π or "pi" is the ratio of the circumference of a circle, to its diameter. https://en.wikipedia.org/wiki/Pi The value of π, as almost universally taught, is not correct, or to be more precise, is only correct to three (as in 3) significant digits only, that is, the fourth digit (3rd decimal place) and onwards is incorrect, and significantly so. :D OK muffas, this is within the ability of many of us to actually measure, so it's time for some science - the original measure pi challenge :) We are taught all sorts of things about pi, and many of them are (and to use the excessively flowery language of the Russian MOD) "not correct". Say what?!#!?? Yes you heard correct - pi as we've all been taught since Archimedes in about 250 B.C. (B.C. already!) is wrong. https://en.wikipedia.org/wiki/Archimedes Oh, and a hat tip to Harry Lear who gave us all the heads up at his Measuring Pi Squaring Phi web cite ;) : http://measuringpisquaringphi.com/ Measure Pi Challenge ==================================================== To kick off this challenge, a meek and humble experiment was in order, no doubt which y'all will handily improve upon, so let's get this ball rolling: One tempered glass tabletop of an apparently circular form and ~1.2m diameter (chamfered) and with visibly extremely smooth surfaces including, most importantly, the external (beyond the chamfer) circumference surface. Also in use are some lengths of wood, set square, metal 1 meter ruler and two metal tape measures (and calculator, pen and duct and paper tape). 0) Ensure length(s) of wood are sufficient length (plus a goodly bit) for the full perimeter/circumference of the table to roll along. 1) Tape a "circumference origin" marker X at one point on the perimeter of circular glass tabletop. Do this on the left hand side of the glass/circle, not the right hand side ;) 2) Stably secure wood length(s) on floor of kitchen/hallway, e.g. with duct tape or through-floor bolts. Again, make a "linear origin" mark Y (with tape and/or pen, or perhaps a jackhammer) near one end of your length of wood on the floor. 3) With a second person to help steady the glass tabletop, carefully align circumference origin X with linear origin Y. 4) Carefully roll the tabletop along the wood, one revolution (i.e., until the circumference origin X lands back on the wood. Mark this second linear point (on the wood), we'll call this point Z. 5) Next, measure the diameter of the tabletop. Once you have measured the distance from Y to Z, you have an approximation of the circumference of your tabletop. Divide circumference by diameter, to get YOUR value of pi! Whatever your results are, they are. For bonus points, use other, better, alternative, or improved, ways to measure pi. Here's the results of tonight's measurements, and see some of reduced size vanity images of this test run. We did more than one set of measurements: - firstly with a 1 meter metal ruler - then with each of two metal tape measures The metal ruler turns out to have markings up to 1.005 meters, that is, it has an extra half centimeter beyond the 1m mark. Neither of us noticed this at first, and the circumference of this tabletop is heading for 4m, and so this extra 5mm was added into the (linear) circumference measurement 3 times when using this metal ruler - in other words, we had an extra 1.5cm on the circumference (not including margins of error), and an extra 0.5cm on the diameter (which albeit partially, but certainly not completely, cancel each other out). Terms: C = circumference D = diameter ~ (tilde) means "about" Initial bodgy (with extra half centimeters included) measurements were: C = 3836.5 mm +/- 2 mm (4 measurements, at +-1/2mm error each) D = 1219.5 mm +/- 1 mm (2 measurements, at +-1/2mm error each) and so when subtracting out the extra 1.5 and 0.5 cm respectively, we get: C = 3821.5 mm +/- 2 mm D = 1214.5 mm +/- 1 mm The assumed error margin of +/- 2mm and 1mm respectively (+/- 1/2 mm per measurement, remembering that errors are cumulative) may seem high, but at least accounts for using a metal ruler with 1mm markings, 4 times for the (linear) circumference measurement and twice for the diameter. pi is simply C/D, so we have (using the program "bc", with scale=10): 3821.5 / 1214.5 = 3.1465623713 And considering the extremes of possible error margins (not counting human error of course): 3823.5 / 1213.5 = 3.1508034610 3819.5 / 1215.5 = 3.1423282599 So the results from these measurements does appear to be above 3.14159, and converges closer to Harry Lear's value of pi 3.1446, than the traditional value of pi 3.1316: 3.1465623713 - 3.144605511 ~= 0.002 3.1465623713 - 3.141592654 ~= 0.005 Harry Lear appears to be on to something. Now using metal tape measure #1: C = 3854 D = 1226 pi = 3854 / 1226 = 3.1435562805 and with say +-0.5mm error: 3854.5 / 1225.5 = 3.1452468380 3853.5 / 1226.5 = 3.1418671015 Tape measures can be considered better, since we don't need to make multiple readings and "line up" our measurement instrument (ruler) for each additional segment, to get our final lengths. And in this case, using the first tape measure and including maximum possible errors applied in opposite directions, although we are closer to traditional pi, we still does not reach it! Comparing the two πs again, we are still closer to new pi than old pi: 3.1435562805 - 3.144605511 ~= -0.00105 3.1435562805 - 3.141592654 ~= 0.00196 And finally a third set of measurements in this test run, using metal tape measure #2: C = 3851 D = 1225 pi = 3851 / 1225 = 3.1436734693 and with say +-0.5mm error, the extremes are: 3851.5 / 1224.5 = 3.1453654552 3850.5 / 1225.5 = 3.1419828641 Again, we are much closer on average to new pi 3.1446, than to the old pi 3.1416, and apparently not reaching old pi, where, in this set of measurements, it is consistently outside our margin of error. Confirming the distance from old and new pi, with tape measure #2, we remain closer to new pi: 3.1436734693 - 3.144605511 ~= -0.00093 3.1436734693 - 3.141592654 ~= 0.00208 ---------------------------------- To be wrong about something so fundamental as the pi ratio of the circumference of a circle to is diameter (or radius), for so long, is simply stunning! The "scientific community" prides itself in its ever increasing accuracy in measurement in many domains, from the micro to the macro, and all the while this one sneaky little constant has been so taken for granted, it has slipped under the radar (intentionally?) for over 2000 years. So, 3 significant digits of pi "as we know it" are correct: 3.14 The fourth significant digit, which begins after the second decimal place, is significantly wrong (and of course, if this is so, so are all subsequent digits). Apparently.
