Pi=4!

"...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.



Much thanks:

http://en.wikipedia.org/wiki/Circle
http://betterexplained.com/articles/prehistoric-calculus-discovering-pi/
http://mathforum.org/~sanders/exploringandwritinggeometry/polygons.htm
http://paramanand.blogspot.com/2010/12/angle-sum-formula-for-polygons.html
http://www.jimloy.com/geometry/pi.htm
http://milesmathis.com/pi3.html
http://dyinglovegrape.wordpress.com/2010/11/17/homology-primer-2-triangulating-a-surface/
http://www.physicsinsights.org/pi_from_pythagoras-1.html
http://www.philosophynow.org/issue81/Mathematical_Knowledge_A_Dilemma
http://www.tutorvista.com/math/infinite-lines-of-symmetry
http://scienceblogs.com/goodmath/2008/10/infinity_is_not_a_number.php
http://www.mathopenref.com/polygoncentralangle.html
http://www.jimloy.com/geometry/pentagon.htm
http://forums.xkcd.com/viewtopic.php?f=3&t=67907
http://www.pi314.net/eng/aleatoire.php
http://jwilson.coe.uga.edu/emt668/EMAT6680.2000/Mylod/Math7200/Project/InscribedCircle.html
http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/integration/archimedes.html
http://www.themathpage.com/atrig/measure-angles.htm

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?

Contemplating Infinity

The word after "infinity" in my dictionary is "infirm," a definition of which is "weak of mind." This is how many of us who are not mathematically inclined feel upon contemplating infinity. (To see how mathematicians and similar thinkers regard infinity, see Working With Infinity: A Mathematical Perspective.)

We feel weak because our finite minds can only go so far with the concept, and because every time we think we're on the verge of securing even a shadowy understanding, we're tripped up by something. A friend of mine once told me that trying to hold her hyperactive toddler was like trying to hold a live salmon. Infinity is like that for us "infirm ones": slippery as a salmon, forever eluding our grasp.

Becoming numb

This is true no matter how you approach the concept. Many of us might consider numbers the most sure-footed way to come within sight of infinity, even if the mathematical notion of infinity is something we'll never even remotely comprehend.

We may think, for starters, that we're well on our way to getting a sense of infinity with the notion of no biggest number. There's always an ever larger number, right? Well, no and yes. Mathematicians tell us that any infinite set—anything with an infinite number of things in it—is defined as something that we can add to without increasing its size. The same holds true for subtraction, multiplication, or division. Infinity minus 25 is still infinity; infinity times infinity is—you got it—infinity. And yet, there is always an even larger number: infinity plus 1 is not larger than infinity, but 2infinity is.

Try another tack: huge numbers. When we play with mind-boggling figures we non-math types may think we're playing in infinity's neighborhood, if not in the same playground. When we're told that the decimals in certain significant numbers, like pi and the square root of two, go on forever, we can somehow accept that, especially when we learn that computers have calculated the value of pi, for one, to over a trillion places, with no final value for pi in sight. (For more on pi, see Approximating Pi.) When we're told that there are 43,252,003,274,489,856,000 possible ways to arrange the squares on the Rubik Cube's six sides, we may feel intuitively (if not rationally) that we must be on our way to the base of that loftiest of all peaks, Mt. Infinity.

One reason we may feel this way is that such numbers are as intellectually unapproachable to the mathematically challenged as infinity itself. Take a Googol. A Googol is 10100, or 1 followed by 100 zeroes, and is the largest named number in the West. The Buddhists have an even more robust number, 10140, which they know as asankhyeya. Just for fun, I'll name a larger number yet, 101000. I'll call it the "Olivian," after my daughter. Now, doesn't an Olivian get me a little closer to infinity than the Googolians or even the Buddhists can get? Nope. Infinity is just as far from an Olivian as it is from a Googol—or, for that matter, from 1.

For many of us uncomfortable with infinity, the word number can be defined as “that which makes numb.”

Perhaps we infirm ones would be wise to take a leaf from the lingual book of Madagascar. The word there for a million is tapitrisa, which means "the finishing of counting." For some tribal groups in other parts of the world, counting stops at three, in fact; anything above that is "many." In some ways this makes sense. How many of us can keep more than a few things in our minds at once? I remember playing a game with myself as a child in which I would think "I'm thinking that I'm thinking that I'm thinking that I'm thinking...." After the third or fourth "I'm thinking," I could no longer retain in my head all the degrees it implies. Such infirmity holds for simple counting as well, as Lewis Carroll reveals so tellingly in Through the Looking Glass:

"Can you do Addition?" the White Queen asks. "What's one and one and one and one and one and one and one and one and one and one?"

"I don't know," said Alice. "I lost count."

"She can't do Addition," the Red Queen interrupted. "Can you do Subtraction?"

For many of us uncomfortable with infinity, the word number can be defined as "that which makes numb," as Rudy Rucker wryly notes in his book Infinity and the Mind (Birkhäuser, 1982). This is especially true when a number is so outlandishly enormous that it smacks, however remotely, of the infinite. Galileo himself felt this way. "Infinities and indivisibles transcend our finite understanding, the former on account of their magnitude, the latter because of their smallness," he wrote in his Dialogues of Two New Sciences of 1638. "Imagine what they are when combined." Rather not, thanks—makes me numb.

Incredible shrinking

Infinities do come in two sizes, of course—not only the infinitely large but also the infinitely small. As Jonathan Swift wrote, "So, naturalists observe, a flea/Has smaller fleas that on him prey/And these have smaller still to bite `em/And so proceed ad infinitum." We may not be able to conceive of Swift's infinitesimal fleas, because reason insists they don't exist, but we can imagine ever smaller numbers without much trouble. It's no hardship, for example, to grasp the notion of an infinity of numbers stretching between, say, the numerals 2 and 3. Take half of the 1 that separates them, we might tell ourselves, then half of that half, then half of that half, and so proceed ad infinitum.

Of course, just when we think we have infinity in the palm of our hands, we watch it evaporate in the harsh light of another of those confounding paradoxes: the numerals 2 and 3 are separated by both a finite number (1) and an infinity of numbers. This conundrum spawned one of the great paradoxes of history, known as Zeno's paradox. Zeno was a Greek philosopher of the fourth century B.C. who "proved" that motion was impossible. For a runner to move from one point to another, Zeno asserted, he must first cover half the distance, then half the remaining distance, then half the remaining distance again, and so on and so on. Since this would require an infinite number of strides, he could never reach his destination, even if it lay just a few strides away.

It wasn't for 2,000 years that Zeno's paradox finally got "solved," for all intents and purposes, by the calculus. Its inventors, Isaac Newton and Gottfried Leibniz, showed us how an infinite sum can add up to a finite amount, that it can converge to a limit. Thus, even though we can't count all the numbers between 2 and 3, we know they converge to 1.

No limits

As Zeno's paradox hints, considering infinity from the perspective of space has much correspondence with that of numbers. We can imagine, for instance, that space, like numbers, is infinitely divisible. We believe Hamlet when he says "I could be bounded in a nutshell/And count myself a king of infinite space." The shortest length physicists speak of is the Planck length, 10-33 centimeters. But might not there be an even shorter length, say, 10-333 centimeters, or 10-an infinite number of 3's centimeters?

Many of us are as queasy around eternity as we are around infinity.

As with numbers, we can also envision space as being infinitely large. After all, if the universe has a boundary, what's on the other side? We might flatter ourselves that we're somehow getting closer to infinity when we consider extremely large distances. On June 12, 1983, while traveling at over 30,000 mph, the Pioneer 10 spacecraft became the first human-made object to exit our solar system. Some 300,000 years from now, unless something interrupts its voyage, the craft is expected to pass near the star Ross 248, a red dwarf in the constellation Taurus. Ross 248 is about 10.1 light-years from Earth, or about 59,278,920,000,000 miles away. Pioneer 10 will still be in the early stages of its journey, though. When our sun bloats into a red giant about five billion years from now and incinerates our planet, our robotic ambassador will still be heading away, knocking off more than 250 million miles a year.

Are we making headway towards an infinite distance with such knowledge? Hardly. An infinite distance, as you've guessed, would be as far from where Pioneer 10 will be in five billion years as it is from the Earth now. If the universe is infinitely large, even the remotest stars we can detect, which are so far away that their light left them some 12 billion years ago, are as far from infinity as we are. (Things get tricky here: as one mathematician pointed out to me, infinity is an abstract concept, appearing only in our mental images of the universe. It is not actually in the universe.)

Forever and a day

Time is another way to contemplate infinity, though many of us are as queasy around eternity as we are around infinity. ("That's the trouble with eternity, there's no telling when it will end," Tom Stoppard writes in Rosencranz and Guildenstern Are Dead.) Yet isn't infinite time somehow easier to swallow than finite time? After all, what can stop time?

Many of us do indeed live our lives thinking that eternity is a given. And again, we may fool ourselves into thinking that we're on the way to eternity when we think of 12 billion years, or of any other frighteningly mind-bending length of time. One of the gamest attempts to define eternity appears in Hendrik Willem Van Loon's 1921 children's classic The Story of Mankind:

High up in the North in the land called Svithjod, there stands a rock. It is 100 miles high and 100 miles wide. Once every thousand years a little bird comes to the rock to sharpen its beak. When the rock has thus been worn away, then a single day of eternity will have gone by.

That passage gives you an inkling for just how gosh-darn long eternity is. But all the usual caveats apply: eternity doesn't have a length, that single "day" of eternity is as far in time from eternity itself as a normal day, etc., etc.

Fear of the infinite

If all this leaves you feeling numb, you're not alone. The Greeks, in fact, invented apeirophobia, fear of the infinite. (The term comes from the Greek word for infinity, apeiron, which means "without boundary.") Aristotle would only admit that the natural numbers (1, 2, 28, etc.) could be potentially infinite, because they have no greatest member. But they could not be actually infinite, because no one, he believed, could imagine the entire set of natural numbers as a finished thing. The Romans felt just as uncomfortable, with the emperor Marcus Aurelius dismissing infinity as "a fathomless gulf, into which all things vanish."


“Infinity is where things happen that don’t.”

The ancients' horror infiniti held sway through the Renaissance and right up to modern times. In 1600, the Inquisitors in Italy deemed the concept so heretical that when the philosopher Giordano Bruno insisted on promulgating his thoughts on infinity, they burned him at the stake for it. Later that century, the French mathematician Blaise Pascal deemed the concept truly disturbing: "When I consider the small span of my life absorbed in the eternity of all time, or the small part of space which I can touch or see engulfed by the infinite immensity of spaces that I know not and that know me not, I am frightened and astonished to see myself here instead of there ... now instead of then." Martin Buber, an Israeli philosopher who died in 1965, felt so undone by the concept of infinity that he "seriously thought of avoiding it by suicide."

Most of us will never feel so put out by infinity that we'll resort to contemplating such extreme measures. We may feel weak of mind, like the anonymous schoolboy who once declared that "infinity is where things happen that don't." But our uneasiness will never get much greater than the schoolboy's delightfully dismissive attitude suggests his got. We can live with that level of discomfort, contenting ourselves with the knowledge that all we can reasonably expect in musing on infinity is to get a feeling for it, like that engendered by this gem from another anonymous sufferer of our common infirmity: "Infinity is a floorless room without walls or ceiling."

~~Contemplating Infinity:
A Philosophical Perspective
by Peter Tyson
http://www.pbs.org/wgbh/nova/archimedes/contemplating.html

Give Me Some More Pi, Please

~~Image: Pi Pie

I'm trying to wrap my head around pi. I mean, what is it exactly? We all know it's a ratio, and one that defines the relationship between the diameter of a circle to its circumference. That relationship is expressed as the number of times the diameter of a circle fits in around its circumference. That's essentially what pi is, but why is it that it is expressed as a seemingly infinite number of digits after the decimal point?

~~Image: Circle illustration showing a radius, a diameter, the centre and the circumference.

Plenty of sites all over the net offer lots of interesting facts about pi, but no matter how many of these you try to cram in, they all still seem to fail in satisfying the pangs for what it is that pi is exactly. For example, below are some facts about pi:

"The sequences of digits in Pi have so far passed all known tests for randomness.Here are the first 100 decimal places of Pi3.141592653589793238462643383279502884…

The fraction (22 / 7) is a well-used number for Pi. It is accurate to 0.04025%.

Another fraction used as an approximation to Pi is (355 / 113) which is accurate to 0.00000849%

A more accurate fraction of Pi is (104348 / 33215). This is accurate to 0.00000001056%.

Pi occurs in hundreds of equations in many sciences including those describing the DNA double helix, a rainbow, ripples spreading from where a raindrop fell into water, general relativity, geometry problems, waves, etc.

There is no zero in the first 31 digits of Pi.Pi is irrational. An irrational number is a number that cannot be expressed as a ratio of integers.

In 1991, the Chudnovsky brothers in New York, using their computer, m zero, calculated pi to two billion two hundred sixty million three hundred twenty one thousand three hundred sixty three digits (2, 260, 321, 363). They halted the program that summer.

The Pi memory champion is Hiroyoki Gotu, who memorized an amazing 42,000 digits.The old memory champion was Hideaki Tomoyori, born Sep. 30, 1932. In Yokohama, Japan, Hideaki recited pi from memory to 40,000 places in 17 hrs. 21 min. including breaks totaling 4 hrs. 15min. on 9-10 of March in 1987 at the Tsukuba University Club House.

Pi is of course the ratio of a circle's circumference to its diameter. If you bring everything up one dimension to get 3D value for Pi, the ratio of a sphere's surface area to the area of the circle seen if you cut the sphere in half is exactly 4."

Do you see what I mean? We can try and digest facts about pi all day long, just as we could try and consume the millions and millions of digits of pi over an entire lifetime, and we would still be left feeling ... empty. The reason as to why pi is an infinite number remains pervasively evasive. The mind, in its search for patterns and relationships, seems unable to relate to pi in any way whatsoever, other than drawing the one obvious conclusion that it is indeed a number. An apparently infinite number. But where do these numbers lead to?

I like the idea that it is a truly random collection of numbers, having been shown to exist without having being formed by any KNOWN pattern, but one that must be sub-ordinate to some higher order that we are as yet unaware of, simply because it is these exact same digits, innumerable as they are, appearing in the exact same order everytime we try to evoke pi. The post below is taken from The Sheila Variations, and offers a splendid insight into just how unrandom the random numbers of pi might be. Extracts used in the post are taken from a New Yorker article entitled The Mountains of Pi, written by Richard Preston, which reveal not only the lost world of homemade super-computers, but also something of man's obsession with identifying what is is that the empyreal pi is trying to convey:

"I knew I had read a profile in the New Yorker years ago about Pi, and then remembered that I have it in one of the New Yorker compilations that I own. It’s called “The Mountains of Pi”, and it’s from 1992, a profile of two brothers (the Chudnovsky brothers) on their quest for Pi. That makes it sound tame and intellectual. No. This is a profile of shared obsession.

I love having a library. “Wasn’t there something about Pi in one of those New Yorker books I have …?”

It’s also online – very fascinating profile of two men driven to extremes by their desire to understand pi. It’s also from a time when something like a “computer” in your house was something of a novelty, let alone a “supercomputer”, built to order. Built to serve Pi and Pi alone.

The Chudnovsky brothers claim that the digits of pi form the most nearly perfect random sequence of digits that has ever been discovered. They say that nothing known to humanity appears to be more deeply unpredictable than the succession of digits in pi, except, perhaps, the haphazard clicks of a Geiger counter as it detects the decay of radioactive nuclei. But pi is not random. The fact that pi can be produced by a relatively simple formula means that pi is orderly. Pi looks random only because the pattern in the digits is fantastically complex. The Ludolphian number is fixed in eternity – not a digit out of place, all characters in their proper order, an endless sentence written to the end of the world by the division of the circle’s diameter into its circumference. Various simple methods of approximation will always yield the same succession of digits in the same order. If a single digit in pi were to be changed anywhere between here and infinity, the resulting number would no longer be pi; it would be “garbage”, in David’s word, because to change a single digit in pi is to throw all the following digits out of whack and miles from pi.

“Pi is a damned good fake of a random number,” Gregory said. “I just wish it were not as good a fake. It would make our lives a lot easier.”

Around the three-hundred-millionth decimal place of pi, the digits go 88888888 – eight eights pop up in a row. Does this mean anything? It appears to be random noise. Later, ten sixes erupt: 6666666666. What does this mean? Apparently nothing, only more noise. Somewhere past the half-billion mark appears the string 123456789. It’s an accident, as it were. “We do not have a good, clear, crystallized idea of randomness,” Gregory said. “It cannot be that pi is truly random. Actually, truly random sequence of numbers has not yet been discovered.”

Our minds just don't seem capable of taking pi in. It is an infinite amount of digits, but ones that do not vanish over some distant horizon, stretched over an infinite distance, as the mind might imagine them doing. No, the infinite numbers of pi do not move further and further away from us, but can be seen to exist in a very finite distance, a space which recedes into nothing more than a point, an infinitesimal dot as it were. I wonder if it might be possible to create a form of pi which might be digested, and ultimately understood by the mind?

Mysterious Pi

"Throughout the many centuries pi (π) has been examined and dissected in countless ways. The fascination with pi continues to the present. To this day no one has noticed anything unusual about pi.

When I was eighteen years I noticed the 3_4_5 right at the start of pi. I thought it odd that the Pythagorean triplet would begin right at the start of pi but gave it no more thought. Years later I noticed the 1_1_2 at the start of the square root of two and thought that this discovery was strange. These two oddities both at the same position fanned my curiosity. During the many years of examining pi (π), √2 and S I found that these three constants have an very odd interwoven relationship.

[..]Pi = 3.14159265358979323... It is very odd that a group of eight small different contiguous primes: 3, 14159, 2, 653, 5, 89, 7, 9323 are right at the start of pi. Many and possibly infinite small (numbers with five or fewer digits) and large (greater than five digits) different contiguous primes may exist after 9323. As it turns out right after the 9323 prime the next contiguous prime is: 846264338327........303906979207, it is 3057 digits long. So if pi started with 8462... the first prime would be 3057 digits long.

It will be interesting to see how many digits the average contiguous prime has. Perhaps more interesting may be to find how scarce are groups consisting of eight small contiguous primes of which none of the prime numbers are duplicated."


~~Extracts taken from "Proof of the Existence of God - The Ingenious Numeration of Three Constants" by Vasilios Gardiakos http://www.artmusicdance.com/vaspi/highlights.htm

Pi, Infinity and Zero

If you visualize pi as the connection between a circumference and diameter, and you release the circumference from the diameter, you have just created reality (male and female, yin and yang, north and south). Pi, as a potential point, and diameter as connection between two real points, creates a mandatory circumference, even though the entire circumference may not show up in one place at one time. This is a mouthful and it takes some time to ponder. But this is how we end up with movement (any entity, process, system). Movement is nothing more, or less, than the connection of two points, a general X and Y. If we weren’t moving, we would not perceive (call) movement as movement. But, for now, this is a diversion.

We can begin with the idea of a point, and assume, from point, that point must assuredly be circle, since point, if it is real, must have a diameter (even if this diameter is too small to discern). Therefore, there are no points that are not also circles, or, another statement that goes with this one is, there are no points, just circles. So no matter how we start, we are at (with) a (the) circle (and also of, to, and from it). If we are the starting point, we are a circle that connects to many other circles (which we may view as points or units or entities, processes or systems). It becomes apparent, quickly, that many words are used to describe the same entity (similar entities). (Also, fortunately and unfortunately, there is no starting point, and there are only starting points.)

But back to pi. Pi, if it connects a circumference and diameter, also connects a half circumference and diameter. Half of the circumference might be showing, and the other hidden. Or, the diameter might be hidden with both halves of the circumference disconnected. If we eliminate any idea of time and-or space, and start out, instead, with pi, as a connector of diameter and half-circumference, we can notice pi is a necessary infinite movement that starts at zero. This allows for any movement anywhere with no limit, in no particular order, and with no particular direction, destination, and-or relationship to the original point. Pi, in this sense, is invisible. (Zero is an invisible circle.) How do we get there?

Imagine pi as a point on an invisible circle. Whatever point you pick is connected in a straight line (diameter) to another point, creating a mandatory circumference or set of circumferences. Assuming we collapse all the circumferences into one, and, further, assuming we divide the circumference in half at the point where the diameter connects one point to another on the same circle, we have created a mandatory second diameter (and third) which will act the same (be connected to another two half-circumferences which also function as diameter connected to another two half-circumferences ad infinitum. There is no limit to this expansion. This is how we get to infinity. It is also how we get velocity and acceleration, because even though the diameter and half-circumference connect the same two points, they are mandatorily different sizes, and this is the origination of time and space.

A constant connects two points via two routes which end up an infinite number of routes, and also a finite number of routes (either you stay on the sphere which is the total set of circumferences of one diameter or you escape it). Zero, in this view, then, would have to be pi. That is, pi is playing the role of both zero and infinity, as it connects a stationary spot to two moving spots, where the two moving spots are not moving at the same rate (or in the same direction, necessarily). This is a very simple analysis that ends up being quite complex, once we recognize, we are certainly NOT limited by our conventional ideas of dimension (sphere). That is, sphere is actually line, because they are both a connection of two circles.

Line means one circle has escaped from the other. Sphere means one circle has trapped the other. In this way, our idea of dimension is flawed. Dimension is more easily thought of in circles (since points must be circles, and thus eventually spheres). You can never say (know) how many circles are within a sphere. As long as you have one circle, you necessarily have two, which may, or may not, appear as a sphere (depending on where YOU are in time and-or space). You cannot disconnect a sphere from a circle or a circle from a sphere, but you must if you want to know what is really going on. One sphere interacts with another as if it were two circles. It is unaware it is a sphere.

We have confused the notions of point, circle, line, and sphere; we can also throw in disk, if we want to be clear. In actuality, we are stuck with circle (of circles) because of how a circle operates. That is, it always begins with pi, at any location, in either time or space, which mandates a diameter and circumference (line and circle) connection somewhere. These can appear in many shapes. All of the shapes, however, must reduce to line, and then circle. Line and circle can also be viewed as line and curve. This is how we get parabola and any open shape. And, also, any closed shape, trapezoid, for instance. What we see around us, as reality, and in our symbolic universes, is a wild set of lines and curves, in no particular order, except, if you look very closely, you will always find a one-two-three relationship (somewhere).

This one-two-three comes from pi-diameter-circumference, but does not necessarily end up in sphere (sometimes, but never always, because diameter and half-circumference are not equal). What it boils down to, though, is line is zero not equal to infinity. Circle is zero equal to infinity. Pi makes the decision, not zero or infinity. Pi is the observer that determines where things start and where they end. There are no starts and ends. Just pi on a continual circle connecting many things to many things. We see this as reality. We name it cosmos, universe, solar system, star, planet, cell, dna, particle, force, speed, acceleration, dimension, and, most important, gravity.

We experience it as ‘I,’ where I is an observer. Pi is the only observer, because observer, once he-she-it makes an observation, must make a circle (between observer and observation). Thus, observer is (always) circle. And what we ‘see’ is a snapshot of something that can never be ‘seen.’ Only a circle can create a circle. However, circle can be (and is) known by many different names. Movement, in this view, is an alternate word for observation (and thus an alternate word for circle). And movement, in physics, and also biology, is quite a convenient unifier. If we notice one entity observes another via relative movement (determines a size and speed) to determine whether or not and how to interact with it, and all entities do this, we can use movement as the entity and process that joins any two entities (or processes) together. (Movement is another word for line is another word for circle.) (Pi is doing the analysis and comparison, not either of the entities.)

Any entity and-or process produces the survival and reproduction of movement, as movement produces the survival and reproduction of any entity and-or process. Movement then, originates at pi (acting as zero) and can go to infinity (depending on the observer). We begin and end nowhere (via mind). Via matter, we have a beginning and end, and this is tied to the beginning and end of our surroundings which also have beginnings and ends. All of us, however, because of pi, have no true beginning, nor end. This is the beginning (and perhaps, in a sense an ending) of an understanding of the mind matter relationship in (via) physics (and biology). http://www.circular-theory.com/pi-infinity-and-zero/

Written by Ilexa Yardley, Conservation of the Circle, and The Circular Theory

The Way of the Tau

The circumference of a circle is 2π times its radius. This is a bit confusing, having to stare at a factor TWO in an expression that is supposed to produce the circumference of ONE whole circle. Also, if you rotate the circle radius ONE full turn (360 degrees), your radian expression will contain an irrelevant factor TWO. Inelegant, clumsy, unnerving.


But these (and plenty more) annoyances can be resolved, if you care to define a different circle constant, τ (Greek letter tau):

τ= circumference/radius

Now, using this new circle constant τ , one full rotation (360 degrees) of a unit radius becomes simply 1 x τ = τ radians. Similarly, turning the radius 60 degrees means turning it one sixth (1/6) of a full turn and hence τ/6 radians, turning it 90 degrees is equivalent to ¼ of a full turn and hence τ/4 radians, and so on. Immensely more intuitive and straightforward, when there is no fooling around with an inexplicable factor 2, isn't it?


The inspiration of using the circumference/radius definition as the proper (and the one-and-only) circle constant first occurred to mathematician Bob Palais (in the article "Pi is wrong!"), but the idea of using the Greek letter τ (Tau) to denote it is due to physicist Michael Hartl. Hartl declared June 28 2010 "Tau Day", and he will probably celebrate the first Tau Day anniversary on June 28, 2011.


2π occurs in a large number of important mathematical expressions (Gaussian probability distribution, Fourier transform, Cauchy's integral formula, etc.), so replacing 2π by τ makes things easier and simpler.


The price of this simplicity is that the expression for the area of a circle becomes A = ½τr2, which contains an inconvenient factor ½. But Michael Hartl maintains that there are a lot of quadratic forms of this type in mathematics and physics (distance fallen: ½gt2, spring energy: ½kx2, kinetic energy: ½mv2, etc.), so the area formula for a circle will be easy to remember.


So τ = 6.283185 ... is the only circle constant worth remembering. Forget all others – there can only be one winner! But victory may take some time, I guess.


Reference:

http://tauday.com/