# Pi Day once more!

In honour of Pi Day (March 14), the chart shows six ways of randomly selecting a point in a unit disc. Four of the methods are bad, for various reasons.

## A. Midpoint of random p, q on circumference

p = (cos(π1), sin(π1)) is a point on the circumference

q = (cos(π2), sin(π2)) is another point on the circumference

x = Β½ cos(π1) + Β½ cos(π2) and

y = Β½ sin(π1) + Β½ sin(π2), for random π1 and π2, define their midpoint.

## B. Random polar coordinates

x = r cos(π)

and y = r sin(π), for random angle π and radius r β€ 1. This gives choices biased towards the centre.

## C. Random y, then restricted x

Random y, followed by random x in the range ββ(1βy2) to β(1βy2). This gives choices biased towards the top and bottom.

## D. Random point on chord in A

Similar to A, but x = a cos(π1) + (1βa) cos(π2)

and y = a sin(π1) + (1βa) sin(π2), for random π1 and π2 on the circumference of the circle and random a between 0 and 1. This gives choices biased towards the periphery.

## E. Random polar with sqrt(r)

Similar to B, but x = βr cos(π)

and y = βr sin(π), for random angle π and radius r. The square root operation makes the selection uniform across the disc.

## F. Random x, y within disc

Random x and y, repeating the choice until x2 + y2 β€ 1. This is uniform, and the selection condition restricts the final choice to the disc.

Oh, and here are some Pi Day activities.

# Pi Day!

Pi Day is coming up again (3/14 as a US date). The number Ο is, of course, 3.14159265… Here are some possible activities for children:

• Search for your birthday (or any other number) in the digits of Ο
• Follow in the footsteps of Archimedes, showing that Ο is between 22/7 = 3.1429 and 223/71 = 3.1408.
• Calculate 333/106 = 3.1415 and 355/113 = 3.1415929, which are better approximations than 22/7.
• Measure the circumference and diameter of a round plate and divide. Use a ruler to measure the diameter and a strip of paper (afterwards measured with a ruler) for the circumference. For children who cannot yet divide, try to find a plate with diameter 7, 106, or 113.
• Calculate Ο by measuring the area of a circle (most simply, with radius 10 or 100), using A = Οr2. An easy way is to draw an appropriate circle on a sheet of graph paper.

You can also try estimating Ο using Buffonβs needle. You will need some toothpicks (or similar) of length k and some parallel lines (such as floorboards) a distance d apart (greater than or equal to k). Then the fraction of dropped toothpicks that touch or cross a line will be 2 k / (Ο d), or 2 / Ο if k = d. There is an explanation and simulator here (see also the picture below). And, of course, you can bake a celebratory pie and listen to Kate Bush singing Ο, mostly correctly!

This picture by McZusatz has 11 of 17 matches touching a line, suggesting the value of 2Γ17/11 = 3.1 for Ο (since k = d).

Actually, of course, Ο = 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 … (digits in red are sung by Kate Bush, accurately, although some have said otherwise).

# Happy Pi Day!

Itβs Pi Day again today (3/14 as a US-style date). The piday.org website seems to not have been updated much, but the celebration in the US seems bigger every year. This year, a great many bakeries and pizza shops seem to be offering discounts.

# Happy Pi Day!

Itβs Pi Day today (3/14 as a US-style date). Oxford Connect is running some interesting events in association with the day, including a large-scale collaborative effort to determine pi using Buffonβs needle (like this):