Pi and phi, what are they? What do they have in common and what they don’t? No student of mathematics would’ve finished their course without coming across pi at least once. And phi, if you admire beauty, all thanks to it. They’ve kept scientists working on them for centuries, unraveling mysteries to the righteous, every now and then. We already have an article on the golden ratio aka phi. If you haven’t read that, you can find it here: The Intriguing divine ratio of Nature
“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
What is Pi?
The ratio of a circle’s circumference to its diameter, denoted by the lower-case Greek letter 𝜋 is one of the famous mathematical constants. For any perfect circle, the ratio of its circumference to the diameter is a constant i.e. pi no matter whether a large or small circle is considered. 3.141592654… is what you get in a calculator not because it’s the value. But, the calculators’ display is often restricted to ten digits. Pi is an irrational number, a non-terminating and non-repetitive decimal. 22/7, 3.14 and 3 are few commonly used approximations.
Mathematicians and their Obsession with Pi:
The search for Pi’s value started long back, that you won’t believe me if I said when. Many renowned mathematicians including Fibonacci, Leibnitz, Gauss, Newton, Johann Heinrich Lambert and several other lesser known mathematicians contributed their life in finding the values of pi. It started 4000 years ago when Ancient Babylonians used the approximated value of circumference of a circle i.e. 3 times its diameter. Evidence states that they’ve used 3.125 which is a close approximation. Greeks have used a formula which approximates pi’s value to be (16/9)2 = 3.165.
Verse 1 Kings 7:23 from the Bible says:
He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line of thirty cubits to measure around it.
Archimedes, a Greek mathematician used an interesting algorithmic approach to prove that the value of pi to be no less than 221/73 and no more than 22/7. It wasn’t until the 1600s that it had a global name. Fast forward, now we’ve pi approximated to 31,415,926,535,897 digits (~ 31 trillion!) To me, the most amazing approximation is by Aryabhata, an Indian mathematician and astronomer who approximated it as 3.1416. Impressive, isn’t it? The admirers of pi have set world records by memorizing the values of pi. You ask how much I know? 100 digits, I’m already proud of myself. But all you need is 39 digits or so to have virtually no error.
Where is pi?
Everywhere around you! Circles, triangles, cones, spheres, cylinders, arcs, name it I see it. Pi is widely used in Mathematics and Physics as in finding area and volume, quantum Mechanics, for building and construction, music theory, air travel and much more. With technological advancements in recent days, their application in real world systems has significantly increased. Used in calculating the trajectories, NASA uses a technique called Pi Transfer in Cassini Spacecraft to complete a maneuver to fly by Saturn’s moon Titan. From tracking population dynamics, studying the structure of the eye to calculating the areas of the skin of aircraft, the value of pi has been widely used. In the measurement of waves, is pi.
Why is that so important?
With people memorizing thousands of digits of it just for fun, it’s nothing ordinary. From the definition of pi comes a new perspective to look at the universe we live in, a new approach in the measurement of angles (radian measure), a more useful way than the familiar degree measure relating the angle with the length. 360° = 2𝜋. Including the Heisenberg’s uncertainty principle, it has a place in many important formulae. The probability that any two random number is relatively prime is 6/𝜋2. Strange! Be it rays or waves, you measure them with pi. Owing to its importance, it has a day dedicated to itself! Guess when? March 14th (3.14).
Pilish, a form of writing developed by the pi aficionados which is based on the length of words, matches the numbers as written in the sequence of digits in pi. There are short stories, poems (piems), pi-kus and also a novel. The maximum length of a Pilish string found in other writings so far is 8.
What is phi?
Denoted by 21st Greek letter is the phi, a ratio obtained when a line segment is divided in a special way. The ratio between two segments of a line segment is said to be phi if their ratio is the same as the ratio of their sum to the larger of the two quantities. Also called the Golden Ratio, Golden Mean, Golden Number, Golden Section, Golden Proportion, Divine Proportion and Divine Section; it accounts for the perfect beauty found in nature. Just like pi, it is also an irrational number. It is approximated to 1.618 .
“The Fibonacci Sequence turns out to be the key to understanding how nature designs… and is… a part of the same ubiquitous music of the spheres that builds harmony into atoms, molecules, crystals, shells, suns and galaxies and makes the Universe sing.”
― Guy Murchie, The Seven Mysteries of Life: An Exploration of Science and Philosophy.
The Flashback:
From Da Vinci to the next door photographer, everyone admires Golden Ratio for its presence makes things beautiful. No one invented the Golden Ratio, they just discovered what was already prevalent in nature. Fibonacci series is closely related to the Golden Ratio. Larger the terms, the closer their ratio gets to phi. In man made marvels, the Golden Ratio can be widely observed in the art and architecture. Dating back to the Parthenon of Greeks, the Pyramids of the Egyptians, we see the Golden Ratio. It was in the 1900s that American mathematician Mark Barr first denoted the golden ratio with the symbol phi (𝜙). Euclid is the first one to use the number 0.618… to divide a line based on its extreme and mean ratio. Though its close relative Fibonacci Series was discovered in the year 1175 AD, its association with the golden ratio might be unexpected.
The Da Vinci Code talks Phi:
Leonardo Da Vinci has used phi in many of his paintings including the most famous Mona Lisa. Not just him but many other renaissance artists. Though it is seen vividly in many art and architectural marvels, we are not quite sure that they thought about that. Many paintings in the renaissance period exhibit phi in one or the other form including the famous Mona Lisa and The Last Supper by Leonardo Da Vinci. Its appearance in and around us has led many writers to give that a key role in their books. Dan Brown is no exception to that. His novel The Da Vinci Code spins around this mysterious number phi. Though not everything he has mentioned in his book is true, the fact that the seeds in a sunflower, leaves around a stem are associated with the golden ratio is true. But, the number’s presence in human beings (ratio of body height to distance from your belly button to the floor) is not factually correct. And not for other animals too.
Its presence in the marvels of the universe could’ve led him to believe this to be true. Though Euclid first used the number, it’s not that others would have not come across that. There are many other constants like Euler’s constant, natural logarithm (e) and much more that play a key role in driving the lives of this universe. In self reproduction appears the Golden ratio, but also several others appear which makes it nothing special. When you derive the characteristic equation for the self reproduction simulations, the roots are algebraic numbers. If you think the golden mean is the only one having a precious metal’s name, check out other metallic means.
Still thinking about what lies common? Nothing I would say except the fact that they are both irrational decimals and play a significant role in their mysterious ways.
Reference:
- Calculating a record-breaking 31.4 trillion digits of Archimedes’ constant on Google Cloud
- A website dedicated for Pi
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