Let's celebrate 𝜋!

Above: The first 54 digits of 𝜋 represented by colored discs. Design inspired by Martin Krzywinski’s 2013 Pi poster featured in the Numberphile video “Pi is Beautiful.”

Happy 𝜋 day! 𝜋 is probably the most familiar of the irrational numbers, and it represents a truly amazing mathematical fact: no matter what size circle you take, if you divide its circumference by its diameter, you always get the same number! That number, 𝜋, can be approximated as a fraction, such as 22/7, or a decimal, 3.14159…

The image above is actually another way to represent 𝜋. Instead of using digits, this image encodes 𝜋 in colors, where each of the numerals from 0 - 9 is matched to a different color.

A digit in the sequence of 𝜋 is then represented by a shaded disc, where the outside is the color corresponding to the digit itself, and the inside is the color of the digit that follows two decimal places later. The discs spiral from the outside to the inside, starting from the top left and moving right. The drawing represents a rich tradition of finding surprising beauty in mathematical randomness. The visualization was inspired by the work of Martin Krzywinski, which is featured in Numberphile’s video “Pi is Beautiful.”

It would be easy to overlook the richness and beauty embedded in 𝜋. The same thing can happen with math and mathiness. Sometimes, we’re quick to pigeonhole others and ourselves into convenient categories of math people or non-math people. It’s easy to decide what math should look like, or make assumptions about who looks like a mathematician and who doesn’t.

This 𝜋 day let’s take a step back from our preconceived notions. Celebrate the beauty and complexity of pi and of math, and remember that there's beauty and complexity in who is and who can be a mathematician, as well!

And if you’d like to celebrate 𝜋 Day with a little extra math, try out the challenge problem below. (The answer appears after the problem.) 

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Challenge problem: Jarek is bored in class and starts putting numbers on his paper like in the following pattern:

If he's really bored and keeps going with this pattern for a long time, what are the 8 numbers that will surround the square containing 1,000,000?

Bonus: What are the eight numbers surrounding the square containing 1,000,010?

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Solution:

The numbers arrange themselves in squares. For example, 4 is in the top-left of a square containing the numbers 1-4, while 16 is in the top-left of a square containing 1-16, etc. There is a pattern of even square numbers going up and to the left, so because 1,000,000=1000^2, it is in the top left of a square containing the numbers 1-1,000,000.

That means that directly to the right of it, there will be the number 999,999, and directly to the left will be the number 1,000,001. That is when the pattern turns down, so below that is 1,000,002 (which is down-left from 1,000,000). So we've found three of the numbers around 1,000,000.

Down and to the right of 1,000,000, there will be the number 998^2=996,004 because another square finishes there. To the left of that (and right below 1,000,000) is 996,005.

Up and to the left of 1,000,000 is 1002^2=1,004,004. To the right of that (and directly above 1,000,000) is 1,004,003. To the right of that (and above-right of 1,000,000) is 1,004,002.

Thus, the eight numbers near 1,000,000 are 996,004, 996,005, 1,000,002, 999,999, 1,000,001, 1,004,002, 1,004,003, and 1,004,004.

Now, for the extra bonus. The number 1,000,010 appears nine spaces below 1,000,001 along the side of the square. Directly to the right of it is the number that is eight spaces below 996,005, which is 996,013. Directly to the left of it is the number that is ten spaces below 1,004,005, which is 1,004,015.

Above and below 1,000,010 are 1,000,009 and 1,000,011. Above and below 996,013 are 996,012 and 996,014. Above and below 1,004,015 are 1,004,014 and 1,004,016.

Thus, the eight numbers near 1,000,010 are 996,014, 1,000,011, 1,004,016, 996,013, 1,004,015, 996,012, 1,000,009, and 1,004,014.