Archimedes’ Pi

Can we sit and meditate and think up all the maths there is? Potentially yes and some mathematicians have done a very good job. Mandalas are geometric patterns that are often used for meditation.  These are geometric patterns often based on circles and squares. The Indian mathematician Ramanujan (played by Dev Patel in the movie the Man Who Knew Infinity) is someone who comes to mind, that did a lot of maths meditation. Another is the ancient Greek, Archimedes.

We know that pi is the ratio of circumference to diameter of any circle. We could get a piece of string and measure it and come to something like 3.14 if we are very careful.  Or we could sit and meditate and think it out to over 99% accuracy or better.  That’s what Archimedes decided to do using only Pythagoras theorem where the square on the hypotenuse (long side of a right angle triangle) is equal to the sum of the squares on the other 2 sides, and a little thought experiment regarding limits.

First he calculated the perimeter of a square outside of a circle where the square is just touching (the sides are tangents to the circle). Here if R=1, then D =2 and the square has a perimeter of 4 x 2 = 8 and we have an upper estimate of pi = C/D = 8/2 =4. Similarly we can look at an inside square where Pythagoras tells us that each edge is  (root 2 ).  This is equal to about 1.414.. giving a perimeter of about 2.828..

So pi has to be greater than 2.82 and less than 4 or in math terms 2.8 ˂ π ˂ 4. We could go further and say that pi is approximately half way between and estimate it to be (2.8+4)/2 = 3.4. Remember π is the perimeter (circumference) of circle with diameter = 1. Not a bad start but can we do better? Sure, just use a polygon with more sides.  If we went to inside hexagons each side would be equal to the radius 1 giving us a perimeter of 6 and an estimate of π =3. Much better already.

Archimedes thought “ what if I divided each side of the inner square in half”? He could then use Pythagoras again to determine the length of each side of the eight sided figure to get an estimate of Pi of 3.0614. He repeated this down to a 96 sided figure to get an estimate of Pi to over 99% accuracy  – and this was over 2000 years ago!!

Essentially, what he figured out was that we can approximate the circumference of a circle with ever smaller straight lines and in the limit we get to pi. There are many ways we can calculate pi, but this is a goodie.

Yes, I realise it’s not quite from scratch as we also need to understand similar triangles and prove that Pythagoras was actually correct. He was not actually the first to propose the idea but he supposedly was the first to prove it.  His was a very simple graphical proof and since then many people have come up with very different ways of proving Pythagoras Theorem, including Albert Einstein! Pythagoras proof from Wikipedia.

(Copy the arbitrary right triangle 4 times and rearrange to show the equivalence.)

So we can meditate to determine π, e and i. No need for any measurements.  Can we meditate to completely describe our universe?  Not as far as we know.  Maths allows anything to be possible (although Kurt Godel showed there are things we also can’t know!).  Physics uses and tests the maths by measuring real life situations and has come up with a number of measurements which are needed to describe our universe.  One which we have seen already is the speed of light C = 299 792 458 m / s. Another I have mentioned is Planck’s constant and yet another is the Gravitational constant (although recent thoughts are that it may not be constant and this may be the answer to dark matter and energy!).

So, to understand the universe we need to understand the maths and we also need to prove the reality (or correlation to real life) of any maths concepts, by experimentation and measurement. If we don’t we are being hoodwinked into believing in nonsense. Pseudoscience is when people take snippets of real data and create their own untested and unproved narrative. It is rife in our current world ruled by the forces of marketing.

Calculus

The ideas of “infinitesimal calculus” have been around for at least a few thousand years. Archimedes understood the principle in taking smaller and smaller divisions until “in the limit” we reach our conclusion (pi).  We also saw this previously with identifying the transcendental number e.

This infinitesimal calculus is now known as integral calculus and is used to calculate the area under (or within) a curve. The ideas of differential calculus were known from at least the 1300s if not earlier.  As previously discussed this is all about derivatives which are about rates of change (eg velocity and acceleration).  In the 1600s a number of mathematicians including Isaac Newton, Gotfried Leibnitz and others determined that these two processes are directly related, one the opposite of the other.

These guys are credited with discovering modern calculus, now used everywhere in mathematical analysis. They put the modern theory together and came up with the modern notation. Greek, Indian and Chinese mathematicians ( and probably others) understood a lot of the essence of calculus but had not put it all together.

To understand the basics we need some concepts and notation. Consider a car travelling at 100 km / hr and travelling for 2 hours.  It would cover a distance of 200 km which can be expressed as a formula

x (distance) = v (velocity) * t (time). The star just means multiply.

To get a more accurate solution, we could also sum the distance travelled over smaller time intervals which we will call delta t or ∆ t.

Now we have ∆x = v * ∆t and in total x = ∑ (v * ∆t) where ∑ is called Sigma and means the sum of.

If we look at infinitesimally small ∆ we use the symbol d instead so we have dx = v dt and over the whole distance x = ∫ v dt ( read as x equals the integral of velocity with delta time or dt).

We also have v = dx/dt where velocity is the derivative of distance over time.

Each small unit of v is considered constant, while time and distance change.

Even though this just looks like a complicated way of measuring velocity, like in our calculating pi example, it actually gives us a very successful tool for accurately determining areas, volumes and all calculations regarding the laws of motion. As always don’t worry about trying to understand the detail just go with the flow of the larger picture.  We need the calculus to accurately determine distance, velocity etc because we are continually adjusting our acceleration.

Derivatives

There are some simple derivatives to remember. Don’t worry about why just yet. First, a simpler way of writing a derivative is using functional notation where y=x becomes f(x) =x (read as a function of x = x).  The derivative is denoted f’(x)

The derivative of any power of x can be found using the formula Or more simply f’ (x³) = 3x² and f’’(x³) = 6x ( this is the second derivative).

Other useful derivatives include f’sinx = cos x and f’ cos x = -sin x

We will get on to what sin and cos (trigonometry) are a bit later.

For now, we sort of know what calculus is, we sort of know about integrals and derivatives and we can find the mathematical derivatives of some important functions. If you look at maths on the internet you will now start to get an idea of what each equation is on about. There are lots of fields of maths that we have not yet dealt with and the different notations (short hand) can be difficult.

Derivatives can also be viewed as the tangents to curves and so relate back to limits. In our unlimited universe some things are still limited.