Five Design Principles from Pi
Greek mathematician Archimedes and a Roman soldier. The last words attributed to Archimedes are: “Do not disturb my circles.”
May I tell a story, imagining us living years ago?
The year is 300 BC. Euclid has just released his treatise “Elements,” the first-ever collection of fundamental principles of geometry [1]. Using only a compass and a straight edge, more than 400 proofs are laid out in a clear and comprehensible way. Some of these proofs involve properties of circles.
It took a few more decades for another Greek, Archimedes, to release his essay, “Measurement of a Circle.” The third proposition in this essay is what many believe to be the first ever calculation of the value of pi [2]. Archimedes’ method involved inscribing and circumscribing polygons around the circumference of a circle. By measuring the perimeter of the inner and outer polygons, he was able to set an upper and lower bound to the value of pi.
Now suppose you are a student of Archimedes. You’ve heard him go on and on about circles and circumference approximations. One day, he asks you to help him with his research. Your goal is to calculate the ratio of a circle’s circumference to its diameter. Circles come in different sizes, but is there a mathematical truth to be found in this relationship?
What might have been your initial hypothesis about this ratio?
In ancient times, many estimated the ratio of circumference/diameter to be 3. However, this can be easily disproved. Maybe a little more than three? How about 22/7? That seems about right. Obviously, the ratio of a circle’s circumference to its diameter should be expressible as a fraction. After all, a ratio is a fraction. This was so obvious that for 2000 years after Archimedes, people believed this fraction existed. It wasn’t until Johann Lambert proved the irrationality of pi in 1768 this notion was put to rest [3]. Pi cannot, in fact, be expressed as a fraction of two integers.
It is understandable that pi’s irrationality was not easily observed. The discovery of the first irrational numbers was quite dramatic. In the 6th century BC, before Euclid and Archimedes, and long before Johann Lambert, Pythagoras’ followers (known as the Pythagoreans) believed in a mathematics-based religion. They believed that numbers (i.e, positive integers) explained the true nature of reality.
The Pythagoreans explored the relationship between numbers and geometry, and ascribed a certain divinity to Pythagoras’ teachings. It is said that when the philosopher Hippasus divulged the existence of irrational numbers (like the square root of 2), the Pythagoreans were so disgusted that they drowned Hippasus at sea [4]. A fitting end, as the Pythagorean gods could not allow such heresy.
After Hippapsus’ death, it is no wonder irrational numbers took a while to become part of mainstream geometry. The concept of a never-ending number would have been disturbing to a Pythagorean. How can the value of a circle’s circumference have infinite digits? How can a finite line have no numerical end?
“Probably no symbol in mathematics has evoked as much mystery, romanticism, misconception and human interest as the number pi”
~William L. Schaaf, Nature and History of Pi
Few numbers have a history so rich. Even in the modern age, every few years a computer scientist generates a world-record calculation of pi, in 2022 up to a length of 100 trillion digits [5]. As you celebrate Pi Day this year, consider five lessons that this transcendental number can teach us about design.
1. Iteration is Key
Archimedes of Syracuse’s early attempts to calculate pi were impressive. His method involved inscribing polygons on the inside and outside of a circle. He increased the number of sides on the polygons, simultaneously increasing the accuracy of his calculations. After every iteration, Archimedes knew that he could do better. So, with every attempt to measure pi he increased the number of sides on the polygon. He analyzed a 6-sided polygon up to a 96-sided polygon [6]. If you are not impressed, grab a compass and a straightedge and try it yourself.
From Archimedes we can learn that no feat of engineering is perfect on the first try. Some things go on and on. He did the best that he could with the tools he had. Mattson and Sorensen provide an introduction to engineering design with the figure below. The steps of the design process are outlined as Understand, Explore, Design, Test, and Refine [7].
2. Some Things are a Waste of Time
The Greeks, in their evident love of geometry, had an unsolved challenge: squaring the circle. Maybe you have heard this as an expression meaning “to do something impossible.” This idea refers to the challenge of drawing a square with the same area as a given circle. However, you are only allowed a compass, a straightedge, and a finite number of steps to solve the problem. It turns out, because pi is never-ending, this is impossible (imagine it: if a circle’s area is 𝝅*r^2, then the square has sqrt(𝝅)*r side length). It may be tricky to find the spot on your ruler corresponding to 1.7724538…
When facing an engineering challenge, learn to recognize when it doesn’t make sense to keep working. The sunk-cost fallacy has prevented many engineers from moving on to more fulfilling projects. That being said, if a problem doesn’t require the square circle, persistence can work miracles.
3. Ask Insightful Questions
The existence of pi, and more generally irrational numbers (not to mention complex numbers), provides a lush playground for philosophical debate.
Consider this: if pi represents the number of diameters in a circle’s perimeter, how can it be infinite?
Asking insightful questions is essential for advancing human understanding. The value of philosophy lies in asking provoking questions. Mathematics and other branches of science originated as philosophies. Ground-breaking research usually begins with a question. All engineering teams benefit from nurturing a spirit of curiosity.
4. Be Rational
How many digits of pi do astronauts require for their ultra-precise computations and trajectory calculations? Maybe 15 [8]. If you were to trace a circle around the observable universe, using 38 digits of pi would allow you to have a measurement accurate to the closest atom [9]. Unlike pi, we need to be rational in our engineering choices. Not only do we need to know when to quit (no. 2), but we also need to recognize how much effort to spend on continuing projects.
In order to prevent racing monopolies, Formula 1 has set limits on how much teams can spend on their staff and supplies. In 2023 the budget cap was set at $135 million per team [10]. The question of how to allocate resources is a big challenge. Knowing how much time, energy, and money to spend on engineering projects is an important skill.
5. Be Creative
It’s true that you could write a computer program to calculate pi, or maybe write out an infinite sum. Even better, you could calculate pi by throwing sausages on the floor [11]. It is rare for the best ideas to be the first ideas. Instead, it can be good to practice ideation by unashamedly letting all your ideas flow like a drain unclogging to allow fresh water through.
Engineering and innovation go hand-in-hand. Because of this relationship, creativity is an under-valued skill for engineers to have. A little bit of design and artistry training goes a long way.
On this 14th of March, we hope you have a happy Pi Day! Celebrate the holiday this year by applying one of these principles to a project you are working on.
By the way, did you notice that the first sentence was written in Pilish [12]?
References
[1] University Library, “Euclid’s Elements,” University Library Website, https://libguides.ncl.ac.uk/euclid accessed Pi Day 2024
[2] Bloomsbury Publishing, “Archimedes, Measurement of a Circle,” Bloomsbury Media Website, https://media.bloomsbury.com/rep/files/primary-source-13-archimedes-measurement-of-a-circle.pdf accessed Pi Day 2024
[3] Mac Tutor, “Johann Heinrich Lambert,” Maths History Website, https://mathshistory.st-andrews.ac.uk/Biographies/Lambert/ accessed Pi Day 2024
[4] Science ABC, “How Were Irrational Numbers Discovered?” Science ABC website, https://www.scienceabc.com/pure-sciences/how-irrational-numbers-discovered-mathematician-killed-hippasus.html accessed Pi Day 2024
[5] Google Cloud, “Even more pi in the sky: Calculating 100 trillion digits of pi on Google Cloud,” Google Cloud website, https://cloud.google.com/blog/products/compute/calculating-100-trillion-digits-of-pi-on-google-cloud accessed Pi Day 2024
[6] Business Insider, “The beautifully simple method Archimedes used to find the first digits of pi,” Business Insider Website, https://www.businessinsider.com/archimedes-pi-estimation-2014-3 accessed Pi Day 2024
[7] Mattson, C. A., & Sorensen, C. D. (2019). Product development: principles and tools for creating desirable and transferable designs. Springer Nature.
[8] Wired, “How Much Pi Do You Really Need?” Wired Website, https://www.wired.com/story/how-much-pi-do-you-really-need/ accessed Pi Day 2024
[9] Jet Propulsion Laboratory, “How Much Decimals of Pi Do We Really Need?” JPL website, https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need/ accessed Pi Day 2024
[10] CNBC, “How an F1 spending cap made racing teams more investable,” CNBC website, https://www.cnbc.com/2023/11/16/f1-spending-cap-made-racing-teams-more-investable.html accessed Pi Day 2024
[11] WikiHow, “How to Calculate Pi by Throwing Frozen Hot Dogs,” wikiHow website, https://www.wikihow.com/Calculate-Pi-by-Throwing-Frozen-Hot-Dogs accessed Pi Day 2024
[12] Cadeic, “Writing in Pilish,” Cadeic website, http://www.cadaeic.net/pilish.htm accessed Pi Day 2024
