# Happy Pi Day!

Mar 14, 2021 By:Vincent Broeren

As a programmer you almost have an inherent connection with math. I’m not saying that math is a need for programming, but it sure makes things a whole lot easier!

## Pi day?

In the ISO standard system you write dates in the correct order, from large to small. An example: the fifth of March 2021 is written as: 2021-03-05. So the 14th of March will be written as: 2021-03-14 which is why we call it Pi-day.
As a programmer Pi is almost a magical number. It fascinates me that there is no apparent ending on the number of decimals in it. And when you also have a love for graphics, the combination of the two is easily made. Let me tell you how I made that combination.

## Visualize Pi

So how can you visualize Pi in a cooler way then just a bunch of numbers? We all know Pi is approximately

3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117
06798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493
03819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602
49141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951
94151160943305727036575959195309218611738193261179310511854807446237996274956735188575272489122793
81830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748
18467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050
79227968925892354201995611212902196086403441815981362977477130996051870721134999999837297804995105
97317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083
81420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927
87661119590921642019893809525720106548586327886593615338182796823030195203530185296899577362259941
38912497217752834791315155

Which is already mesmerizing to look at, but it could be visualized even cooler. If you give each digit a distinct color, then you can create a “painting” of the number Pi like the one below (click on it for full size). Every colored dot is a digit of Pi, and there are a lot of them in that painting.

## Calculating Pi

When I was writing a small program to create the Pi-painting I used the internet as a source to get the needed digits of Pi. But as a true professional you should always check your sources. Which made me realize I NEVER calculated Pi by myself! I always trusted my math-professors and my calculator when they told me Pi was 3.1415 etc…

I took on the adventure to calculate Pi myself, just to make sure my calculator is telling the truth! First of all we need the formula for the calculation, for which I used the highly popular “Chudnovsky algorithm”. I chose that algorithm, because it is also used by the super computers to calculate the 50,000,000,000,000 number of digits of Pi (the current world record, which took 303 days to calculate!). I don’t have the time or computing power to even approach that kind of digits, but the first 100.000 should be enough (calculated on my laptop).
The used algorithm is a series, meaning that the longer you calculate the more decimals you should get. But after implementing the algorithm all I got was: 3.1415926535897417108201077343, which goes wrong after 13 digits already. 13 digits may sound as a nice precision but I want 100.000 digits of precision!

#### So I needed to analyze the algorithm…

Without getting too much into the nitty gritty math stuff there are four numbers used in the series and in math we give numbers cool names when we talk about them. So let me introduce you to C, Mk, Lk and Xk. C is a constant, which is nice and easy. Mk is large number which keeps getting larger and larger when we increase the precision. This also counts for Lk. But the fun starts with Xk, because that number becomes incredibly small and approaches 0 really fast. And that is where my implementation broke, because I used only a decimal type in the calculation. So I ran into the imperfections of that primitive type really fast.
And that turned my 30 minute playtime implementation into a much larger undertaking, because I needed to solve that imperfection problem. Maybe you think “well just leave it there, 13 digits is already quite nice”. But we are programmers aren’t we!? “Cannot” doesn’t exist in our world!
Maybe you guys know I’m also a lecturer at one of the best Universities in the Netherlands. And I’m fortunate enough that I can use some of my boss’s time as “professional/personal development”. And math is a must for lecturers, right!? So I can use some of that time to research into a better implementation of my pi-calculations.
Fortunately the .NET framework has a BigInteger implementation which can be used to store arbitrarily large integers. Which we can also use to store infinitely small numbers. And that was exactly what I needed to solve my precision problem.

## Conclusion

I’m happy to inform you that I’ve checked the popular representation of Pi and the first 100,000 digits are indeed correct!
This is an example of a small pet-project which you can do in your own down-time. What is important is to “Question everything” and just keep programming as much as possible. Always try to find cool new projects to create and extent your love for programming!