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(𝜃1), sin(𝜃1)) is a point on the circumference

q = (cos(𝜃2), sin(𝜃2)) is another point on the circumference

x = ½ cos(𝜃1) + ½ cos(𝜃2) and

y = ½ sin(𝜃1) + ½ sin(𝜃2), for random 𝜃1 and 𝜃2, define their midpoint.

B. Random polar coordinates

x = r cos(𝜃)

and y = r sin(𝜃), for random angle 𝜃 and radius r ≤ 1. This gives choices biased towards the centre.

C. Random y, then restricted x

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

D. Random point on chord in A

Similar to A, but x = a cos(𝜃1) + (1−a) cos(𝜃2)

and y = a sin(𝜃1) + (1−a) sin(𝜃2), for random 𝜃1 and 𝜃2 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 = √r cos(𝜃)

and y = √r sin(𝜃), for random angle 𝜃 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 ≤ 1. This is uniform, and the selection condition restricts the final choice to the disc.

Oh, and here are some Pi Day activities.

Timeline of mathematical notation

Following up on my earlier timelines about zero and about Hindu-Arabic numerals, here is a timeline for some other mathematical notation, starting with the square root symbol (click to zoom).

Pi Day!

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

  • Search for your birthday (or any other number) in the digits of π
  • Follow in the footsteps of Archimedes, showing that π 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 π by measuring the area of a circle (most simply, with radius 10 or 100), using A = πr2. An easy way is to draw an appropriate circle on a sheet of graph paper.

You can also try estimating π using Buffon’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 / (π d), or 2 / π 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 π, mostly correctly!

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

Actually, of course, π = 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).

Pumpkin time!

It’s Halloween time, at least in the USA. There have been some interesting carved pumpkins around with scientific and mathematical themes, like Nathan Shields’ interpretation of “Pumpkin Pi” above. The legendary Will Gater also excelled himself last year, and Lenore Edman produced a great “DNA-o-lantern” a few years ago (photo below).

Greetings to everybody who celebrates the day!

Why study mathematics?

Like John Allen Paulos, I am often asked why mathematics is worth studying. In his book A Mathematician Reads the Newspaper (Basic Books, 1995), Paulos gives an excellent answer:

As a mathematician, I’m often challenged to come up with compelling reasons to study mathematics. If the questioner is serious, I reply that there are three reasons or, more accurately, three broad classes of reasons to study mathematics. Only the first and most basic class is practical. It pertains to job skills and the needs of science and technology. The second concerns the understandings that are essential to an informed and effective citizenry. The last class of reasons involves considerations of curiosity, beauty, playfulness, perhaps even transcendence and wisdom.

The second and third answers are reflected in the words inscribed on the door of Plato’s Academy: “Let no one ignorant of geometry enter” (Ἀγεωμέτρητος μηδεὶς εἰσίτω):


The first answer relates to the critical importance of mathematics in several fields of human endeavour, including science, engineering, medicine, and finance. For example:

A stressed ribbon bridge is strong if its shape is that of the mathematical curve called a catenary.

The spread of an infectious disease can be predicted by a set of three differential equations, relating three variables: S, I, and R (left). Real-world disease outbreaks show a similar pattern (right).

Many people list this as the only reason for studying mathematics, but it only applies to a minority of students – those keeping open the option of entering those fields. The second answer relates to the importance of mathematics in decision-making by ordinary citizens, and this applies to everybody. Some of those decisions by citizens require quantitative thinking. For example, which groceries are the best value for money? If two studies on 20 people report that a certain vegetable causes cancer, and one study on 1,000 people report that it doesn’t, is the vegetable safe? More subtly, training in mathematics helps in thinking clearly even about non-quantitative issues. Plato seemed to think that mathematics was essential training, and I would agree. Bertrand Russell put it this way: “One of the chief ends served by mathematics, when rightly taught, is to awaken the learner’s belief in reason, his confidence in the truth of what has been demonstrated, and in the value of demonstration.

What is the best value for money – the melons at $5 each, the grapes at $4 per kg, or the blueberries at $3 per punnet?

This classic book has applications to more than mathematics.

The third answer relates to a famous remark of Debussy – “La musique est une mathématique mystérieuse dont les éléments participent de l’infini” (“Music is a mysterious mathematics whose elements partake of the Infinite”). It works the other way around too. Mathematics is a mysterious and beautiful music that puts one in touch with the Infinite. As Plato would have said, mathematics reminds us that more things exist than just the finite and physical. This particularly applies to those parts of mathematics which relate to infinity, such as the number π, or the Mandelbrot set:

π = 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989 …
Some of the (infinitely many) digits of π.

The Mandelbrot set contains an infinite amount of detail (click for zoom animation).

Rudy Rucker’s little book The Fourth Dimension and How to Get There is also a great mind-stretcher. And, of course, having one’s mind stretched like that is a lot of fun.