By Marco March

As information is passed down over the years, it is only right that we listen to our elders and take their advice seriously. How many times have you been pleasantly surprised by a seemingly-ancient technique from the 1970’s that has bailed you out of a sticky situation? For instance, when the old guy knew a coat hanger would prove to be a fitting replacement for the bathroom door handle you pulled off, saving you from being stuck in the room — it’s that kind of wisdom that will live on for an eternity.

But of course there is a lot of outdated information being shared around the world, much of which is taken as fact when it is untestable and unobservable. So where do we draw the line between the old and new schools and between fact and fluff?

From a mathematical perspective, this shouldn’t be a problem as maths might be the purest way to consider the universe and its subsidies. That is to say, one can most accurately assess constructions and how they interact together with maths. And yet, when we try to apply established theories in practice, we are often surprised when the results are not as we expected. This could be due to a human error such as a failure to consider a pertinent factor, or it could be due to the imperfections of the theory itself.

Take, for example, the disk brake of a car. The more perfectly the centre of the brake pads align with the tangent on the circumference of the disk brake, the more effective the friction is, and subsequently there will be more stopping power. To get the circles to align perfectly, the number Pi is used because it gives us the ‘exact’ ratio of the circumference and the diameter of the circle, thereby giving two mathematically ‘exact’ circles of relative sizes. But there is a problem in that Pi is a transcendental number, meaning that it is infinite. Because of this, it produces clumsy results leading to an imperfect circle — not good news for mechanics.

There are two notable counter theories, one of which has a movement behind it that aims to ‘kill’ Pi and replace it with ‘Tau’ (2) because it produces ‘cleaner’ results. The other claims that the so called ‘golden ratio’ is a better representation than Pi because it uses the number derived from the Fibonacci sequence (1.6180…) to more precisely determine the circumference using the equation = 3.1446… as opposed to Pi which is 3.1415… That tenth of a percent could be the difference between a Mercedes and a Ferrari, and since most would agree that Mercedes perform much better in the corners because of their superior brakes (amongst other aspects), it is a minor factor that should not be overlooked… if you’re trying to win a Grand Prix, that is. If you are making a pink Hummer limousine, though, it really doesn’t matter which formula you use; it doesn’t like corners. And this is exactly the attitude schools demonstrate in the matter. They don’t like corners. They don’t like to use three potential constants when one will ‘do’ and has been ‘done’ for 2,500 years. So why change it?

Consider another example from Ancient Greece. Eratosthenes was a famous mathematician and geographer who proved that the earth was spherical over 1,500 years before Christopher Columbus set foot in the Americas. He paid one man a substantial amount of money to pace out the distance between Alexandria and Syene, and then to mark if there was a shadow hanging behind a tall tower when the sun was at its highest. There wasn’t.

He then checked if there was a shadow behind a tower of the same size in Alexandria at the same time of day, and there was. The only reasonable explanation for this phenomenon was that the earth was spherical. But people are often unreasonable, and as you would expect, Eratosthenes encountered adversaries who said that it couldn’t be possible for the earth to be spherical because there was nothing to keep its contents held down (this was later disproven by Newton).

We are often more interested in ‘what could be’ rather than ‘what is’, which can lead to ambitious claims such as a spherical earth and baseless opinions such as which method of measuring a circle is ‘best’. But how can we be expected to blindly follow a supposed theory, as it is considered by the majority of people who have a basic understanding of the topic, when there is no observable evidence? We can’t. The best way to look at it, certainly in terms of positivist theory, is to submit to the notion that there is merit in all theories, right or wrong. Newton may not have reached his conclusion that gravity existed if he wasn’t absolutely sure that the earth was spherical. It only became recognised as a fact in the Western world after Columbus’ voyages and Magellan’s circumnavigation of the globe. It is one thing to create a whole new measuring system and another to create it when the evidence surrounding it is suspect.

Eratosthenes not only gave mathematical proof that the earth was spherical, he even predicted the circumference of the earth which we have only recently discovered is remarkably close to the circumference we use today. The reason why there was scepticism at the time was the relatively crude method that he used to measure it. A work of genius as we can now see, but at the time, mere speculation.

The same sort of thing is happening today with the speculation over whether to use π, ϕ or 2π to deduce the circumference of a circle. It’s tempting to use the established theory until it is proven wrong, but what if that won’t be for another 1,500 years when we finally have a more precise way of measuring it? Students need to be aware of these irregularities because they couldn’t begin to figure out a way to fix them if they don’t know they exist. Whether it is on an exam paper or not, if a teacher thinks it’s worth teaching, they should teach it.

References:

www.youtube.com/watch?v=B8QWuSn_Wxw

https://en.wikipedia.org/wiki/Eratosthenes

http://www.goldennumber.net/pi-phi-fibonacci/

https://www.explainxkcd.com/wiki/index.php/1292:_Pi_vs._Tau

http://goodmath.scientopia.org/2010/12/08/really-is-wrong/

http://www.veteranstoday.com/2015/02/05/pi/

http://www.goldennumber.net/pi-phi-fibonacci/

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