The Last Stoic


Posted in Uncategorized by munty13 on July 1, 2011

“…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:

Circling The Square

Posted in Uncategorized by munty13 on April 16, 2011

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?

Give Me Some More Pi, Please

Posted in Uncategorized by munty13 on April 9, 2011

~~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?

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

Posted in Uncategorized by munty13 on January 15, 2011

“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.

~~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.

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”

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.”

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.”

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:

First Matter

Posted in Uncategorized by munty13 on November 30, 2010

“In the beginning God created the heaven and the earth.”
~~Book of Genesis 1:1

~~The NASA Hubble Space Telescope photo of V838 Monocerotis (February 8, 2004)

Before physicists, the men of science, so to speak, were alchemists. Unlike modern theory today, the doctrine of the alchemists was very much centred around theology. To them, God was not some abstract philosphical concept, but rather, God was understood, and accepted as being something from which everything is made. In other words – God is everything – God is the Universe itself. In this sense, the alchemist was more closely related to the mystic than a scientist.

“Alchemists agree with chemists that there is unity in matter, but whereas chemistry teaches that atomic particles are the smallest units in matter, alchemists believe in an ultimate source they call Ether, or the universal fluid. Matter to the alchemist is therefore compact energy, which can be dissolved into free energy or force. For the alchemist energy and matter are the same thing, namely substance. The substance is the Absolute, the One that Hermes Trismegistus describes in the Emerald tablet. This One is divided into three parts: Intelligence or force, Energy and Matter.”
~~The complete book of spells, ceremonies, and magic By Migene González-Wippler

~~The Alchymist by Joseph Wright of Derby (1734-97) 1771

Alchemists hoped to move closer to God by developing a greater understanding of God, not only in the spiritual sense, but in a real, physical sense too. If God truly was everything, and everything owed its existence to God, then it must be possible to find the existence of God in any thing, in any place we care to look. What the alchemist was seeking in all material things, and that includes himself, was the very substance of God. There is a real possibility that some alchemists may have found it. Even the word “alchemy” seems to hint at something of the nature of this substance.

“The word alchemy is derived from the words Al and chemia.

Al is an ancient word meaning ‘God’ in the sense of the ‘All’, the ‘Absolute’. As part of the word alchemy it means ‘divine’ or ‘universal’. The word was used in many ancient languages and cultures, including the Egyptian, Mesopotamian, Hebrew and Celtic. Later, the Hebrew form of the word came to be written as El, which in the Christian bible is translated as ‘God’. In Islam the word appears as Allah.

Chemia is from the Greek word khemia, which itself is derived from the ancient Egyptian word kemit, meaning ‘black earth’.”

In other words, the term “alchemy” could be interpreted as meaning “God is black earth.” These choice of words to describe God may at first seem a bit off-key – one might have expected something a bit more poetic, more holy, more ethereal – such things as “God is mountain dew,” or “God is sunshine.” Instead, for those seeking the substance of God, the alchemist tells us to ignore all this fluffy idealism, and points only to the dirt at our feet. According to the Bible, this dirt, or dust is also the same foodstuff which the serpent of Eden is condemned to eat forever.

“And the LORD God said unto the serpent, Because thou hast done this, thou art cursed above all cattle, and above every beast of the field; upon thy belly shalt thou go, and dust shalt thou eat all the days of thy life.”
~~Genesis 3:14

When God banishes Adam and Eve from the Garden of Eden, God makes a point of telling Adam that the dust of the ground served as the material from which he was created. This at first seems only slanderous, the final insult to be added to Adam’s degradation – but this being the word of God, it is told only as a truth. It is intriguing that the only substance required by God to create intelligent life is little more than ordinary dust from the ground.

“By the sweat of your brow you will eat your food until you return to the ground, since from it you were taken; for dust you are and to dust you will return.”
~~Genesis 3:19

Thus, a significant clue to the nature of God, and the powers of creation, are all contained in nothing more than a speck of dust. In searching for the substance of God, it is likely that these same thoughts must have occured to the alchemist. If one could gain knowledge of this biblical dust, then one would know the form which God takes before creation. In other words, the dust mentioned in the Bible, woefully unflattering as it is, could be interpreted as being the definitive description for the uncreated substance of God.

“Uncreated Independent Substance: thing that is not dependent upon the causal power of any other thing in order to exist or to remain in existence, and is not a property of any other thing.(The only thing that satisfies this definition of “uncreated substance” is God.)”

The alchemist believed that all material bodies are formed from only one substance, indeed, “the substance.” The entire material of the Universe was said to consist of this one single element – the materia prima (first matter.) For some, the materia prima represented God in its most purest form. It represents the primordial matter from which all things, and especially living things, are made.

Using the Bible, Paracelsus and others, connected prima materia to God; “before Abraham was made, I am.” (John 8:58) Since prima materia is supposedly the [philospher’s] stone, also, this also demonstrated the stone is without beginning or end. Jung noted many Christians hearing this would not believe their ears, but it was plainly stated in the Liber Platonis quartorum, “That from which things arise is the invisible and immovable God.”

If the alchemist was able to gain mastery over this substance, then it should, at least in theory, be possible for them to take dominance over the will of God. The forces of God would be subjucated to the whim of the alchemist, giving him/her the power to fulfil all their earthly desires instantaneously. The alchemist would at last become master of their own fate, and the need for a God will have been entirely displaced.

The goal of Alchemy is the Great Opus or the Great Work which is the purification of the lesser or gross and its elevation to the greater or more refined, whether in plants, metals, or inconsciousness. The ultimate goal of the alchemist is to find the Prima Materia or the First Matter of nature as the dark, passive, unformed and raw virgin and universal stuff of creation. Through the alchemical process the alchemist transforms this Prima Materia into the Philosophers Stone. This accomplishment is most commonly known as the transformation of Lead into Gold, the heaviest, darkest, densest most earthbound, least valuable metal becoming Gold: Incarnated Light; the most glittering, luminous, valuable metal; symbol of the sun and of spiritual attainment and consciousness, spiritual illumination as cosmic consciousness which is the ultimate goal of the human evolution.

Certainly, one of the goals of alchemy was to achieve great wealth (after all, regardless of how close they moved towards God, there was always the danger of getting deeper into debt!). Possessing the philospher’s stone would allow the alchemist to turn lead into gold, which quite literally, would give them the power to print their own currency. This remarkable prize however, pales in significance when compared to the true treasure which awaits the seeker. It is the one thing which is universally prized by humans, and desired above all other things, though seldom believed possible – the gift of immortality. The alchemist would finally be allowed to lift the veil, and to enter the world hidden from view, and recieve the key which will give him/her the power, and ultimately freedom, to choose the day they die. To become immortal would signify the alchemist’s completion of the Great Work.

This philosopher’s stone is a metaphor – which means that it has both an inner and outer reality, neither of which can be taken for granted or understood exclusively. The development of the philosopher’s stone could only occur through a refinement of the initially untransformed base material of the world – the “prima materia” or black earth, which is simultaneously the alchemist’s own psyche, both conscious and unconscious (Jung, 1967, 1978, 1993) 4, as well as the actual underlying physicality of all the world’s substances.

Descriptions of how the philospher’s stone appears physically, are understandably evasive, but that does not mean that the stone, “a stone which is not a stone,” is necessarily hidden from view. Quite the contrary, it was sometimes said to be a common substance, found everywhere but unrecognized and unappreciated.

As you can see, the importance of identifying the substance of the materia prima is essential to the creation of the philosopher’s stone. So, what exactly is the materia prima? Which one of the elements from the periodic table is meant to be the hidden substance of God?

Many thanks:
Alchemy: an introduction to the symbolism and the psychology By Marie-Luise von Franz

Are We Ready?

Posted in Uncategorized by munty13 on November 4, 2010

“It’s a waste of energy to be angry with a man who behaves badly, just as it is to be angry with a car that won’t go.”
~~Bertrand Russell

Nowadays, the idea that energy could be a substance is actively discouraged, and very often ridiculed. However, reducing the term “energy” to what amounts to nothing more than an abstract concept, is really only a recent development in terms of human history. Before the advent of modern theory, our knowledge was based on traditions, stemming back to ancient times, which percieved energy as a substance. Indeed, it was supposed that the entire Universe is immersed in it. In an enlightening article taken from Centerpointe’s newsletter “Mind Chatter” – “What is Reality (and why should you care?)” – Bill Harris, writes a fascinating account of how the mystics came to interpret “reality.” As Harris explains, what the mystics percieved as energy, also had a lot to do with what they saw as the Divine. Below, is an extract taken from the article:

“For thousands of years, mystics have said that there is one energy in the Universe, that the Universe and everything in it is the play, the dance, the vibration, of that one energy. Underneath the seeming multiplicity, they say, everything is made of the same substance. This energy, they say, is everywhere and “everywhen.” This principle is sometimes described as Omnipresence or God. The Hindus and Buddhists call this principle, Sat-one energy, everywhere, making up everything, always, past, present, and future.

Quantum mechanical physicists, for several decades, have been saying the same thing. They notice that on the sub-atomic level, particles come into being, seemingly out of nothing, and dissolve, and disappear back into nothing, that two or more particles collide, and one, two, three or more particles, of a different kind, appear from the collision, or all the particles cease to exist. There is a “something” that everything comes out of, and returns to, and which makes up, or is the background of, everything.

The mystics, however, went one step further. In adddition to noting that this one energy is Omnipresent, they also said something else that I think is rather startling. They said that this one energy is aware of itself being everything and everywhere and everywhen: that it is conscious, that it has consciousness. The mystics called this second characteristic of reality Chit.”

I suspect that one of the reasons as to why modern theory is so reluctant to imagine energy as a substance, is because admission immediately arouses unwelcome religious fervour. Science wants only to deal with the stuff in the Universe that it can quantify, and substantiate – it’s not overly concerned with an invisible, intangible God, nor for that matter, any unseen, immaterial, imponderable substances which might, or might not be energy, or consciousness, or whatever. To science, the questions and answers to these concepts are irrelevant – mere distractions from the job at hand. Entering philosophical and theological discourse will quite often do nothing to enhance their sums.

If science were ever tempted to admit energy as a substance, conceding to the idea that it can be quantified in someway, it would fling the door wide open to suggestions that the stuff of energy, which it is weighing, measuring, and collecting, comes very close to being the pure substance of God. It would allow practically every person on the planet to point out to scientists, that the substance of energy, which they are pouring from their flasks into test tubes, also amounts to unequivocal proof in the existence of God. Now, it is not simply energy which we can reach out and touch, all with our very own fingers, but God’s personal Being.

I suspect that there are some physicists who are more than familiar with the God-energy conundrum, but are hesitant to openly discuss it – not because they are wilfully obstructive, or even particularly dispassionate – but because they are aware of the chain of events such an admission will unleash. The question which rises to the forefront most is not: is science willing to admit to the existence of God? The question we should really be asking is: is humanity ready to recieve all that power?

Is There Such A “Thing” As Energy?

Posted in Uncategorized by munty13 on October 5, 2010

“There is a theory which states that if ever anybody discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There is another theory which states that this has already happened.”
~~Douglas Adams


Previously, we’ve discussed force as an “interaction energy.” The use of the word “interaction” suggests that this energy is found BETWEEN two objects. The implications of this are twofold: there is such a “thing” as energy – a real physical thing – and that it is possible for this thing to exist outside matter. Under modern theory mind, all sorts of complications arise when you start saying this kind of stuff. It seems to break every convention that physics has. That’s mainly because energy is not seen as a substance, or indeed, anything in particular. It is observed strictly as an abstract concept.

With perhaps the exception of energy in the form of light, energy is not a thing per se. Rather, energy refers to a condition or state of a thing.

Getting to the core concept of energy is no mean feat, because we are now messing with fundamental laws of physics. These dictate that energy cannot exist outside matter because it is solely a property of matter. Properly defined, matter has energy, but in itself, is not energy per se. Matter needs energy, for without it, matter cannot exist.Described in this way, it suggests that energy is a different “thing” from matter.

A kind of cyclic argument ensues, which says that even if it was possible for energy to exist outside matter, it cannot physically exist as a thing, because energy only refers to the condition or state of an object. Consistently treating energy as only describing the energetic state of a material object, ensures that it remains indescribable as an independent substance. Once you do decide to try and seperate energy from matter, then out pops a big, wrestling squid in the shape of the laws of conservation. Basically, the laws all subscribe to the idea that energy has to belong to an object or body, and because of this, it is impossible for energy to simply enter, or exit a system, without it belonging to another object or body. This concept is known as the conservation of energy.

The conservation of energy is a fundamental concept of physics along with the conservation of mass and the conservation of momentum. Within some problem domain, the amount of energy remains constant and energy is neither created nor destroyed. Energy can be converted from one form to another (potential energy can be converted to kinetic energy) but the total energy within the domain remains fixed.

The law behind the conservation of energy refuses to admit that energy can exist outside matter. This means that the space around matter is strictly space – with nothing in it – nada, zip, donut, etc. However, something is still needed to explain how energy is transferred from one body to another. Because the energy is moving from one body to the other, regardless of how small that distance is, there must come a moment where energy is seen to exist by itself. This should give us the perfect opportunity to finally get a glimpse of “energy.” What we find though is not energy in the buff – but energy dressed-up in a new outfit. To help explain energy transfer, modern theory devised a vehicle to carry the energy – the photon. Happily for modern theory, the photon is still maintained to be a particle of matter, even though this particle has no mass whatsoever.

Physics experiments over the past hundred years or so have demonstrated that light has a dual nature. In many instances, it is convenient to represent light as a “particle” phenomenon, thinking of light as discrete “packets” of energy that we call photons. Now in this way of thinking, not all photons are created equal, at least in terms of how much energy they contain. Each photon of X-ray light contains a lot of energy in comparison with, say, an optical or radio photon.

Conveniently, because the photon has no mass, it means that it does not have to obey the confines of space-time, and therefore does not experience our idea of “time.” This allows the photon to quite literally arrive at its destination before it has even departed from its source – including distances which span the entire Universe!Because the photon can cross the divide between bodies instantaneously, modern theory is not obligated to explain how the photon might appear in the void. Modern theory persists in refusing to define energy, and maintains that the photon is only an energy carrier and not energy per se. Fortunately for us, this presents something of a crack for us to try and pry open.

Now that the photon has been emitted and begins its flight, we are purely in a relativistic mode. Einsteins equations for space distortion and time dilation tell us that the path in front of the photon shrinks to zero and the time of flight shrinks to zero as well. Now that the photon has been emitted and begins its flight, we are purely in a relativistic mode. Einsteins equations for space distortion and time dilation tell us that the path in front of the photon shrinks to zero and the time of flight shrinks to zero as well. This has always raised a troubling problem because we know that some photons take billions of years to fly across the universe and move about 1 nanosecond a foot of travel.

As the author of “Quantumweird” points out, how is it possible for instantaneous energy transfer to take place if the photon is physically restricted by a maximum speed limit? If the photon cannot physically travel faster than light, how is it possible for energy to do so? The only thing which makes any sense is to admit that it is only energy, and not the photon, which is capable of instantaneous transfer. Quite what this means is very open to debate, but I like the idea of a surrounding medium, a medium of surrounding energy, being responsible for the transfer of energy between bodies.

“As far as a photon is concerned the passage from point A to point B is instantaneous – and it always has been. It was instantaneous around 13.7 billion years ago when the entire universe was much smaller than a breadbox – and it still is now.

But once you decide that the speed of light is variable, this whole schema unravels. Without an absolute and intrinsic speed for relatively instantaneous information transfer, the actions of fundamental forces must be intimately linked to the particular point of evolution that the universe happens to be at.

For this to work, information about the evolutionary status of the universe must be constantly relayed to all the constituents of the universe – or otherwise those constituents must have their own internal clock that refers to some absolute cosmic time – or those constituents must be influenced by a change in state of an all-pervading luminiferous ether.”

Science has long been able to avoid confronting the question of what energy is exactly, by using a formula which allows it to inter-change energy for mass. The formula, probably the most famous equation of all time, is unequivocal proof that energy and mass are equivalent. I am of course referring to E = mc2.

Einstein correctly described the equivalence of mass and energy as “the most important upshot of the special theory of relativity” (Einstein, 1919), for this result lies at the core of modern physics. According to Einstein’s famous equation E = mc2, the energy E of a physical system is numerically equal to the product of its mass m and the speed of light c squared. It is customary to refer to this result as “the equivalence of mass and energy,” or simply “mass-energy equivalence,” because one can choose units in which c = 1, and hence E = m.

If science ever needs to draw upon a physical represention of energy, then it does so in the form of matter. What this does though, is screw around with our previous statement where we said “without energy, matter cannot exist.” Because energy and matter are inter-changeable, the statement must also be able to read “without matter, energy cannot exist.” Which is not what we agreed on at all. We know that matter cannot exist without energy, but nowhere have we said that energy cannot exist without matter. The problem with E=mc2 is that it can be manipulated in such a way, it forces us to accept the notion that energy cannot exist without matter, when in-fact, there is no direct proof that this is true.

If you remember, we agreed that “matter has energy, but in itself, is not energy per se.” In this way, energy and matter can be seen as two different entities. If they are different, then it must be possible to seperate them. On one hand, we have matter which we know needs energy to exist, but on the other, we might find that energy, pure unadulterated energy, is more than able to exist as an entity in its’ own right. Detractors will say that trying to seperate energy from matter, is a bit like trying to seperate butter from hot crumpets. I mean, why would you want to? Crumpets taste lovely with butter. To seperate the two is simply pointless. But of course it’s not. Mankind’s final understanding of what energy is, could represent our greatest adventure yet.

Many thanks:

Newton’s Smart Ass

Posted in Uncategorized by munty13 on September 25, 2010

“If I have seen a little further it is by standing on the shoulders of Giants.”
~~Sir Isaac Newton

Much like mass or volume, energy is a property of an object. Energy is the potential for an object to exert force. Therefore, it can be said that a simple machine, such as a lever or pulley system, stores energy which then gives it the “potential” to do something. Here it seems that we have also given ourselves a pretty good description of the term “potential energy.” That is, potential energy is energy stored in matter.

Energy differs from force, in that I need energy to exert a force. Energy is needed to exert force. If I don’t have any energy, I will not be able to do anything. It is energy which is the currency for performing work. You need energy to do work. If I have no energy, I can’t use force to make something move. Which thus brings me to the somewhat unsatisfying conclusion that energy can exist without force, but that force cannot exist without energy. Or, to put it another way, energy is not force exactly, but force on the other hand, could be described as being energy in one form or other. There. Clear as mud. To try and build a better picture, let’s take a closer look at what is meant by the term “force” exactly.

A force is a push or pull upon an object resulting from the object’s interaction with another object. Whenever there is an interaction between two objects, there is a force upon each of the objects. When the interaction ceases, the two objects no longer experience the force. Forces only exist as a result of an interaction.

Our modern understanding in the workings of force and motion is derived from observations made by Sir Isaac Newton (1642-1727.) His most outstanding contribution to physics was in formulising the exact forces which act on objects, and how these define an object’s motion. Essentially, it amounted to a mathematical representation of how energy was involved in moving objects, and indeed, stationary objects. Thus, Newton was able to arrange his findings to develop a mechanical model of the Universe, giving us the expression “classical mechanics.”

The term classical mechanics was coined in the early 20th century to describe the system of mathematical physics begun by Isaac Newton and many contemporary 17th century natural philosophers, building upon the earlier astronomical theories of Johannes Kepler, which in turn were based on the precise observations of Tycho Brahe and the studies of terrestrial projectile motion of Galileo, but before the development of quantum physics and relativity. Therefore, some sources exclude so-called “relativistic physics” from that category. However, a number of modern sources do include Einstein’s mechanics, which in their view represents classical mechanics in its most developed and most accurate form.

What Newton was trying to establish was a set of equations which illustrated how motion is governed by the Universe. In theory, these formulas amounted to a firm set of rules which could be applied to motion anywhere in the Universe – from everyday objects to distant planets. In the following three hundred years, it should be duly noted that the Universe, being something of a stickler for rules and regulations, has always been seen to obey them. In 1686, Newton set-out the formulas in his paper, “Principia Mathematica Philosophiae Naturalis,” and presented them as the Three Laws of Motion.

Newtonian concept of force is given in three simple mechanical laws:

1) A body remains in its kinetic state, either at rest or in motion, unless there is an external force acting on it; which is known as the law of inertia.

2) If a (net) force F is applied to an object with mass m, then the acceleration a of the object caused by F follows the relation F = ma

3) If object A exerts a force F on object B then object B, then object B exerts a force -F on object A, equal in magnitude and opposite in direction to F

The definition of “force” can be found in Newton’s Third Law, which states that for “every action there is always opposed an equal reaction.” According to the Third Law, a force is simply a mutual interaction between two objects that results in an equal and opposite push or pull upon those objects. This illustrates the fact that forces always occur in pairs. Forces never act in isolation, and when two objects interact, action and reaction forces of equal magnitude are always paired and act on the objects in opposite directions.

Newton’s 3rd Law states that for every action, there is an equal and opposite reaction. This basically means that forces always occur in pairs. In any situation where one object is exerting a force on the other, then the other object is also exerting a force that is equal in magnitude but opposite in direction on the initial object. So for example, The Sun exerts a gravitational pull on Earth, and likewise, the Earth exerts the same gravitational pull on the Sun. But since the mass of the Earth is so much smaller than that of the Sun, the effects of the force on the Earth is much greater, and so we orbit around the Sun, while the Sun merely does a small wobble in reaction to the forces exerted by the Earth.

The most important provision is that for forces to occur there must be TWO OBJECTS. Forces only exist as a result of an interaction between two objects. Therefore, if energy can be percieved as a property of one object, then perhaps we can think of force as being a property of two objects. Force is never found to be the property of one object. To develop a greater understanding of force, it is important to know how to apply the Third Law correctly.

In physics, there is a supposed paradox relating to the Third Law, in-which a wise, but particularly lazy donkey refuses to move because he claims that the action of him pulling the cart, only results in the cart pulling him back with an equal opposing force. Never mind how much he strains to pull away, the cart pulls him back with the same magnitude of force, thereby leaving him stuck to the spot. There’s nothing for the donkey to do, except shrug his shoulders and ask for another carrot.

In truth though, there is no paradox. That’s because the donkey and the cart are not a straight forward action-reaction pair. There is also a third object that is conspicuous by its absence from the donkey’s reasoning – the ground! If you imagine the donkey and cart floating in the vacuum of space, with donkey kicking his heels in vain, it might reveal just how vital the ground really is for this type of motion.

To get moving, the donkey has to get the cart moving too. Strictly speaking, it is not as simple as saying that the cart opposes the motion of the donkey. Rather, it is the forces which the donkey generates which are in opposition to the forces generated by the cart. In trying to explain this process a little better, it is perhaps best to think of the donkey and cart as consisting of two conflicting systems. One system is the donkey and the ground making up one action-reaction pair, while the other system consists of the cart and ground in another action-reaction pair.

If we begin with the system of the donkey and ground, we have the donkey which is applying an action force downward and backward, while the ground has a reaction force which is acting forward and upward. The donkey resists the ground moving forward, and the ground resists the donkey moving backward. It is noticeable that the applied backward push on the ground is dependent on a very significant force in order for it to be converted into forward motion – friction.

In ancient times, Aristotle had maintained that a force is what is required to keep a body in motion. The higher the speed, the larger the force needed. Aristotle’s idea of force is not unreasonable and is in fact in accordance with experience from everyday life: It does require a force to push a piece of furniture from one corner of a room to another. What Aristotle failed to appreciate is that everyday life is plagued by friction. An object in motion comes to rest because of friction and thus a force is required if it is to keep moving. This force is needed in order to cancel the force of friction that opposes the motion. In an idealized world with no friction, a body that is set into motion does not require a force to keep it moving. Galileo, 2000 years after Aristotle, was the first to realize that the state of no motion and the state of motion with constant speed in a straight line are indistinguishable from each other. Since no force is present in the case of no motion, no forces are required in the case of motion in a straight line with constant speed either.

We can think of friction as a force that impedes motion – it is always a resistance to the motion of things. If the ground was slippery for example, the conversion of the action-reaction forces would be greatly reduced by the lack of friction. Friction fulfils the very definition of force, in that it describes an interaction between two objects, such as the wheel and the ground for example, and that both bodies generates forces which act on the other body.

Friction is not a fundamental force but occurs because of the electromagnetic forces between charged particles which constitute the surfaces in contact. Because of the complexity of these interactions friction cannot be calculated from first principles, but instead must be found empirically.

Needless to say, friction is also required by system number two, the action-reaction pair of the cart and the ground, to keep the cart moving along the ground. The force of the wheel pushing backward is dependent on friction to convert it into a reaction force which pushes the cart forward. If friction did not exist between the wheels of the cart and the ground (it might help to imagine the wheel and ground smothered in oil – but never include yourself in the picture – that’s just wrong!) the wheels would simply never get a grip on the road (and very likely give up energy in the form of heat, and noise) and the cart would never get off the spot.

Friction helps people convert one form of motion into another. For example, when people walk, friction allows them to convert a push backward along the ground into forward motion. Similarly, when car or bicycle tires push backward along the ground, friction with the ground makes the tires roll forward. Friction allows us to push and slide objects along the ground without our shoes slipping along the ground in the opposite direction.

The motion in both these systems, and therefore the system over-all, is dependent upon frictional forces. If we overlay these two systems on the over-all system of donkey, cart, and ground, we can see that what we effectively need in-order to see motion, is for the donkey to produce a reaction force which is greater than the cart’s resistive force. In other words, the cart will move forward when the frictional force between the horse and ground, is greater than the frictional force between the cart and ground. Because both these reaction forces are so dependent upon friction, it also reveals something of the force which is truly responsible for motion (at least walking motion) – electromagnetic force.

Now, there are some who explain the paradox of the lazy donkey, but still fail to acknowledge that the donkey and cart are NOT an action-reaction pair. If the donkey and cart were an action-reaction pair, motion would only be possible if the action of the donkey was somehow greater than the reaction of the cart. But how is this possible, if according to Newton’s Third Law, the opposing force is ALWAYS of the same magnitude as the applied force?

“If a force acts upon a body, then an equal and opposite force must act upon another body.”

Some commentators still maintain that motion is possible because the reaction force and the action force are working independently from one another. This means the reaction force does not cancel the action force because they are both acting on different bodies – the donkey is acting on the cart, and the cart is reacting to the donkey. Unfortunately, their reasoning blatantly contradicts what defines an action-reaction pair in the first place – that is, they always come in pairs! It seems inconceivable that one can operate with a strength which is totally independent from the other.

As we have seen, motion is not dependent, in anyway, upon an “interaction” between the donkey and cart. Sure, there is something at work between the donkey and cart because they are tied together, meaning that forces must exist between them, but as such, this arrangement is not responsible for the forces which drive motion. Rather, it is the interaction with the ground which is driving the motion of the donkey and cart; the forces which occupy the harness, and wagon tree between the donkey and cart are a response to that interaction.

In an “interaction,” forces appear between two objects, and exert forces of the same magnitude in opposite directions. In effect, what we are seeing is each body being repelled in the opposite direction by a force that is exerted by the other body. What we see is one body moving away from the other body – one body goes left while the other goes right. A good way of visualising this is to imagine two ice-skaters who are leaning one against the other with their hands. This presents itself as a good example because the lack of frictional forces between the skaters and the ice, allows us to dismiss the “interaction” between the ice-skaters and the ground, and to concentrate more on the ice-skaters acting as an action-reaction pair.

On one level, the difference between dancing on a floor and skating on ice is the lack of friction. Smooth ice provides very little resistance against objects, like ice skates, being dragged across its surface. Compared to, say, a wooden floor, ice has much less friction.

Thus, ice-skater A pushes against ice-skater B, and a force will emerge that also pushes back against A. They “interact.” Both A and B are sent in opposite directions AWAY from each other, they repel one another, and they do so with the same magnitude of force. This means that if A and B both have the same mass, then at least in theory, they shall both glide the same distance away from each other.

Using the example of two-ice-skaters, brings to the forefront the importance of the term “interaction.” At the conclusion of my last post, I basically said that it was possible to think of force and energy as describing the same entity, even though it seems to break a well-founded tradition in physics that supposes they are not. What is exciting is that by developing an understanding of what exactly is meant by “interaction,” the discreet affair between “force” and “energy” is finally exposed.

Energy is the property of one object, whereas force is never the property of one object, but ALWAYS two objects. However, strictly speaking, force is not the property of two objects either. A very distinctive feature of force is that it describes the “interaction” between two objects. In other words, force is the “interaction.” Quite what this means, evaded me somewhat, until I came across a post on the blog “Gravity and Levity,” and at once, everything was revealed. Some extracts from the post are featured below:

Force and energy: which is more real? This question sounds ridiculous, and maybe it is. So if you’re not in the mood for philosophy right now, you can skip this post.

Nonetheless, I think it makes sense to talk about our general attitude toward the concepts of force and energy. “Which is more real?” may not be a very well-defined question, but I think it is a very natural one that cuts to the heart of how we think about forces and energies. Furthermore, it is one to which my answer has changed over the years. The change was a difficult one: force and energy are such profoundly important concepts in physics that to change your view of them is to change your view of all topics that are built upon them (basically, everything). But for me it has been extremely important. Shifting my position from “force is more real” to “energy is more real” was essential for understanding and enjoying advanced topics in physics.

…If you believe that energy is more “real” than force, then you stop talking about “forces acting on objects” and instead talk about “interactions between objects.” Your starting assumptions for describing the universe must be the “interaction energies”.

…So which viewpoint is more correct? In a sense, it doesn’t matter: both give results that are perfectly consistent with our observations of nature. The second one seems a little crazier, but in fact it requires fewer assumptions. It also manages to explain why all forces come in pairs: between two objects there is only one interaction energy, to which both objects will respond. In my mind, the “energy is more real” viewpoint is much more compatible with advanced physics concepts like thermodynamics (where energy is really the only consideration), quantum mechanics (where our force laws are no longer strictly obeyed, but energy remains absolute) and field theory (where we are given a way of picturing where the energy is really stored). Perhaps most importantly, I find the energy viewpoint much more conducive to wonder.

The question of whether force is simply another term for describing energy, though presenting itself as a bit of a stumbling block for me, is something which seems to be already familiar with both world-weary physicists and philosophers (and bloggers!) alike. The authors of “Gravity and Levity” (unfortunately, I couldn’t find their real names anywhere on the blog) are to be congratulated for such a great post. They have approached the problem in such an effective way, that they have succeeded in producing an expression of the relationship between energy and force in one turn of phrase, and it is one that is so astute, so sublime, that it becomes, quite frankly, life-changing:

“…Between two objects there is only one interaction energy, to which both objects will respond.”

There. Isn’t that just beautiful? Force is an interaction energy. But of course there’s more. Do you see? The interaction energy is describing something that exists OUTSIDE the two objects. It is describing the existence of energy ouside bodies, and it is this energy which acts on bodies, meaning that there must be energy outside all bodies, meaning that we may aswell go the whole hog and say that energy exists EVERYWHERE.

If you recall the definition of potential energy given at the top of this page, it describes energy as being the property of an object. Saying that energy can exist outside objects, or bodies, is not without controversy in today’s climate. Heavily influenced by relativity, and for too many reasons to divulge in this post, the entire field of physics is sold on the concept that all energy is contained inside matter. Even the photon, which is essentially massless, is still considered to be a particle of some description. The void of a vacuum is supposed to be just that – void! There’s not supposed to be anything there, least of all energy! The idea that energy can exist outside matter would present all sorts of problems for modern theory, because it would be forced to admit to the existence of energy, not as an abstract mathematical concept, but as a real, physical substance.

If energy was a real substance that the Universe was immersed in, it would mean, at least in theory, it should be possible to reach out and grab it and use it, and do so, (here comes the dirty word) for “free.” That’s right, I’m talking about “free energy.” No-one respectable in physics likes to discuss free energy. Talking about the possibility of free energy amounts to breaking one of the greatest taboos in science. It’s the definitive “no-no.” If you demand to talk to a physicist about free energy, they’re likely to grab your wrist and give you a nasty chinese burn. Free energy, they will tell you, is impossible. And they’ll probably call you a “crackpot” too. For the moment though, there’s no need for us to enter the forray, and so we’ll stick more closely to the term “interaction energy.”

In comparing the two ice-skaters to the donkey and cart, we are seeing some obvious differences in the way energy interacts. With the ice-skaters, energy acts upon them and repels one from the other in opposite directions. Something quite different happens to the donkey and cart, because unlike the two ice-skaters, these two move together in the same direction. The forces are moving in opposite directions, but now they pass one another, like trains in the night, so to speak. Forces which move from the donkey to the cart, appear to be combined in some way with those forces moving from the cart to the donkey. There is an intimate dance of energy, where one force can be seen moving from left to right, while the other moves from right to left, as it were.

Between the donkey and cart we find there exists a transmission of forces, and with our new-found wisdom, we might also describe it as a transmission of energy. In other words, energy is being transmitted. It is the harness and wagon tree which is responsible for transmitting forces from the donkey to the cart, and from the cart to the donkey. Remember, these forces are not directly derived from an action-reaction pair, which means that they are not the result of an “interaction” between the donkey and the cart. No, the forces are derived from two seperate systems, namely: the donkey and the ground, and the cart and the ground. What we have are two independent sources for the forces, which means that the magnitude of the forces can differ – something that is not possible with an action-reaction pair. This gives a much more rounded explanation as to how the donkey is able to overcome the resistive force of the cart.

Zeno and the Buddha

Posted in Uncategorized by munty13 on September 25, 2010

September 29, 2007, 9:05 pm
Filed under: Buddhism, Philosophy

“Man conquers the world by conquering himself.”

“Greater in combat
Than a person who conquers
A thousand times a thousand people
Is the person who conquers oneself.”
(Dhammapada 8:103)

“Get rid of the judgement, get rid of the ‘I am hurt,’ you are rid of the hurt itself.”
-Marcus Aurelius

“All things are not-self”
Seeing this with insight,
One becomes disenchanted with suffering.
This is the path to purity.
-Dhammapada 20: 279

“Where is the good? In the will. Where is the evil? In the will. Where is neither of them? In those things which are independent of the will.”

“All the suffering of this world arises from a wrong attitude.
The world is neither good or bad.
It is only the relation to our ego that makes it seem the one or the other. “
-Lama Anagorika Govinda

While stoic has come to mean unemotional or indifferent to worldly things, it was a lively and popular philosophy to the ancient Greeks and Romans, influential to it’s many adherents. It appealed to many people of different swaths of life, from the slave Epictetus, to the Roman emperor Marcus Aurelius. And as we’ve seen, it shares some similarities and perhaps even connections with Buddhism. So what is Stoicism and how similar is it to Buddhism? Of course, one person’s definition of Stoicism and Buddhism will differ from another’s so you will have to bear with me as I try to make the connections.

Stoicism was founded by Zeno of Citium in 301 B.C.E. He was a merchant who studied under the Cynics and carried with him some of their thoughts. He spoke from his porch (the stoa), and taught his students to ease their suffering by becoming indifferent or “apathetic” to the rising and falling of emotions and desires, our passions. Interestingly, passion is also equivalent to suffering in Christianity, although the meaning and use of the word may be different. While Christians are to live in line with God, Stoics were to live in line with the rational Universe, which to them was virtuous. Zeno taught that by becoming indifferent to the passions that arise in us, we learn wisdom and learn to live in line with the Universe. Ignorance of this leads to suffering.

By becoming less moved by our emotions, it seems, we learn to be restrained without emotion instead of restrained by emotions. This has translated to the modern definition of stoic. For the Stoics, living stoicly was not just to become a stubborn rock, unmoved by anything. Living in such a way means to live peacefully, with a sense of inner tranquility. This is similar to the Buddhist arahats who become awakened through renunciation and letting go of attachments and desires. Both claim to achieve a state of inner peace from suffering, which are the two traditions’ common goal. The Stoics as well as the Buddha and his disciples sought to free people of suffering. The difference however is that Zeno taught the suppression of the passions whereas the Buddha spoke of becoming open and free from passions.

Apparently, the Stoics also used meditation to practice. The Stoics used “contemplation of death, training attention to remain in the present moment” to gain that indifference that they are now so famous for. Contemplation of death is also a practice used by wandering forest monks in Buddhism. They would settle into cemeteries and contemplate the similarities of their own aging bodies with that of the corpses, the only difference being that of time. And the practice of training attention to remain in the present moment is common to all of Buddhism. Known as dhyana, practitioners begin by focusing on the in and out of the breath, eventually extending their attention to other sensations in their body and mind. It is said that a peaceful state of bliss arises during dhyana, and is emphasized in the tradition of Zen (chan, seon, thien, etc).

Stoics had one major difference in their beliefs with Buddhism. Stoics upheld, paradoxically, both determinism and free will. They believed that the Universe was rational and that it carried people along with it. While people had the free will to act in whatever way they saw fit, voluntarily conforming to the Universe led to the freedom of suffering.

It has been said that Marcus Aurelius was the last Stoic philosopher. Stoicism has since then kept itself in the background, but it’s ideas have re-emerged or taken a different representation. Examples include the French expression, “C’est La Vie”, or the Christian serenity prayer: “God grant me the serenity to accept the things I cannot change; courage to change the things I can; and wisdom to know the difference….”

P.S. After a little search on the Interwebs, it seems some have already beat me to the punch. There’s even a Japanese Zen monks who wrote a blogpost noting the similarities (though I wish he had a greater command of the paragraphical indentation). There are even some class notes for a philosophy class comparing the two.

Learning Maxwell’s Rope

Posted in Uncategorized by munty13 on September 21, 2010

“A person needs a little madness, or else they never dare cut the rope and be free.”
~~Nikos Kazantzakis

In his 1861 paper, “On Lines of Physical Force,” Maxwell elaborates on his desire to investigate “the mechanical results of certain states of tension and motion in a medium,” so that they might be compared with “the observed phenomena of magnetism and electricity.” A copy of the paper, and fascinating it is too, can be seen here, thanks to Wikisource:

Maxwell, just as Faraday had done before him, treated the lines of magnetic force (as revealed by iron filings spilled over paper held above a magnet,) as being real entities. Maxwell describes each filing as being “magnetized by induction,” and that they unite to form “fibres,” and that these fibres will then “indicate the direction of the lines of force.” Maxwell considers the lines of magnetic force as “existing in the form of some kind of pressure or tension, or, more generally, of stress in the medium.” He imagines the stress evoked by the lines of magnetic force as representative of tension, “like that of a rope.” Rope? I wonder what exactly was Maxwell getting at?

If you look at a piece of string under a magnifying glass as you pull on the ends more and more strongly, you will see the fibers straightening and becoming taut. Different parts of the string are apparently exerting forces on each other. For instance, if we think of the two halves of the string as two objects, then each half is exerting a force on the other half. If we imagine the string as consisting of many small parts, then each segment is transmitting a force to the next segment, and if the string has very little mass, then all the forces are equal in magnitude. We refer to the magnitude of the forces as the tension in the string, T.

Tension usually arises in the use of ropes or cables to transmit a force. It is the opposite of compression. Tension is the force with which a rope or line pulls. If we hang a length of rope from the ceiling, and add a weight to the end of it, the pull force created by the weight is called the tension force. The tension force on the rope is equal to the weight of the object.

In general, low-mass objects can be treated approximately as if they simply transmitted forces from one object to another. This can be true for strings, ropes, and cords, and also for rigid objects such as rods and sticks.

Tension force, or tensile force, is an example of a pulling force, and is typically measured in pounds (Ibs) or newtons (N.) Tension force will act on opposite ends of the rope and pull it tight. The force is applied in the direction of the rope. Objects on both ends of the rope will experience a pulling force equal to the tension force. Tension force in the rope is of equal magnitude throughout the rope.

If you think about it, a rope without tension is remarkably useless when you need to get something moved (unless of course, you use it as a whip and order someone else to move it!) For example, a length of rope is not much use in pushing an object across a flat surface – it can’t be used like a rod or stick. However, if the rope is tied round said object, and I pull on it, then I stand a chance of shifting the thing. Tension generated in the rope thus allows me to transmit force. Strings, ropes, cables and chains can only be used in instances where there is pull force. Essentially, somebody, or something has to pull the rope.

If we translate these ideas back to lines of magnetic force, and imagine them as ropes or chains that are fashioned in a series of continous loops, then something must be pulling at them. What is required is a mechanism of some description, one that resides in the material of the magnet, and acts upon these “ropes” by pulling on them. What emerges is something that is highly reminiscent of a pulley system.

A pulley is a simple machine consisting of a string (or rope) wrapped around a wheel (sometimes with a groove) with one end of the string attached to an object and the other end attached to a person or a motor. Pulleys may seem simple, but they can provide a powerful mechanical advantage so lifting tasks may be done easily.

Rope-pulley systems are used when there is a need to transmit rotary motion. The advantage of ropes and chains is that they can transmit force without their performance being affected by length. A common misconception holds that simple machines, such as levers and pulleys, increase forces – but this is not quite correct; levers and pulleys transmit forces, and if the masses of the levers or pulleys are negligible, the input and output forces are equal. In other words, simple machines make it possible to lift heavy objects because they reduce the magnitude of the required input force, but it shall be seen that regardless of whether we use a pulley system or not, the amount of over-all effort needed to move an object always remains the same. A box that weighs 100 Newtons is always going to need an upward force greater than 100 Newtons to lift it.

The effect of a pulley is analogous to the gears of a bicycle; a lower gear makes it easier to turn the pedals, but they must be turned more often for the bicycle to travel the same distance. Likewise, a pulley system makes it easier to lift a load, but the length of cable used to lift the load is greater than the distance the load is lifted. The more pulleys that are used, the easier it is to lift the load, but the longer the length of cable needed to lift the the same distance.

In practise, rope-pulley systems are generally regarded as inefficient due to the force of friction. Ropes tend to slip and stick along pulley wheels, meaning that energy is lost from the system. Another disadvantage with the system is that rope can permanently stretch under tension, robbing the rope of its elasticity and strength. This is why in some installations it is better to use a chain system, whose rigid design allows it to retain tension, and prevents stretching. Chains can also be made to fit on gears so that slipping is not a problem. For these reasons, if I were to try and imagine how lines of force might physically appear, then I would come up with a chain system. I then picture these chains as being pushed and pulled by a multitude of sprockets and gears, all whizzing frantically away inside the magnetic material.

A sprocket is a profiled wheel with teeth that meshes with a chain, track or other perforated or indented material. It is distinguished from a gear in that sprockets are never meshed together directly, and differs from a pulley in that sprockets have teeth and pulleys are smooth.

Sprockets are used in bicycles, motorcycles, cars, tanks, and other machinery either to transmit rotary motion between two shafts where gears are unsuitable or to impart linear motion to a track, tape etc.

Effectively, what a chain or rope-pulley system allows us to do is “trade” force for distance – which is the exact same principle by which a simple lever works. Pulleys lengthen the distance of the rope, thereby increasing the distance over which the force acts. By increasing distance, the system is able to increase the amount of energy that it stores. I tend to imagine it somewhat as the total force that is required to lift an object, as being broken down into smaller, more manageable units, and then spread over a greater distance.

With four wheels and four ropes, a pulley cuts the lifting force you need to one quarter. But you have to pull the end of the rope four times as far.

It is interesting that as a direct consequence of the increase in distance, we are now seeing that action takes place over a longer time. This is in compliance with the “golden rule” of mechanics, in that the mechanical advantage derived will always be accompanied by a loss in displacement, or in other words, time. This further suggests that force, regardless of the amount, always seems to be transmitted at the same speed. It means that Archimedes claim to be able to lift the world, given a lever that was long enough, is theoretically possible – just as long as we’re prepared to wait a few million years!

Archimedes knew that by applying a lever, one could lift the heaviest of weights by applying even the weakest of forces. One had only to apply this force to the levers longer arm and cause the shorter one to act on the load. He therefore thought that by pressing with his hand on the extremely long arm of a lever he would be able to lift a weight, the mass of which would be equivalent to that of the earth.

Let us imagine for a moment that Archimedes had at his disposal “another earth” and also the point of support he sought. Further imagine that he was even able to manufacture a lever of the required length. I wonder if you can guess the amount of time he would need to lift a load equivalent in mass to that of the earth, by at least a centimeter? Thirty million million years- and no less!!

When you push down on a lever, the force you push with is multiplied by the length of the lever to produce a torque. The torque of a force is the turning effect of the force about a point. Torque is, by definition, the product of a force applied in a rotational motion or twisting force. It is a turning force. It is the force that produces rotation. Because pulley systems use rotational forces, they too use the principle of torque.

Torque, also called moment or moment of force, is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist.

Whenever I think of torque, I’m always reminded of the cordless screwdriver I would wield at work. You may have noticed that most cordless screwdrivers have a torque setting. To remove a screw from the wall, one that was wedged in especially tight, it is necessary for the torque to be switched over to a high setting. If the torque is not high enough, the screw will not budge, because the drill is simply not strong enough. It is torque which is defining the drill’s strength. If the screw is especially tight, and the torque setting is strong enough, you might find that the screw will still not move, but now it is your arm that the drill wants to spin round!

Both levers and the inclined plane lower the force required for a task at the price of having to apply that force over a longer distance. With wheels and axles the same is true: a poweful force and movement of the axle is converted to a greater movement, but less force, at the circumference of the wheel. In a circular geometry, torque is a more useful concept than force and distance. The wheel and axle can be thought of as simply a circular lever, as shown in Figure 5 [below]. Many common items rely on the wheel and axle such as the screwdriver, the steering wheel, the wrench, and the faucet.

If you want to increase the load that a pulley system may carry, then it is necessary to increase torque. The greater the torque, the more energy stored. Torque units contain a distance and a force. Torque is, in effect, the product of the force and the length of the lever arm. Understood in this way, it is clear that there are two ways of increasing torque; either increase the force or increase the length of the lever arm. As torque is a product of force and distance, one may be “traded” for the other.

Torque, like work is measured in pound-feet (lb-ft) However, torque, unlike work, may exist even though no movement occurs. A good example of this is the torque exerted when you try and loosen a very tight nut. As you are pulling on the wrench you are exerting a force, but not until the the nut moves has this torque resulted in work.

Work is done when we use a force (a push or pull) to move something over a distance. Energy is needed to move a force through distance. In calculating work done, two things need to be measured: the amount of force and the distance that it moves. Thus, it is important to make the distinction that work is a measure of what is done, not the effort applied in trying to move it. Where energy is the capacity for doing work, it means that both these quantities are measured by the same unit – in a sense the two are equivalent. Work and energy are merely different ways of looking at the same thing.

Because of the way it is defined, there will be plenty of times in my life (some might say the story of my life!) where I will spend energy, but no work is done. That’s because it’s possible to for me to exert a force where we see nothing move. If I remain still, holding a dumbell above my head, a physics textbook will tell me that I have done no work because my actions do not involve motion – no distance, no work. A physicist might be cheeky enough to tell me that I’m not doing any work – but given a few minutes, it won’t stop my arms from feeling like they want to fall off! The fact is, I am still spending energy even though there is no evidence of “work done.”

People often confuse energy, power, and force. Force is a push or a pull on an object or body. The strength of the force used and the distance through which it moves determine the amount of work done. Two factors determine the amount of work done. One factor is the amount of force applied. The other is the distance the object moves. In physics, work occurs only when the force is sufficient to move the object. In other words, work is a measure of what is done, not the effort applied in attempting to move the object. People do work when lifting, pushing, or sliding an object from one place to another. They do no work when holding an object without moving it, even though they may become tired.

If work and energy are both a measure of force times distance, what does this mean if we extract distance from the equation? Of course we don’t even have to go as far as removing it, but simply reduce its quantity to zero. Thus, energy now becomes equal to force times zero distance, surely meaning that force is only describing energy in some way? Looking at this equation again, it is also interesting that distance too emerges as being another means of describing energy.

Returning to the idea that a lever, or pulley system are able to reduce the magnitude of the necessary force by applying force over a longer distance, it seems that this “trade” of force for distance would suggest that the system is storing force in some way. Indeed, if we were to expand on this a little further, it might be said that distance itself is responsible for storing this force.

Now, I must tread carefully in how I explain the idea of how a system might “store force.” As any physics textbook will tell you, it is not possible to “store force” as such – rather, it is energy which is stored, and not force. This is because force, at least in terms of how some physicists describe it, is not energy. Energy and force are treated as being something of different concepts, but this does not mean they are incompatible. I think it’s possible to argue that energy and forces are merely different guises of the same entity.

Now there may well be some who are more familiar with physics, whose sensibilities are more delicate than others, and whom after reading that last sentence, might just have sprayed the wall with coffee. To these persons I apologise profusely for any mess caused, but I believe that there are good grounds for such a theory, and I’m going to discuss them in my next post (those that are consuming drinks have been warned!)

Many thanks to:
Physicists look back: studies in the history of physics By John Roche
What is electricity? By John Trowbridge, I. Bernard Cohen