"...I’ll place a straight ruler next to it and measure it and so I'll make a square out of a circle and place a marketplace in its centre where all the roads will be straight… It’ll be like a star which, though round, all its rays go off in all directions in straight lines."
~~Aristophanes, The Birds
We know π as the ratio of how many times the diameter of a circle fits in its circumference - somewhere in the region of 3.14159....ad infinitum. Because it is an infinite number, and therefore unknowable in its entirety, it means that in any situation where we use π, the answer we get is also incomplete. Everytime we use π, such as to determine the area of a circle, we only ever achieve an approximation - regardless of how accurate that approximation might be.
We imagine that in-order to increase the accuracy of π, we have to add more and more numbers to the millions and millions of digits that we already have. This might increase the accuracy of π, but it still only amounts to a more accurate approximation. If we want to know the exact answer, what we really need, is to KNOW pi.
Some argue that in certain situations, the infinite figure we have for pi is wrong, and that pi is really 4. This would mean that the number of times the diameter of a circle fits into its circumference is not three and a bit - but 4. Typically, we cannot draw a circle which has a circumference four times its diameter. We could try, but we would fail - and that failure would look a lot like a square.
If we were to roll-out the circumference of a circle, and compare it to the perimeter of a square with the same diameter, it would ably demonstrate that the length of the circle's circumference is roughly three-quarters that of the square. This being the case, how is it possible to produce a square from a circle which shares the same diameter?
In order to reach a length which is four times that of the diameter, it is necessary that the length of the circle's circumference is increased; that is, it is extended in some way. In effect, the line of the circumference has been stretched. Pi has mysteriously increased from the infinite figure of 3.14159... to the more finite figure of 4. What is more, if 4 can be produced by stretching π, it would further suggest that π is in some way, a compressed form of 4.
The number 4 is nowhere near as slippery as π to work with - it is neither transcendental, nor irrational. It is not weighed down by an infinite amount of digits after the decimal point. 4 is a nice whole number. In using 4, we no longer get approximate answers, we get THE answer. But how has the number 4 come into existence?
The new number of 4 for pi is not quite as bizarre, nor as impossible, as it at first might appear. The theory goes that a square can be manipulated in such a way that it effectively takes on the semblance of a circle. This circular shape is known as an infinite-sided concave polygon. Concave, as in the sides of the polygon are made to quite literally "cave-in."
"To remember what concave means it’s best to split the word up like this – “con” + “cave”. The important part is the “cave” part – the word concave is used to describe shapes that have something looking like a cave in them. When you talk about concave polygons, the cave is on the outside of the polygon. Another way of spotting concave polygons is to look out for any interior angles that are larger than 180°. Remember that angles larger than 180° are called reflex angles."
~~Image: The reflex angle of a concave polygon has an exterior angle which looks a bit like a "cave".
If a polygon has a reflex angle, then it is said to be a concave polygon. A reflex angle is greater than 180 degrees and less than 360 degrees. An infinite-sided concave polygon will have an infinite number of sides which "cave-in", producing an infinite number of reflex angles.
A simple concave polygon gives the impression of a shape which is awkward in nature. Increasing the number of "caves" (exterior angles) seems to make the polygon appear only more complex, and irregular. However, just as increasing the number of sides of a regular polygon will see it assume more and more of a circular shape - so too does a concave polygon whose reflex angles are increasing in number, and in uniform fashion. In other words, if we keep increasing the number of sides, ensuring that each side is the same length, and that they occur at regular intervals on the perimeter, then the concave polygon will begin to adopt a shape which is more circular. If you were to try and picture how this might look so far, then it is perhaps easier to imagine the perimeter as the blade of a circular saw.
~~Image: Circular saw blade on antique portable sawmill.
In order to cram as many sides into the perimeter as possible, it is imperative that every side has the same length. The sides act as the legs of the reflex angle, and the space formed between them, the "cave" as it were, must be infinitesimally small so that an infinite number of "caves" might occupy the circumference. These exterior angles are fundamental to the shape, as they serve to prevent the legs of the reflex angle from ever touching each other. In the case of an infinite-sided concave polygon the distance between each side is as infinitesimally small as can be imagined. If these "caves" or spaces did not exist, it would be impossible for a perimeter to shake its length, and dissolve into a circumference.
An infinite-sided concave polygon will have an infinite number of reflex angles. The sum of the reflex angles must surely provide some dazzling figure for the number of degrees inside the shape. Funny enough mind, regardless of the infinite figure in the interior, it can be argued that the sum of a concave polygon's exterior angles, infinite-sided or not, shall always remain at 360 degrees. This shares a remarkable symmetry with convex polygons, whose exterior angles undergo the exact same phenomena - they too always add up to 360 degrees.
If we observe each indivual reflex angle in the infinite-sided concave polygon, we find that it is imperative that the number of degrees inside the angle only ever approach 360 degrees. The reflex angle must maintain an angle which is as close to 360 degrees as permissable, so that the exterior angle remains as acute as possible. It is important that the reflex angle is never allowed to fully complete 360 degrees, otherwise the exterior angle would come to equal zero degrees, and the "cave" would simply "pop" out of existence.
If we were to return to our previous analogy, then I suppose we'd be looking at a circular saw blade with an infinite number of tiny teeth - each tooth seperated from the other by an infinitesimal gap. If we ourselves were infinitesimally small, we would be able to see that the circumference of the blade followed a distinct zig-zag pattern. If we were to increase the distance between us and the saw, then it would challenge our previous conception, as the blade now appears perfectly smooth. It's interesting to think that our judgement is made so bias by perspective.
The entire process described above, is brought to dramatic effect in the illustration below. Known to some as "π = 4! Problem Archimedes?", it's already gained some notoriety on various maths and physics forums, where some others have come to call it - undeservedly perhaps - "Troll Pi":
~~Image: "π = 4! Problem Archimedes?"
What the illustration shows, and rather well, is how the perimeter of a square might be reduced to occupy the circumference of an inscribed circle. Ordinarily, this would be deemed impossible because a square's perimeter is four times its diameter, while a circle's circumference is just over three times its diameter. To get a square to squeeze into a circle, the length of the square's perimeter will need to be shortened by as much as a quarter.
It begins with the square's corners being "removed" or inverted, so that it adopts the shape of a concave polygon. Accordingly, more corners are inverted, meaning that more sides are added, and the perimeter pretty much resembles a ziggurat made up of "Lego" blocks. More and more of the perimeter is encroached upon as the number of sides increase. As their numbers swell, we see the size of the blocks decrease - so much so - that by the time we arrive at numbers which are infinite, the perimeter is now a curve which possess a row of barely imperceptible jagged little teeth. Under its new guise, the perimeter seems able to fit perfectly inside the circumference of the circle.
One might imagine the square as having been panel-beaten, then concertinaed into the shape of a circle. The circle has been created by nothing short of square-mangling. If this is true, then we might also assume that the circle is just as easily taken and bent back into the shape of the square. Flipping between these shapes - from circle to square, and from square to circle - we can see that nowhere in the entire process is anything either added, or taken away. At no-time is the line cut or dismantled. The only thing which is observed is MOTION.
For a square to transform into a circle, its sides are allowed to crumble and fall into rubble around the circle's circumference. Those once imposing corners of the square have somehow been lost - buried - in the defining line of the circumference. The square can be rebuilt however, and it begins by sweeping the rubble back into piles - piles which grow bigger and bigger - until they are at last gathered into four mountainous heaps.
To summarise then, "π = 4! Problem Archimedes?" is a revelation in how the circumference can act as a perimeter, and how the perimeter can change into a circumference. This simple illustration explains very well how this transformation takes place, clearly showing how it is possible for the length to remain unchanged. The length of the square's perimeter is ALWAYS the same - and remains so - even when it is asked to assume the role of a circle's circumference.
All this is very good in theory, but if we were to examine a circle on a page - how do we go about peeling the circumference of a circle from the paper, so that we might use it to make a square of the same diameter? Surely, everytime we unravelled the circumference and measured it, the answer will always be π, not 4? To reach 4, something needs to be added to π. From where has the required extra bit materialised?
What is required is a reliable mathematical method to show how the extra length is gained legitimately. In a landmark paper, "The Extinction of π", Miles Mathis proposes that he has discovered one such method. Mathis appears to have resolved the problem of pi=4 by including time in his geometric analysis. It can be seen that his geometry is no longer describing some static abstraction, but something much more physical. Something real.