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/

In The Beginning (and the bit that came before it.)

"In the nineteenth century, scientists had thought that the cosmos was made up of ninety-two basic elements, such as hydrogen, oxygen and iron, which were indestructible. This implied that the universe had a diversity of independently existing materials. However, during this century research had revealed that all elements were in fact made up of a single energy. The cosmos was therefore intrinsically one, whether it appeared as a speck of dust, a tree, a Nobel Prize-winning genius or a black-hole beyond the galaxies. The differences were merely appearances. Our senses give us a knowledge of what is apparent, but not of the underlying one reality of the cosmos. This one energy which permeates the whole of creation was what Hinduism calls ‘brahma’. Long before physics discovered it, Shankara had argued that the world of sense experience, that is the world of matter, was a world of appearance (maya), because at the root of each individual existence is the same energy which forms the cosmos. The human self (atman) is ultimately not distinct from the universal self (brahma). Duality is illusion. Reality is not dual, but one. Science has yet to catch up with what the seers in India had already understood over 2500 years ago. While Greece is the country of my birth, India is the country of my soul."
~~Queen Frederika

What was there before the absolute beginning of creation? It is a question which has haunted humanity since ... well, for want of a better word, the beginning. We've always been curious as to where and what exactly it is that the Universe unfurled from. Modern theory holds to the idea that the fabric of the Universe - four dimensional space-time - was born from the Big Bang. But what was before the Big Bang? Because it is impossible to concieve of either space or time before the Big Bang, it makes the question terribly difficult to answer, except of course in terms of "nothing." In his paper, "What Happened Before the Big Bang?" Paul Davies sets about explaining some of the frustration in why it is that "nothing" (at least in terms of modern theory) is the only, though unsatisfactory, reply. An extract from the paper is given below:

"Well, what did happen before the big bang? Few schoolchildren have failed to frustrate their parents with questions of this sort. It often starts with puzzlement over whether space "goes on forever," or where humans came from, or how the planet Earth formed. In the end, the line of questioning always seems to get back to the ultimate origin of things: the big bang. "But what caused that?"

Children grow up with an intuitive sense of cause and effect. Events in the physical world aren't supposed to "just happen." Something makes them happen. Even when the rabbit appears convincingly from the hat, trickery is suspected. So could the entire universe simply pop into existence, magically, for no actual reason at all? This simple, schoolchild query has exercised the intellects of generations of philosophers, scientists, and theologians. Many have avoided it as an impenetrable mystery. Others have tried to define it away. Most have got themselves into an awful tangle just thinking about it.

The problem, at rock bottom, is this: If nothing happens without a cause, then something must have caused the universe to appear. But then we are faced with the inevitable question of what caused that something. And so on in an infinite regress. Some people simply proclaim that God created the universe, but children always want to know who created God, and that line of questioning gets uncomfortably difficult.

... Many people feel cheated. They want to ask why these weird things happened, why there is a universe, and why this universe. Perhaps science cannot answer such questions. Science is good at telling us how, but not so good on the why. Maybe there isn't a why. To wonder why is very human, but perhaps there is no answer in human terms to such deep questions of existence. Or perhaps there is, but we are looking at the problem in the wrong way.

Well, I didn't promise to provide the answers to life, the universe, and everything, but I have at least given a plausible answer to the question I started out with: What happened before the big bang? The answer is: Nothing.
http://www.fortunecity.com/emachines/e11/86/big-bang.html

~~Image: The 'ghost' of the Big Bang. A striking image showing the ghost of the Big Bang has been captured by a new space telescope. The Planck satellite was launched by the European Space Agency in May 2009 to study the early universe.
http://www.independent.co.uk/news/science/space-telescope-captures-images-of-big-bangs-afterglow-2018909.html

But I wonder, can it be possible for the human mind to concieve of something that does exist outside space and time, without leaning on something as unsubstantial as the term "nothing"? In terms of human history, is it possible to find any evidence of anyone ever having tried to point at the existence of such a thing? There is one source which comes to mind, and it's a place where the line between myth and metaphysics can get very fuzzy indeed - the Bible. Quite aptly, it is at the very beginning of the Bible, in the first words of the Book of Genesis, that we are introduced to a version of how things looked before the moment of creation:

"In the beginning God created the heaven and the earth."
~~Genesis 1:1

This would seem to suggest that God stands alone, and seperate from the heaven and earth which He created, forming a trinity as it were, of heaven, earth, and Himself. We think we know what heaven and earth are, but where exactly does that leave our understanding of God? The way in-which God enters the Bible from some unmentioned, hidden realm, guarantees His presence is virtually impossible to define. He has effectively been drawn from nothing, having appeared from nowhere, so that before creation, He is seen as being empty and invisible - a concept of meaningless proportions. This concept has been dragged over into our understanding of God in the time after creation too, having so far proved Himself impossible to substantiate in anyway, shape or form - either physically, or philosophically. All this has assured God a place in humanity's collective consciousness as someone, or something, whom is distant, and unapproachable, simply because He is unknowable. To some, God might be the Universe, but to others, He might as well not bother to exist at all.

Quite unexpectedly, what we find is that Big Bang theory seems to correspond with the creation story told in Genesis, in that the Universe was born from "nothing." The Universe might be here now, for whatever reason, but both the Bible and Big Bang theory seem to agree, you would not find anything which existed before it - that includes space-time, and apparently, that goes for God too.

However, our understanding of what was around previous to creation, at least in terms of Western religion, is based on but one interpretation of what amounts to a few choice words from the Bible. Is it at all possible to interpret them differently? Professor Van Wolde of Radboud Univerity in The Netherlands, claims to have done so, and her interpretation of the first sentence of the Book of Genesis, gives us the perfect opportunity to piece together an entirely new perception of God, the Universe, and lest we forget, ourselves:

...Prof Van Wolde, 54, who will present a thesis on the subject at Radboud University in The Netherlands where she studies, said she had re-analysed the original Hebrew text and placed it in the context of the Bible as a whole, and in the context of other creation stories from ancient Mesopotamia.

She said she eventually concluded the Hebrew verb "bara", which is used in the first sentence of the book of Genesis, does not mean "to create" but to "spatially separate".

The first sentence should now read "in the beginning God separated the Heaven and the Earth"
http://www.telegraph.co.uk/news/newstopics/religion/6274502/God-is-not-the-Creator-claims-academic.html

If you remember from previous posts, we've discussed how the doctrines of the alchemists acknowledged the materia prima as the uncreated substance of God. The materia prima, though never fully disclosed, was also known to them as "black earth." Taking the view that the black earth of the alchemists' is the same sort of stuff as the earth mentioned in Genesis, then it appears that our modern understanding of the creation story is not as cast-iron as we think. Apparently, God did not just happen to mysteriously materialise into existence at the point of creation, but rather, He'd been hanging out the entire time previous to it too, in the guise of something non-too-disimilar from common, everday dirt.

Maybe the Book of Genesis was not written quite so simply as we percieve it today. Perhaps there is more to it than it simply being a story of God's magical appearance, and how He pulled heaven and earth out of a hat. What Moses may have been trying to convey, was an explanation of how a rather impressive Being, whom having made some sort of decision to create the Universe, did so by separating into a duality containing both heaven and earth. This is not describing dualism in the sense of two opposing substances in conflict with one another, but more of a monoism, in-which just one substance creates the illusion of duality by opposing itself.

"The concept of duality seems to be an integral part of all life as we perceive it on this planet. To those who have become increasingly aware of a deeper place of being, it is still, at times, difficult to comprehend the connection or better said, oneness, between these “seemingly” two realms of relative reality. We have called it “heaven/earth”, spirit world/natural world, or material world/nirvana, but the same veiled reality exists within all these concepts. One world we would call “natural”, is a phenomenal world perceived by the natural senses of sight, hearing, touch, smell, and taste. The other world seems to be more a dimension of “spirit” and is “sensed” through a different set of senses which in most are hopelessly obscured behind an elusive veil. The subject of being able to reasonable discern the spirit realm to the extent that its peace, delightfulness and abundance may be experienced as a part of Life on earth has been the quest of people for millennia and has been woven into every form of religion, cult, or psychic group imaginable. Yet the premise that we are living in one realm “A” and needing to get to another realm “B” is foundation upon which all these formulae are based."
http://www.aplaceofbeing.com/Letters/articles/letters/on_duality.htm

If God is a substance of some description, it must mean that heaven is a substance too. Importantly, the substance of heaven is the exact same stuff, the exact same single element, from which God Himself is made. This one same substance, the materia prima, is the thing from which the Universe was created, and subsequently, it might be supposed that it is also the one thing which constitutes the entire Universe in this very moment too. It's exciting to imagine that if heaven is describing a substance, it means, at least theoretically, that we should be able to find it around here someplace. Maybe, just maybe, it's possible to enter heaven within the limits of our own lifetime.

If we choose to, God can now be seen, and fully understood, not as some errant dictator, but as an ever-present, all-pervading entity from which all of creation spilled forth. It's intruiging that this perception of God as Universe, now moves us much closer to the philosophy expounded by Eastern mystics. For them, God is not some separate entity to be worshipped, but One whom is very much entwined with our everyday lives.

" One of the main philosophical trends in Hinduism is known as the Vedanta. Several streams of thought emerge from Vedantism, of which one is referred to as non-dualistic. While God enters the soul for an intimate communion with the saint in Christianity, God is also a separate entity with an existence apart from the mystic, hence the dualistic nature associated with Christian mysticism. In the East, the world is a manifestation of God (pantheism); in the West, the world is the creation of God (theism). In theism the mystic never is or becomes identical with God; there is always a “great gulf” between God and man. When Meister Eckhart claimed that “God and I, we are one,” he was accused of heresy by the church. In Hindu mysticism and Plotinus, mysticism seeks to go beyond all dualism and rest in an absolute undifferentiated unity. To these mystics it appears that there is within their mystical consciousness no division whatsoever, there is no God outside the Self; God is the Self. The secret is realizing that the individual self, the pure unity of the finite ego, IS the Universal Self, the Absolute. Where there is consciousness of the Self, individuality is no more. It is not that the individual self Becomes the Universal Self. It always was the self. It comes to realize this truth in the moment of illumination."
http://www.soultrek.com/3%20eastern%20mys.html

If we were to accept that God, and heaven, and earth are all describing the same substance, then it seems only natural to want to know what this substance is precisely. The next line from the Book of Genesis does not tell us directly, but it does offer something of a clue to the nature of this biblical earth, the only thing to exist before creation and the supposed uncreated substance of God.

"And the earth was without form, and void; and darkness was upon the face of the deep. And the Spirit of God moved upon the face of the waters."
~~Genesis 1:2

Sure, it's not exactly brimming with adjectives, is it? We are told that the biblical earth is something without form - a void, or chasm or somesuch - to be found only in utter darkness. For all intents and purposes, these could be the same choice of words that a blind-folded person might use when asked to describe what they see. Despite knowing that the materia prima is a real, physical substance, we are resigned to a fate where it is never discovered, because it seems we are unable to elaborate further on its constitution, other than using words which seem only to add upto "nothing." However, Moses does appear to be trying to describe SOMETHING. Something moves. Something is there. It's not much to go on, but I wonder, might it be enough?





Many thanks:

http://www.theosociety.org/pasadena/sunrise/46-96-7/ev-prent.htm
http://apocryphile.org/jrm/articles/paradox.html
http://www.nonduality.com/whatis9.htm
http://www.kheper.net/topics/worldviews/dualism.htm
http://www.themystica.com/mystica/articles/d/dualism.html
http://www.booksie.com/religion_and_spirituality/article/juggernaut/collection-of-thoughts-from-hindu-mystics-partii
http://en.wikipedia.org/wiki/Buddhist_atomism
hhttp://www.suite101.com/content/hun-and-po-the-duality-of-the-taoist-soul-a253866
http://scripturetext.com/genesis/1-2.htm