## Why Pi?

If you measure the ratio of the circumference to the diameter of any circular object you get the number Pi (π). Everyone who has done any maths or physics at all knows this. Some people who have gone on to do more maths knows that Pi is an irrational number, which is, looked at one way, merely the category into which Pi falls.

There are other irrational numbers, for example the square root of the number 2, which are almost as well known as Pi, and others, such as the number e or Euler’s number, which are less well known.

Anyone who has travelled further along the mathematical road will be aware that there is more to Pi than mere circles and that there are many fascinating things about this number to keep amateur and professional mathematicians interested for a long time.

Pi has been known for millennia, and this has given rise to many rules of thumb and approximation for the use of the number in all sorts of calculations. For instance, I once read that the ratio of the height to base length of the pyramids is pretty much a ratio of Pi. The reason why this is so leads to many theories and a great deal of discussion, some of which are thoughtful and measured and others very much more dubious.

Ancient and not so ancient civilisations have produced mathematicians who have directly or indirectly interacted with the number Pi. One example of this is the attempts over the centuries to “square the circle“. Briefly squaring the circle means creating a square with the same area as the circle by using the usual geometric construction methods and tools – compass and straight edge.

This has been proved to be impossible, as the above reference mentions. The attempts to “trisect the angle” and “double the cube” also failed and for very similar reasons. It has been proved that all three constructions are impossible.

Well, actually they are not possible in a finite number of steps, but it is “possible” in a sense for these objectives to be achieved in an infinite number of steps. This is a pointer to irrational numbers being involved. Operations which involve rational numbers finish in a finite time or a finite number of steps. (OK, I’m not entirely sure about this one – any corrections will be welcomed).

OK, so that tells us something about Pi and irrational numbers, but my title says “Why Pi?”, and my question is not about the character of Pi as an irrational number, but as the basic number of circular geometry. If you google the phrase “Why Pi?”, you will get about a quarter of a million hits.

Most of these (I’ve only looked at a few!) seem to be discussions of the mathematics of Pi, not the philosophy of Pi, which I think that the question implies. So I searched for articles on the Philosophy of Pi.

Hmm, not much there on the actual philosophy of Pi, but heaps on the philosophy of the film “Life of Pi“. What I’m interested in is not the fact that Pi is irrational or that somewhere in its length is encoded my birthday and the US Declaration of Independence (not to mention copies of the US Declaration of Independence with various spelling and grammatical mistakes).

What I’m interested in is why this particular irrational number is the ratio between the circumference and the diameter. Why 3.1415….? Why not 3.1416….?

Part the answer may lie in a relation called “Euler’s Identity“.

$e^{i \pi} + 1 = 0$

This relates two irrational numbers, ‘e’ and ‘π’ in an elegantly simple equation. As in the XKCD link, any mathematician who comes across this equation can’t help but be gob-smacked by it.

The mathematical symbols and operation in this equation make it the most concise expression of mathematics that we know of. It is considered an example of mathematical beauty.

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The interesting thing about Pi is that it was an experimental value in the first place. Ancient geometers were not interested much in theory, but they measured round things. They lived purely in the physical world and their maths was utilitarian. They were measuring the world.

However they discovered something that has deep mathematical significance, or to put it another way is intimately involved in some beautiful deep mathematics.

This argues for a deep and fundamental relationship between mathematics and physics. Mathematics describes physics and the physical universe has a certain shape, for want of a better word. If Pi had a different value, that would imply that the universe had a different shape.

In our universe one could consider that Euler’s Relation describes the shape of the universe at least in part. Possibly a major part of the shape of the universe is encoded in it. It doesn’t seem however to encode the quantum universe at least directly.

I haven’t been trained in Quantum Physics so I can only go on the little that I know about the subject and I don’t know if there is any similar relationship that determines the “shape” of Quantum Physics as Euler’s Relation does for at least some aspects of Newtonian physics.

Maybe the closest relationship that I can think of is the Heisenberg Uncertainty Principle. Roughly speaking, (sorry physicists!) it states that for certain pairs of physical variables there is a physical limit to the accuracy with which they can be known. More specifically the product of the standard deviations of the two variables is greater than Plank’s constant divided by two.

In other words, if we accurately know the position of something, we only have a vague notion of its momentum. If we accurately know its velocity we only have a vague idea of its position. This “vagueness” is quantified by the Uncertainty Principle. It shows exactly how fuzzy Quantum Physics.

The mathematical discipline of statistics underlay the Uncertainty Principle. In a sense the Principle defines Quantum Physics as a statistically based discipline and the “shape” of statistics determines or describes the science. At least, that is my guess and suggestion.

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To return to my original question, “why Pi?”. For that matter, “why statistics?”. My answer is a guess and a suggestion as above. The answer is that it is because that is the shape of the universe. The Universe has statistical elements and shape elements and possibly other elements and the maths describe the shapes and the shapes determine the maths.

This is rather circular I know, but one can conceive of Universes where the maths is different and so is the physics and of course the physics matches the maths and vice versa. We can only guess what a universe would be like where Pi is a different irrational number (or even, bizarrely a rational number) and where the fuzziness of the universe at small scales is less or more or physically related values are related in more complicated ways.

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The reason for “Why Pi” then comes down the anthropological answer, “Because we measure it that way”. Our Universe just happens to have that shape. If it had another shape we would either measure it differently, or we wouldn’t exist.

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## The number of the universe.

Anything that can be measured can be encoded in a single number. Take for instance the trajectory of a stone thrown into the air. Its position in relation to the point of launch and the time it has taken to reach that point can be encoded into a set of numbers, three for the spacial dimensions and one for the time dimension. This can be done for all the points that it passes through. These individual numbers can then be encoded into a single number that uniquely identifies the trajectory of the stone.

Or, a physicist can describe the motion of the thrown stone by using generic equations and plug in the starting position and starting velocity of the stone, which can then be encoded, probably in a simpler fashion than the above point by point encoding.

If we can imagine a set of equations that describe all the possible physical processes (the “laws of nature”?) and we can imagine that we can measure the positions of all the particles (including photons,’dark matter’ and any more esoteric things that might be out these), then we could encode all this in a huge number which we could call the ‘number of the universe’. Such a number would be literally astronomical and I do mean ‘literally’ here.

The most concise expression of the state of the universe over all time is probably the universe itself and the laws that govern it. Each individual particle has its own attribute, like charge, mass, position and so on as well as things like spin, charm and color. Some of these change over time and some are fundamental to the particle itself – if they change so does the nature of the particle. The rest of the universe consists of other particles which have a lesser or greater effect on the particle, all of which sum together to describe the forces which affect the particle.

There are a couple of things which might derail the concept of the number of the universe. Firstly there is Heisenberg’s Uncertainty Principle and secondly there is the apparent probabilistic nature of some physical processes.

What follows is my take on these two issues. It may make a physicist laugh, or maybe grimace, but, hey, I’m trying to make sense of the universe to the best on my abilities.

People may have heard of the Uncertainty Principle, which states that there are pairs of physical properties which cannot both be accurately known at the same time. You may be able to know the position of a particle accurately, but you would not then be able to tell its momentum, for example.

It is usually explained in terms of how one measures the position of something, which boils down to hitting it with something else, such as a photon or other particle. The trouble here is that if you hit the particle with something else, you change its momentum. This is, at best, only a metaphor, as the uncertainty principle is more fundamental to quantum physics than this.

Wikipedia talks about waveforms and Fourier analysis and an aspect of waves that I’ve noticed myself over the years. If you send a sound wave to a frequency analyser you will see a number of peaks at various frequencies but you cannot tell how the amplitude of the wave changes with time. However, if you display the signal on an oscilloscope you can get a picture of the shape of the wave, that is the amplitude at any point in time, but not the frequencies of the wave and its side bands. Err. I know what I mean, but I don’t know if I can communicate what I mean!

The picture above shows a spectrum analysis of a waveform. I don’t have the oscilloscope version of the above, but below is a time-based view of a waveform.

In any case, the uncertainty doesn’t imply any indeterminacy. A particle doesn’t know its position and momentum, and these values are the result of its properties and the state of the rest of the universe and the history of both. This means that the uncertainty principle doesn’t introduce any possible indeterminacy into the number of the universe.

On the second point, some physical processes are probabilistic, such as the decay of a radioactive atom. I don’t believe that this has any effect on the number of the universe. The number incorporates the probabilistic nature of the decay, including all the possibilities.

There is an interpretation of quantum physics called the “Many Worlds Interpretation“, where each possible outcome of a probabilistic process splits off into a separate world, resulting in an infinity of separate worlds. I don’t believe that this tree of probabilistic worlds is a useful view of the situation.

No, I think that there is a probabilistic dimension, just like time or space. All the things that can happen, ‘happen’ in some sense. The probability of you throwing 100 tails in a row with a fair coin is very small, but it is possible. As I see it the main objection to this view is the fact that we only see one view of the universe and we don’t appear to experience any other possible views of the universe, but this is exactly the same with the dimensions of space and time. We only experience one view of space at a time as we can’t be in two places at the same time. While we could be in the same place at two times they are two distinct views of the universe.

In any case the number of the universe encompasses all probabilities so if you still adhere to the single probability model of the universe, our universe and all possible universes are encoded by it. The question then becomes how you can extract the smaller number that encoded the single universe that we experience. I believe that that is not a question that needs to be answered.

The question that does remain open is – why is that number the number of our universe? Why not some other number?