watch out to do it correctly, for example you must have the fall spot probability as random as possible, not biased in any way), or you can similarly make a square from wood, then cut out its inscribed circle, weight both parts and compute pi (with the same formula as for Monte Carlo).
+
+{ I tried this -- I took a pizza box, cut out four squares, then  used a pencil on string to draw quarter circles on each, cut them and weighted both groups. All the circle parts weighted 61 grams, the rest weighted 16 grams, this gives me a nice estimate value of pi of about 3.16. ~drummyfish }
+
 An ugly engineering [approximation](approximation.md) that's actually usable sometimes (e.g. for fast rough estimates with integer-only hardware) is just (something like this was infamously almost made the legal value of pi by the so called Indiana bill in 1897)
 
 pi ~= 3