Circling The Square

While looking at some facts about pi, there was one in particular that really engaged me. It is not a fact about pi as such, but more a statement about the nature of pi...

"Most people would say that a circle has no corners, but it is more accurate to say that it has an infinite number of corners."

I like this fact because it forces us to examine the nature of a circle. If a circle has corners, then surely, it must be related to the square in a fundamental way. The circle is proof that mere appearances can be deceptive. In a mathematical sense, a square and circle are not two entirely different shapes as such. Those severe bends that we see forming the four corners of a square, can surprisingly, be found to also exist in those soft curves which form a circle. The best way to get from a square to a circle is by increasing its number of corners.

~~Image: Red Square Painting (2009 Digital Remix) by Nigel Tomm

A square has its four sides and its four corners, and if we add another side to it, to create a five-sided polygon, then it also generates another corner. If everytime we keep adding more sides, and everytime we make all the sides the same length - then we can go on to form a fantastic array of regular polygons.

If we keep on increasing the polygon's number of sides, and keep on increasing them, then theoretically, the sides of the polygon will eventually reach numbers which are infinite. It shall produce a polygon with an infinite number of sides - and for all intents and purposes - a shape that looks convincingly like a circle. It is by creating polygons that the early pioneers of geometry, such as Archimedes, were able to gain more and more accurate approximations of pi. The following extract below comes from this site, which also offers some excellent working demos of how pi can be approximated - one by unravelling circles, and another by inscribing polygons:

"π is an irrational number, which means that we can never write the value of it completely accurately. So how do we calculate it? After all, it is difficult to measure round the edge of a circle. You could get an approximation by winding a piece of string round a tin, then measuring the string and across the tin, but this will not be very accurate. Another way is to fit a polygon (like a square or a hexagon) to the circle, either inside or outside. We can calculate the edge of a polygon. As we increase the number of sides in the polygon, it fits the circle better and better, so its edge becomes closer and closer to the circumference of the circle. What is more, the outer polygon will have a longer edge than the circle, and the inner one will be less. So we can get two approximations for for each polygon, one too big and one too small."

As you increase the number of sides of a polygon, you increase the sum of the interior angles of the polygon. Each time we increase the number of sides by one, the sum of the angles increase by 180 degrees. For example, a square (quadrilateral) has the interior sum of 360 degrees, while a five-sided polygon (pentagon) has a sum of 540 degrees. Going further, we see that the sum of the interior angles of a ten-sided polygon (decagon) are 1440 degrees. The sum of the interior angles of a polygon are calculated by inscribing triangles (triangulating). If we know that the sum of the interior angles of a triangle are ALWAYS 180 degrees, and we can count the number of triangles being used to form the shape of the polygon - then we have the perfect formula for calculating the sum of the polygon's interior angles (n being the polygon's number of sides):

(n-2) × 180° = sum of interior angles

Basically, everytime we add a side to a polygon, we generate a new triangle inside the polygon, and increase the sum of the interior angles by 180 degrees. A square, for example, can be made up by two triangles (hence 2 × 180° = 360°), while a pentagon can be made up by three triangles (3 × 180° = 540°).

~~Image: There are two triangles in a square.

~~Image: A pentagon has five sides, and can be made from three triangles.

If a circle can be described as a polygon with an infinite number of small sides, then we must assume that the sum of the interior angles of such a circle will too approach figures somewhere in the infinite. If we were to try and triangulate such a polygon to try and reach the sum of the interior angles, we would have to deduct two from the number of sides to give us the number of triangles. This means we would be left trying to tackle the rather troublesome sum of infinity minus 2 (n-2) to achieve the answer.

Trying to add or subtract to infinity is always a little awkward. After all, infinity is considered to be a concept rather than a number - you can't just go around ripping bits off it, or for that matter, slapping things on it. In order to deduct 2 from infinity to get a number, it means that we would have to ask infinity to be a little less infinite, and be a bit more finite, which is probably asking the impossible. Or is it?

Assuming that the sum of the interior angles will reach amounts which end somewhere in the infinite, it remains that the sum of the exterior angles of a such a polygon, a polygon with an infinite number of sides, if measured, will still be found to equal 360 degrees. This is because the sum of the exterior angles of any convex polygon will ALWAYS add upto 360 degrees. Essentially, all the exterior angles amount to one full revolution (360°). In other words, adding all the exterior angles together is the mathematical equivalent of taking the shape and rotating it one complete turn.

~~Image: In this diagram the exterior angles have been given different colours. You can see how they can be put together to make a full circle.

If we add up the interior angle and the exterior angle of a regular polygon, we get a straight line - 180 degrees. The interior and exterior angles are distinctly related. We can increase the polygon's number of sides to figures which are infinite, and with it, we will also see an increase in the sum of the interior angles. In theory, the number of degrees should become infinite - infinitely big. However, each interior angle cannot be seen to be equal to, or exceed, the boundary of 180 degrees, otherwise we will encroach upon the space of the exterior angle.

~~Image: Internal angle + external angle = 180°

Increasing the number of sides will see the sum of the interior angles grow,and grow, but this growth is wholly reliant on each exterior angle, the one at each vertex, becoming smaller, and smaller - infinitely smaller. In other words, the growth of the sum of the interior angles is severely restricted. The infinite sum of the interior angles are by no means boundless.

" The sum of the exterior angles of a polygon are 360 degrees regardless of the number of sides. That means that the measure of each exterior angle must get smaller as the number of sides increases. There is no "least possible measure" because even though the limiting value is 0 you can never achieve a 0 degree exterior angle and still have a polygon. You can get as close to zero as you like, but as close as you get, someone else can always come along and get closer. Another way to look at it is that a zero degree exterior angle measure implies that there are an infinite number of sides. And an infinite number of sides implies a circle, not a polygon."

Personally, I would argue that a circle, or infinite sided polygon, does not possess a zero degree exterior angle. That's because an infinite amount of nothing will still give you nothing. Nevermind how much nothing you get, you'll still be left holding nothing. The sum of the exterior angles, regardless of the fact that they are infinite in number, shall always add upto 360 degrees. Therefore each exterior angle must be seen to amount to something, even if it is something infinitesimally tiny, in order to reach the sum of 360 degrees. Coincidentally, this exact same restriction we find outside the circumference of any polygon, or circle, is also at work in the shape's centre.

One of the defining properties of a circle, and indeed, any regular polygon, is that its entire central angle ALWAYS measures 360 degrees. If we were to add a central vertex, or central point to a pentagon for example, and inscribe triangles in the same way that we might slice up a pizza, then our pentagon would produce 5 triangles - all sides would have the same length, and all the interior angles would be the exact same size. The central angle of each triangle will also be the same, and the sum of these shall ALWAYS add upto 360 degrees.

~~Image: The central angle of a regular pentagon (5 × 72° = 360°)

Using this same method, we can imagine inscribing triangles to an infinitely sided polygon, to create an infinite number of infinitesimal triangles. But if we add the sum of these infinitesimally tiny central angles together, they produce the sum of 360 degrees. It doesn't matter how many triangles we have, infinite number or not, they shall always add up to 360 degrees. An infinite sided polygon does not have a central angle whose sum reaches an infinite number of degrees - it produces only an infinite number of ways to percieve the finite sum of 360 degrees.

I have always imagined infinity as an entity which fulfils the very definition of freedom. But where is all the freedom that I was hoping to embrace? No, infinity offers only the illusion of freedom. Infinity can never escape the confines of the finite. The term infinite is not actually describing the phenomena of ever-expanding space - it only pretends to.

We imagine that in order to behold the infinite we have to travel to some far-flung, incomprehensible horizon - but the reality is, for us to comprehend the infinite, we don't so much as have to leave the spot. Infinity is always describing the exact same space - a space chopped up into an infinite number of ways, an infinite number of ways in-which to percieve it - but it is the exact same space nonetheless.

If the state of infinity were truly free, then surely, there should be no restrictions to its growth whatsoever. Here however, we can see that infinity is shackled to enormous constraints. We can try to develop a sense of the infinitely big - building a polygon with infinite sides - but we find that that growth is constricted by a number of finite limits.

Infinite growth is restricted by constraints imposed both inside and outside the circle. The sum of the central angle, and the sum of the exterior angle can never exceed 360 degrees. Also, the linear pair of the exterior and interior angles can never exceed 180 degrees. Infinity is dependent upon how big we make each interior angle, and at the same time, how small we make each exterior angle. Each interior angle can never extend to, or beyond 180 degrees, and at the same time, the exterior angle can never be allowed to fall to zero. If any of these restrictions are breached, well, then you no longer have a perfect circle.

A circle may well be a polygon with infinite sides, but at its heart, it is still very much a square - a square bent infinitely out of shape - but a square nonetheless. A square of finite proportions.

The sum of the interior angles, the angles which exist inside the circumference of the circle as it were, can reach figures which are infinite - but this stands only as an expression of how limited our understanding is. The sum of the interior angles could never truly, unrelentlessly expand into infinite space - there is a limit in place. A limit so vast that it is unknowable - but a limit nonetheless. We may not be able to know the number of that limit, but we can see it. We see it all the time. That's because the limit itself is a construct of a remarkably simple shape - the circle.

~~Image: The human eye - one of the most outstanding examples of a circle that we see everyday.

The state of infinity perhaps, might best be described as a place that exists somewhere between a square and a circle. If this were true, what exactly does it mean for the supposedly infinite ratio of pi?