Maths – Page 4 – Me on the net

Puzzles

Pieces of a puzzle (Photo credit: Wikipedia)

I’ve been musing on the human liking for puzzles. I think that it is based on the need to understand the world that we live in and predict what might happen next. A caveman would see that day followed night which followed the day before, so he would conclude that night and day would continue to alternate.

It would become to him a natural thing, and in most cases that would be that, but in a few cases an Einstein of the caveman world might wonder about this sequence. He might conclude that some all powerful being causes day and night, possibly for the convenience of caveman kind, but if his mind worked a little differently he might consider the pattern was a natural one, and not a divinely created phenomenon.



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Puzzling about these things is possibly what led to the evolution of the caveman into a human being. Those cavemen who had realised that the world appear to have an order would likely have a survival advantage over those who didn’t.

The human race has been working on the puzzle of the Universe from the earliest days of our existence. Solving a puzzle requires that you believe that there is a pattern and that you can work it out.



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The Universal pattern may be ultimately beyond our reach, as it seems to me that, speaking philosophically, it might be impossible to fully understand everything about the Universe while we are inside it. It’s like trying to understand a room while in it. You may be able to know everything about the room by looking around and logically deducing things about it, but you can’t know how the room looks from the outside, where it is and even what its purpose is beyond just being a room.

Solving a puzzle usually involves creating order out of chaos. A good example is the Rubik’s Cube. To solve it, one has to cause the randomised colours to be manipulated so that each face has a single colour on it.

A jigsaw puzzle is to start with is chaos made manifest. We apply energy and produce an ordered state over a fairly long time – we solve the jigsaw puzzle. After a brief period of admiration of our handiwork we dismantle the jigsaw puzzle in seconds. Unfortunately we don’t get the energy back again and that’s the nature of entropy/order.

Many puzzles are of this sort. In the card game patience (Klondike), the cards are shuffled and made random, and our job is to return order to the cards by moving them according to the rules. In the case of patience, we may not be able to, as it is possible that there is no legal way to access some of the cards. Only around 80% of of patience games are winnable.

Other games such as the Rubik’s Cube are always solvable, provided the “shuffling” is done legally. If the coloured stickers on a Rubik’s Cube are moved (an illegal “shuffle”) then the cube might not be solvable at all. A Rubik’s Cube expert can usually tell that this has been done almost instantly. Of course, switching two of the coloured stickers may by chance result in a configuration that matches a legal shuffle.

When scientists look at the Universe and propose theories about it, the process is much like the process of solving a jigsaw puzzle – you look at a piece of the puzzle and see if it resembles in some way other pieces. Then you look for a similar place to insert your piece. There may be some trial and error involved. Or you look at the shape of a gap in the puzzle and look for a piece that will fit into it. One such piece in the physics puzzle is called the Higgs Boson.

The shape is not the only consideration, as the colours and lines on the piece must match the colours and lines on the bit of the puzzle. In the same way, new theories in physics must match existing theories, or at least fit in with them.

Jigsaw puzzles are a good analogy for physics theories. Theories may be constructed in areas unrelated to any other theories, in a sort of theoretical island. Similarly a chunk of the jigsaw could be constructed separately from the rest, to be joined to the rest later. A theoretical island should eventually be joined to the rest of physics.



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Of course any analogy will break down eventually, but the jigsaw puzzle analogy is a good one in that it mirrors many of the processes in physics. Physical theories can be modified to fit the experimental data, but you can’t modify the pieces of jigsaw to fit without spoiling the puzzle.

The best sorts of puzzles are the ones which give you the least amount of information that you need to solve the puzzle. With patience type games there is no real least amount of information, but in something like Sudoku puzzles the puzzle can be made more difficult by providing fewer clues in the grid. A particular set of clues may result in several possible solutions, if not enough clues are provided. This is generally considered to be a bad thing.

Some puzzles are logic puzzles, such as the ones where a traveller meet some people on the road who can only answer “yes” or “no”. The problem is for the traveller to ask them a question and deduce the answer from their terse replies. The people that he meets may lie or tell the truth or maybe alternate.

Scientists solving the puzzle of the Universe are very much like the traveller. They can question the results that they get, but like the people that the traveller meets, the results may say “yes” or “no” or be equivocal. Also, the puzzle that the scientists are solving is a jigsaw puzzle without edges.

Everyone who has completed a jigsaw puzzle knows that the pieces can be confusing, especially when the colours in different areas appear similar. For scientists and mathematicians a piece of evidence or a theory may appear to be unrelated to another theory or piece of evidence, but often disparate areas of study may turn out to be linked together in unexpected ways. That’s part of the beauty of study in these fields.



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Cycling through life

I’ve been thinking about cycles. A cycle is something that repeats, like the rotation of a wheel, or the rotation of the earth. A true cycle never has an end until something external affects it, and the same is true for the start of a cycle in that something external to the cycle has to happen to start the cycle off.

Conceptually, a perfect cycle would be something like a sine or cosine wave. It’s called a wave because if plotted (amplitude versus time) it resembles a wave in water, with its peaks and troughs. It’s fundamental constants are the distance between the waves and the amplitude of the maximum of each cycle.



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The sine and cosine waves are derived from a circle – when a radius of the circle rotates at a constant rate, the sine and cosine can be measured off a diagram of the circle and the rotating radius. The point where the radius touches the circle is a certain distance above the horizontal diameter of the circle and is also a certain distance to the right of the vertical diameter of the circle. If the radius of the circle is one unit, then the sine is the height and the cosine is the distance to the right.

As the radius sweeps around the circle the sine of the angle it makes to the horizontal diameter goes from zero when the angle is zero and the radius lies along the horizontal diameter to one unit when it is at 90 degrees to the horizontal diameter. When the angle increases further, the sine decreases until it is again zero at 180 degrees, and as it sweeps into the third quadrant of the circle it goes negative, increasing to one unit again at 270 degrees (but downwards) and finally returning to zero at 360 degrees. 360 degrees is (simplistically) the same as zero degrees and so the cycle repeats.

The cosine starts at one unit at zero degrees, decreases to zero units at 90 degrees, decreases further to one unit downwards (conventionally called minus one) at 180, then increases to zero again at 270 degrees and finally to complete the cycle, it increases to one unit at 360 degrees.

When plotted against the angle, the sine and cosine produce typical wave shapes, but shifted by 90 degrees. If the radius rotates at a constant speed, the sine and cosine can be plotted against time, which produces a curve like the track of a point on a wheel as it is rolled at constant speed.

animation of rolling circle generating a cyclo... — animation of rolling circle generating a cycloid; black and white, anti-aliased (Photo credit: Wikipedia)

While these curves are pleasingly smooth and symmetrical, in the real world we can only get close to these ideals. A wheel will slip on the surface that it is turning on, friction on axles slows a freely spinning wheel, lengthening each “cycle” by small amounts, altering the curves so that they are minutely different at different times.

If an ellipse is drawn inside the circle such that it touches the circle at the points where circle touches the horizontal diameter, the radius will cut the ellipse at some point and it turns out that the curves plotted from the intersection point are still sine and cosine curves. However the heights or amplitudes of the curves are different.

An ellipse is approximately the shape of the orbit of a planet about a sun for reasons that I won’t go into here. It isn’t an exact ellipse, mainly because of the effects of other bodies, though it is accurate enough that things like the length of a planet’s year doesn’t vary significantly over many lifetimes. The most accurate atomic clocks can be used to measure the differences but they only need to be adjusted infrequently by very small to keep in line with astronomical time.

To account for these errors the astronomer Ptolemy devised an ingenious scheme. An ellipse can be looked on as result of imposing a smaller cycle of rotation on a larger one, a bit like having a jointed rod, with the larger part connected to the centre of a circle and the smaller part connected to the end of the larger part. If the smaller rod rotates at a constant speed at the end of the larger rod then the tip of the smaller rod draws out a more complex path. If the correct rotation rate is chosen, as is the correct starting angle between the two rods, then the tip of the smaller rod will draw out an ellipse.

Circles on an old astronomy drawing, by Ibn al... — Circles on an old astronomy drawing, by Ibn al-Shatir (Photo credit: Wikipedia)

Ptolemy suggested that the variations from an ellipse could be modelled by imposing other smaller cycles on the first two cycles, and indeed this does result in more accurate descriptions of the orbits.

Ptolemy got a bad press because he believed that these cycles were real manifestations of reality, and his system of epicycles on epicycles on epicycles was hugely complex, but his system can be extended to model any physical system to any degree of accuracy required. It can be proved mathematically that his process exactly matches any equation if the process is taken to infinity. It’s one method of fitting a curve to arbitrary data.

In particular Ptolemy was able to use his methods to calculate the distance of the planets, which was a singular success for his method. It is the sort of technique which is used today to calculate the orbits of newly discovered comets – when it is discovered the astronomer has only one point of location so he/she cannot predict the orbit. When the comet’s next position is measured, the astronomer can start to predict the orbit. A third observation can vastly improve the accuracy of the calculation of the orbit.

Subsequent observations allow the orbit to be refined even more until the astronomer can accurately predict the complete orbit of the comet and its periodicity using something like Gauss’ method as described in the link. In essence the procedure of observation, calculation and prediction/re-observation is the same as Ptolemy used, even though the underlying physics and philosophy is different. Ptolemy’s ideas may seem quaint to us, but in his time we knew much less about the universe, and, given the era in which he was working his ideas were not that outlandish. He did not even know that the planets revolved around the sun. He didn’t know about gravity as a universal force.

In the Zone

(Ugh! I forgot to post this last week. My apologies)

Programming, as I’ve probably said before is a strange occupation. You start with a blank sheet, steal bits and pieces from where ever you can find them and glue them together modify them, add some bits of original (to you) code and try to think of all the possible ways your program can go wrong.

Then you try and break your code (and usually succeed at first). Programming is still very much an art form. Of course things have changed a lot over the years, and we are able to use the work of others to help us in our endeavours, but my first paragraph is still true.

In the beginning there was “Hello World”. This is probably the simplest program that does something visible. It doesn’t take any information in and its output, the words “Hello” and “World” are not very useful in themselves. Actually, I’d say that there is an even simpler program that takes no input, produces no output, and in the process changes nothing. A “null” program if you like.

A programmer writing a new program may well jump in and start coding by grabbing some other code that he or she has access to, but that stolen code was developed, ultimately, from “Hello World” or the null program.

Picture of "hello world" in C by Use... — Picture of “hello world” in C by User:aarchiba. (Photo credit: Wikipedia)

A good programmer is one who steals code from elsewhere and modifies it to do what he or she wants. There is no stigma of plagiarism attached to this process, and it is in fact strongly encouraged that programmers share code. A spoof news item that I came across stated that all programming courses would be replaced with a course on how to find code on “Stack Overflow“. I’ve been unable to find the link again, but I believe that the item was on “The Onion“, a well known satirical website.

Of course, such a process may propagate errors or bugs across many programs, but it is such an effective strategy that it is used more often than not. If code exists to solve a problem then it would be silly to pass it by and write it ones self, maybe introducing bugs to the code. The advantage of “borrowing” code is that while errors and bugs may be in the borrowed code, many eyes will have looked at the code and there is nothing more that programmers like than pointing out bugs in the code of other programmers.

Stack Overflow allows anyone to post code and comment (up to a point), so code posted may not be top quality, but other programmers are quick to jump if they see bugs or inefficiencies in code. Contributors will also point out code which doesn’t follow standards or conventions in the programming language being used. This is considered useful, as the code, if modified, can be accessed and understood more easily, and may often be safer and free of more bugs than unconventional code.



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When a program is written it starts out as literally a few lines of code or even an empty file. Any programmer knows that a program grows swiftly and in ways that can’t be foreseen until it may be of enormous size. It won’t be all written in one sitting but is usually written in stages. I personally like to write my programs in very small chunks, building on what has gone before. I think that many programmers use this process, though there may be others who write a sizeable chunk of code before testing it.

Ah, testing! Testing is the less enthralling parts of writing programs. Any program must be tested, to ensure that it does all that is required and nothing else. Generally the program being written doesn’t do all that is required and does things that shouldn’t happen, and initially it is likely to crash or produce cryptic error messages under some conditions.



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Testing is supposed to reduce the number of such unwanted happenings, and the programmer may do some rudimentary testing and may handle at least some errors. However the programmer will realise that users who are unfamiliar with how the program is written may well do something that he has not expected.

So clever people have developed ways of automatically testing programs. To do this they have had to write the programs that are used to test programs. And of course those testing programs may have bugs. You can see where that leads to!

Zebra (programming language) (Photo credit: Wikipedia)

When a programmer knows a programming language really well, he is able to literally think in that language. The word “literally” has been devalued in recent time, but I am using it in the true sense of the word. This is hard for some people to understand as they think of language as something like French or Tagalog, and they can’t understand how one can think in a programming language, which is qualitatively different from a spoken language.

An interesting thing happens when a true programmer is programming something. His thought processes become so involved in the process of programming and in thinking in the programming language that he loses track of the outside world. That’s why programmers are whimsically thought to subsist on fizzy energy drinks and dialled in pizza. It is because those things are easily acquired and the programmer can keep programming.



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A programmer “in the zone” is so embedded in the world of the program that he or she may often be reluctant to leave that world and respond to irritations like bodily needs and colleagues. I doubt that there is a real programmer who has not surfaced from a deep dive into the depths of a programming problem and realised that all his colleagues have left and it is late at night or very early in the morning. That’s the reason programmers stay after all other people have left – they know that they can slay the current bug with just a few more changes and a few more runs of the program.

The zone has similarities to the state of meditation. While meditation is passive though, programming is an active state. In both cases the person basically disconnects from the world, so far as he or she can, and the concentration is directed internally. Now that I think about it, any deep thought, be it meditation, programming, or philosophising, even playing a sport at a very high level, needs such concentration that much of the world is disregarded and the exponent enters the zone.



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Time Travelling

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Time stories have been around for years, and yet we have seen no time travellers. If there are time travellers, then they are hiding from us. That would be pretty hard, since society and language are changing all the time, and they would most likely look out of place.

If they came from the past, their mannerisms, clothing and use of language would likely give them away. Imagine someone from the time of Sir Francis Drake appearing in the current era. He would quickly be spotted.



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For someone travelling from the past, the issue is that he or she would not know what to expect as the present is the future of times past, so they would not be able to prepare themselves for the future, as they would consider it.

If a citizen of the future where to travel to the current time, he or she could presumably prepare his/herself for what would be the past to him/her. The time traveller could learn about our era and equip him/herself with clothes, money and other things from our era and would be able to learn the idioms of the language of the current time, as well as the ethics and morals of the era.

There are web sites on the Internet which claim to have proof of time travel (I’m not going to link to one – a Google search will bring up many of them). In most cases the evidence is far from compelling, relying on blurry photographs and dubious eye witness accounts.

I’ve recently been scanning my old photographic slides and in one of them, from the early 1980s or late 1970s, the person in the picture appears to be holding an iPad! What in fact she is holding is a place mat, with a cream coloured border and relatively dark picture on it. This demonstrates how easily “evidence” of time travel can be found if you look hard, and if you strongly believe that time travel is possible.

Every person, every object, even every elementary particle has position which can be measured (leaving aside the issue of Heisenberg’s Uncertainty Principle and Quantum Physics for now) at any moment in time. The four dimensions, three of space and one of time uniquely identify an event in the life of the person, object or particle.

These four coordinates represent a single point in a four dimensional space. Since we find four dimensions hard to visualise, this space/time is usually represent by a depiction of a three dimensional space of two space and one time coordinate axes. The path of a person, object or particle through life consists of a single unbroken line in the four dimensional space.

Note that in time dimension, if time travel is not possible, for every value of the time coordinate there will only be one point of the person’s life line. In other words, while the person could visit and revisit the same three dimension spot in space many time, they will only pass through a particular time once and once only. A person’s now is unique.

Time travel means that a person could pass through the same moment in time multiple times, and the possibility arises of loops in time. It seems obvious that the same event could not appear on the time traveller’s life line. In other words the loops in a life line would not cross.

To get from one event in space/time to an earlier event in space/time, the person could either travel through the intervening times or just jump from the first event to the second. In other words time travel if it is possible would be represented by a line going backwards in time or it could be discontinuous, with a break the in the person’s time line.

From the point of view of an observer, a discontinuous time line for the time traveller would be seen as a sudden appearances from nowhere and a later sudden disappearance. If the time line is contiguous, the observer would again see a sudden appearance of the time traveller, and then two instances of the time traveller both apparently travelling into the future at one second per second.



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One will be the time traveller doing just that, and the other will be the observer’s view of the time traveller as he travels backwards in time. Eventually the observer will see one of the instances (people!) merge with another instance of the time traveller and disappear. Since we don’t normally observe such sudden appearances and disappearances it’s very tempting to say that time travel does not happen.

To see what I mean, take a piece of paper and draw a line from top to bottom with a look in it. Now horizontal lines across the page represent time as seen by the observer. If you move a ruler down the page, at first there is a single line, the time line of time traveller. But at the point that the time travellers has travelled back to, suddenly there are (apparently) three lines travelling down the page. At the point that the time traveller travels from, two of the lines merge and disappear.

English: Meter stamp catalog image, three stac... — English: Meter stamp catalog image, three stacked horizontal lines (Photo credit: Wikipedia)

If time travel is discontinuous the observer would first see one instance of the traveller, and then another instance would pop into existence. The two would coexist for a time, and then one instance (the original one) would disappear, with the second continuing to exist.

As I said, it’s tempting to say that this proves that time travel is not possible. Certainly, a macro level we don’t see people appearing and disappearing so it is definitely very unlikely, though reasons for this not to be noticed can be constructed.

At an atomic level though, we do see spontaneous generation of particles, into a particle and its anti-particle. If the anti-particle the meet another particle and they annihilate one another, this could be construed to be a single particle with the anti-particle being the particle travelling back in time.

This is bad news for time travellers. To travel back in time by this method, the time traveller would have to be zapped into a burst of energy and an anti-traveller who would then travel back in time to the earlier time when a burst of anti-energy would be required to zap the anti-traveller into another instance of the time traveller. These occurrences would be likely to destroy the integrity of the time traveller’s body. That is it would destroy it.

Dwelling in the past

Some accounts have the wizard Merlin living his life from future to past, in the opposite direction to the rest of us. This meant that to him, what would a final farewell to us would be a joyous first meeting for him, and a first meeting would a sad goodbye. He remembered the future, but the past was a complete mystery to him.

The trolls in Terry Pratchett’s Diskworld had similar ideas. They considered that they travelled through time from past to future, as is normally understood, but they also considered that we were facing backwards in time as we travelled through it. This was conjectured by the trolls, to explain the fact that we can see where we are going when we travel in whatever direction we choose but we cannot see where we are going in time. Similarly we can’t see where we have been when walking from one place to another, but we can see where we have been in time.

Scientists have no difficulties with direction in time. In an equation with a time variable in it, one can trace the changes to other variables from the nominal zero point to see what would occur in the future, merely by incrementing the time variable. A scientist can predict the trajectory of a thrown item (a parabola) merely by substituting later times into the time variable in the equation.

y =ax² + bx + c

The scientist should compare these results to experiment and find, that this more or less works. Lets say though, that the scientist is an astrophysicist investigating an asteroid or other object on a parabolic trajectory around a larger object, like a moon or planet. (A parabolic trajectory is the trajectory which divides objects in hyperbolic trajectories which are not bound by the larger body’s gravity, from those in elliptical trajectories where the object is bound by the larger body’s gravity).

The astrophysicist would be able to track the trajectory of the body backwards in time, simply by substituting negative values for time into his equations. He would be able to see where it had been. However this process of substituting negative time values into the equations only works so far. At some point the body will feel the influence of other objects and the retrograde trajectory will deviate from the values predicted by the parabolic equation.



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Of course, if the trajectory is projected forwards far enough similar considerations arise. Eventually some other body will divert the body away from the parabolic trajectory. However in the region in which the parabola applies, the behaviour is symmetrical with respect to time. From a film of the event one could not tell whether or not the film was being run forwards or backwards.

Of course, if a film is run backwards for any length of time, it becomes obvious that something is wrong. Things fall upwards, and broken crockery comes back together again. People walk backwards.



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On a closer look, people can intentionally walk backwards and it is possible that a spring or other mechanism could be used to shoot things upwards, but it is a lot harder to imagine a way of reversing the breakage of the crockery. It implies that some process involved in the breaking of crockery is not reversible at a macro level.

It is likely that the process in question is at the molecular level or slightly above. To rejoin two broken surfaces spontaneously would presumably require that the molecules be in the correct positions and that a little burst of energy (equivalent to the little burst of energy that comprises the sound that the crockery makes in breaking and any heat release) be supplied at an instant in time. The weak bonds between the parts of crockery would need to be created, and that is really difficult.



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There’s therefore a discrepancy between the scientist’s view of the world, through his equation which time-symmetrical, and the man in the street’s view of the world, which is asymmetrical with respect to time. In fiction this asymmetry is used to good effect, when the protagonist may “wind back time”, to write a wrong or divert history to an alternate course.

When we consider space we usually imagine a three dimensional space. Events happen at locations in this space and three coordinates are enough to locate an event in space. Every possible point in space has its set of unique coordinates. It is common to add an extra dimension for time, making the space four dimension and consequently difficult to imagine successfully.

All events in space and time have a location and time and are represented by a point in the four dimensional space. The path of a particle is a line within this space, and any point on the line represents the position of the particle at a particular time. Positions on the line are either before or after this point, so it constitutes a “now” point for the particle. There is no actual motion over the line, since in the 4-D space all points represent the past and the future of the particle. They are already there.

The Klein bottle immersed in three-dimensional... — The Klein bottle immersed in three-dimensional space. (Photo credit: Wikipedia)

To move to an earlier time, all we need to do is move to an earlier spot on the line. We would need to either travel back along the line at a speed of so many seconds per seconds where the first “seconds” is seconds measured in the space and the second “seconds” is some other time scale or we could jump out of the space-time completely and return to the requisite point earlier in time. We’d need to do the latter in some other space-time that embeds the original space-time, adding a number of extra dimensions to the mix.

Both options require the addition of extra dimensions, which while possible complicates the situation unacceptably to my mind. The process of adding extra dimensions could be repeated and go on forever, so we end up with infinite dimensions. I believe that it is correct to employ Occam’s razor at this point and declare that it appears unlikely that we could either roll back time or jump to an earlier point in time because of the implication that we would needs infinite dimensions as a result.

Sinistralism

English: Photo of the sinistral shell of Achat... — English: Photo of the sinistral shell of Achatina fulica. Locality: Mauritius. (Photo credit: Wikipedia)

At one time banks and post offices used to tie down the pen that they provided for people to sign things, such as cheques, with a piece of string or chain to prevent customers from stealing the pens. These days you are more likely to have a bunch of pens (with the bank’s logo) pressed on you.

Anyway, I was always discomforted by these tied down pens as I am left-handed and the string or chain was always fixed to the right of the counter and rarely was long enough for me to easily sign my cheques with left hand. As a result I was cramped up and twisted round as I manoeuvred the cheque book closer to the right hand side.

This is only one of the many times that my sinistralism has left me disadvantaged. Corkscrews and scissors, even power tools, wood screws and nuts and bolts are all designed for the majority who are right-handed. (I believe that the correct term would be dextralist). Can openers are the things that I find most tricky to use.

However, sinistralists living in a dextral world soon learn to use right handed tools, to at least some level of competence. There are tools made specifically for left-handed people and it is quite funny to give one to a right-hander. They don’t have a clue! This is because they have not had to learn to use tools with a sinistral configuration, whereas a sinistral has at least a level of competence with right-handed tools since they are everywhere.

The Universe obviously has a basic chirality or handedness which is a bit odd to say the least. It’s interesting to wonder why chirality exists. Is it just so that we can easily remove corks from bottles of wine? “Thanks, Deity, that just what we needed. Will you take a glass?”

Chirality makes it hard for some creatures. Imagine that you are a mollusc with a left-coiled shell and all the other mollusc are right-coiled. You’d find partners to be rare. Of course, it can’t make reproduction too difficult or the left-coiled molluscs would become extinct within a generation or so. (That’s probably too simplistic, but let’s not quibble).

Chirality allows us to have streams of traffic travelling both ways on a single road, but I’m sure that Deity would have considered that and arranged for some way that we could duplex our highways. Allowing a way for solid matter to inter-penetrate would be one, but perhaps “both directions” implies chirality anyway and in a non-chiral universe there would only ever be one way.

In some universes (and maybe in this one for all I know) there may be cases of even more complex situations. One reflection of a chiral object changes it into another object that can’t be superimposed on the original object. A second reflection does allow it to be superimposed on the original. Imagine a universe where three or more reflections are need to allow the superimposition! In that universe the motorways and autobahns would need at least three carriageways.

The spin of some sub-atomic particles has an interesting spin characteristic. The linked Wikipedia article says in one place:

However, if s is a half-integer, the values of m are also all half-integers, giving (−1)^2m = −1 for all m, and hence upon rotation by 2π the state picks up a minus sign.

Rotation through 2π is a rotation through 360 degrees. So a particle with a spin of ½ turns into its mirror reflection on one full rotation. At least I think that’s what that means. Two full rotations and it is back to its original orientation (or at least, its original spin value).

In the macro world two reflections of an object create an image that can be superimposed on the original. That means that if you look at the reflection of your reflection you see yourself as others see you rather than your mirror image. You are so much more used to seeing your reflection that it can be difficult to comb your hair while looking at a doubly reflected image of yourself.

Chiral objects are not symmetrical. Reflections of symmetrical objects can be superimposed on the original object however (after being moved and turned). Symmetry and asymmetry are part of the fabric of the universe that we live in and I think that having both allows for much more complexity in our universe than would be possible in a universe without the symmetry/asymmetry dichotomy.

Volvo fire engine of the "Bedfordshire an... — Volvo fire engine of the “Bedfordshire and Luton Fire & Rescue Service”, with “FIRE” in mirror writing (“ERIF”) (Photo credit: Wikipedia)

One of the great questions of philosophy is “Why does anything exist?” or “Why does something exist rather than nothing?” The real answer to this question is that no one knows and no one is likely to know. I suppose the Deity might if the Deity exists Itself. Given that the universe exists why is does it embody the concept of left and right? Again, no one knows or is likely to know, or so it appears at this time.

One reason may be that if asymmetry did not exist one could probably not travel from A to B. If one started from A to travel to B in a universe without symmetry and hence without asymmetry, and someone else was travelling in the opposite direction then to pass one another the two travellers would have to pass on one side or the other of the other traveller.

It seems to me that the concept of “side” implies the concepts of “left” and “right”. Asymmetry comes in because, to either traveller, one passes to the left or right of the oncoming traveller who passes to the right or left of you. Symmetry comes in because, to pass one another both traveller has to keep to their left or their right to successfully pass one another.

Of course, that is only true for our universe. It is conceivable that a universe could exist where symmetry and asymmetry do not exist, but it is a universe which we would, most likely, be unable to conceive of, except in the broadest terms.



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Documentation

Documentation. The “D word” to programmers. In an ideal world programs would document themselves, but this is not an ideal world, though some programmers have attempted to write programs to automatically document program for them. I wonder what the documentation is like for such programs?

To be sure if you write a program for yourself and expect that no one else will ever look at it, then documentation, if any, is up to you. I find myself leaving little notes in my code to remind my future self why I coded something in a particular way.

Such informal documentation can be amusing and maybe frustrating at times. When reading someone else’s informal documentation such as “I’m not sure how this works” or “I don’t remember why I did this but it is necessary”. More frequently there will be comments like “Finish this bit later” or the even more cryptic “Fix this bit later”. Why? What is wrong with it? Who knows?

The problem with such informal in code documentation is that you have to think what the person reading the code will want to know at this stage. Add to this the fact that when adding the comments the programmer is probably focussing on what he/she will be coding next, the comments are likely to be terse.

Add to this the fact that code may be changed but the comments often are note. The comment says “increment month number” while the code actually decrements it. Duh! A variable called “end_of_month” is inexplicably used as an index to an array or something.

English: Program Hello World Česky: Program He... — English: Program Hello World Česky: Program Hello World (Photo credit: Wikipedia)

Anyone who has ever done any programming to a level deeper than the usual beginner’s “Hello World!” program will know that each and every programmer has tricks which they use when coding, and that such tricks get passed from programmer to programmer with the result that a newcomer looking at code may be bamboozled by it. The comments in the code won’t help much.



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Of course such programming tricks may be specific to the programming language used. While the same task may be achieved by similar means at a high level, the lower level of code will be significantly different. While that may seem to impose another barrier to understanding, I’ve found that it is usually reasonably easy to work out what is going on in a program, even if you don’t “speak” that particular language, and the comments may even help!

While internal documentation is hit and miss, external documentation is often even more problematic. If the programmer is forced to write documents about his programs, you will probably find that the external documentation is incomplete, inaccurate or so terse it is of little help in understanding the program.

In my experience each programmer will document his/her programs differently. Programmers like to program so will spend the least possible amount of time on documentation. He/she will only include what he/she thinks is important, and of course, the programmer is employed to program, so he/she might get dragged away to write some code and conveniently forgets to return to the documentation.

If the programmer is at all interested in the documentation, and some are, he/she will no doubt organise it as he/she thinks fit. Using a template or model might help in this respect, but the programmer may add too much detail to the documentation – a flowchart may spread to several pages or more, and such flowcharts be confusing and the source of much frustrated page turning.

Of course there are standards for documentation, but perhaps the best documentation of a program would be to specify the inputs and specify the outputs and then a description of how the one becomes the other at a high level. As I mentioned above a programmer will probably give too much detail of how inputs become outputs.

Documentation tends to “decay” over time, as changes are made to the program and rarely is the documentation revisited, so the users of the program may need to fill in the gaps – “Oh yes, the documentation says that need to provide the data in that format, but in fact that was changed two years ago, and we now need the data in this format”.

The problem is worse if the programmer has moved on and gone to work elsewhere. Programmers tend to focus on the job in hand, to write the program to do the job required and then move on to the next programming task, so such code comments as there are will be written at the time that the programmer is writing them. Such comments are likely notes to the programmer him/herself about the issues at the time that the program is being written.



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So you get comments like “Create the app object” when the programmer wants a way to collect the relevant information about the data he/she is processing. Very often that is all that one gets from the programmer! No indication about why the object is needed or what it comprises. The programmer knows, but he/she doesn’t feel the need to share the information, because he/she doesn’t think about the next person to pick up the code.

I don’t want to give the impression that I think that documentation is a bad thing. I’m just pondering the topic and giving a few ideas on why documentation, as a rule, sucks. As you can imagine, this was sparked by some bad/missing documentation that I was looking for.

Open source software is particularly bad at this as the programmer has an urge to get his program out there and no equal urge to document it. After all, a user can look at the code, can’t he/she? Of he/she could look at the code, but it is tricky to do so for large programs which will probably be split into dozens of smaller ones, and the user has to be at least a passable programmer him/herself to make sense of it. Few users are.

Screenshot of the open source JAVA game Ninja ... — Screenshot of the open source JAVA game Ninja Quest X (I am one of the programmers) (Photo credit: Wikipedia)

So I go looking for documentation for version 3.2 of something and find only incomplete documentation for version 2.7 of it. I also know that big changes occurred in the move from the second version of the program to the third undocumented version. Ah well, there’s always the forums. Hopefully there will be others who have gone through the pain of migration from the second version to the third version and who can fill in the gaps in documentation too.

Philosophy and Science



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Philosophy can be described, not altogether accurately, as the things that science can’t address. With the modern urge to compartmentalise things, we designate some problems as philosophy and science, and conveniently ignore the fuzzy boundary between the two disciplines.

The ancient Greek philosophers didn’t appear to distinguish much between philosophy and science as such, and the term “Natural Philosophy” described the whole field before the advent of science. The Scientific Revolution of Newton, Leibniz and the rest had the effect of splitting natural philosophy into science and philosophy.

Science is (theoretically at least) build on observations. You can’t seriously believe a theory that contradicts the facts, although there is a get-out clause. You can believe such a theory if you have an explanation as to why it doesn’t fit the facts, which amounts to having an extended theory that includes a bit that contains the explanation for the discrepancy.

Philosophy however, is intended to go beyond the facts. Way beyond the facts. Philosophy asks question for example about the scientific method and why it works, and why it works so well. It asks why things are the way they are and other so called “deep” questions.



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One of the questions that Greek philosopher/scientists considered was what everything is made of. Some of them thought that it was made up four elements and some people still do. Democritus had a theory that everything was made up of small indivisible particles, and this atomic theory is a very good explanation of the way things work at a chemical level.

Democritus and his fellow philosopher/scientists had, it is true, some evidence to go and to be fair so did those who preferred the four elements theory, but the idea was more philosophical in nature rather than scientific, I feel. While it was evident that while many substances could be broken down into their components by chemical method, some could not.

Antoine Lavoisier developed the theory of comb... — Antoine Lavoisier developed the theory of combustion as a chemical reaction with oxygen (Photo credit: Wikipedia)

So Democritus would have looked at a lump of sulphur, for example, and considered it to be made up of many atoms of sulphur. The competing theory of the four elements however can’t easily explain the irreducible nature of sulphur.

My point here is that while these theories explained some of the properties of matter, the early philosopher/scientists were not too interested in experimentation, so these theories remained philosophical theories. It was not until the Scientific Revolution arrived that these theories were actually tested, albeit indirectly and the science of chemistry took off.

Model for the Three Superior Planets and Venus... — Model for the Three Superior Planets and Venus from Georg von Peuerbach, Theoricae novae planetarum. Image enhanced for legibility. The abbreviations in the center of the diagram read: C[entrum] æquantis (Center of the equant) C[entrum] deferentis (Center of the deferent) C[entrum] mundi (Center of the world) (Photo credit: Wikipedia)

Before that, chemical knowledge was very run by recipes and instructions. Once scientists realised the implications of atomic theory, they could predict chemical reactions and even weigh atoms, or at least assign masses to atoms, and atomic theory moved from philosophy to science.

That’s not such a big change as you might think. Philosophy says “I’ve got some vague ideas about atoms”. Science says “Based on observations, your theory seems good and I can express your vague ideas more concretely in these equations. Things behave as if real atoms exist and that they behave that way”. Science cannot say that things really are that way, or that atoms really exist as such.

Indeed, when scientists took a closer look at these atom things they found some issues. For instance the (relative) masses of the atoms are mostly pretty close to integers. Hydrogen’s mass is about 1, Helium’s is about 4, and Lithium’s is about 7. So far so tidy. But Chlorine’s mass is measured as not being far from 35.5.

This can be resolved if atoms contain constituent particles which cannot be added or removed by chemical reactions. A Chlorine atom behaves as if it were made up of 17 positive particles and 18 or 19 uncharged particles of more or less the same mass. If you measure the average mass of a bunch of Chlorine atoms, it will come out at 35.5 (ish). Problem solved.

English: Chlorine gas (Photo credit: Wikipedia)

Except that it has not been solved. Democritus’s atoms (it means “indivisibles”) are made up of something else. The philosophical problem is still there. If atoms are not indivisible, what are their component particles made of? The current answer seems to be that they are made of little twists of energy and probability. I wouldn’t put money on that being the absolute last word on it though. Some people think that they are made up of vibrating strings.

All through history philosophy has been raising issues without any regard for whether or not the issues can be solved, or even put to the test. Science has been taking issues at the edges of philosophy and bringing some light to them. Philosophy has been taking issues at the edge of science and conjecturing on them. Often such conjectures are taken back by science and moulded into theory again. Very often the philosophers who conjecture are the scientists who theorise, as in famous scientists like Einstein, Schroedinger and Hawking.

The end result is that the realm of philosophy is reduced somewhat in some places and the realm of science is expanded to cover those areas. But the expansion of science suggests new areas for philosophy. To explain some of the features of quantum mechanics some people suggest that there are many “worlds” or universes rather than just the one familiar to us.

This is really in the realm of philosophy as it is, as yet, unsupported by any evidence (that I know of, anyway). There are philosophers/scientists on both sides of the argument so the issue is nowhere near settled and the “many worlds interpretation” of quantum mechanics is only one of many interpretations. The problem is that quantum mechanics is not intuitively understandable.

The “many worlds interpretation” at least so far the Wikipedia article goes, views reality as a many branched tree. This seems unlikely as probabilities are rarely as binary as a branched tree. Probability is a continuum, like space or time, and it is likely that any event is represented on a dimension of space, time, and probability.

I don’t know if such a possibility makes sense in terms of the equations, so that means that I am practising philosophy and not science! Nevertheless, I like the idea.

All things are connected

English: computer network IP address (Photo credit: Wikipedia)

There are networks everywhere. Not just the Internet or the LAN at work, but everywhere. A network could loosely be defined as being comprised of a number of nodes and a number of connections between them. A node is a point or thing which is connected through a connection to another node. A connection is what joins nodes together. This rather circular definition will do for now.

A family can be described by a network. Let’s consider a typical average nuclear family with parents and 2.4 kids. Errm, on second thoughts, lets make that 3 kids. If each person in the family is a node, we can’t really have 0.4 of a kid.

A Date with Your Family (Photo credit: Wikipedia)

So there are multiple connections between any one family member and another. The father has a connection with his wife, his daughter and his sons. The daughter has connections with her father, her mother and her brothers. One way that this could be shown in a diagram is to draw a pentagon, each vertex of which is a member of the family and lines between the family member showing the relationships.

That’s a total of 15 interrelationships in a small family. Actually depending on the way you look at it, there may be more, as the father is the father of his daughter but the daughter is not the father of the father (obviously). This can be looked at at two relationships, one from father to daughter (A is B’s father) and another from the daughter to the father (B is A’s daughter), or one relationship between father and daughter.



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If you consider that there are two relationships between any two family members, then each relationship can be considered to have a direction and a value. “A is B’s brother” and “B is A’s brother”, for example. Alternatively the relationship could be simple viewed as “brothers”, in which case the relationship has a value, but is non-directional.

I’ve described the familiar relationships in detail to hopefully bring out the facts that relationships between nodes and connections can be complex or describe complex situations. It’s entirely a matter of what you want the network to show.

The Internet is what people tend to think of when someone says “network” and it is indeed a complex network with myriads of interconnections across the globe, but in another way it is quite simple. Basically you have a computer, say your desktop or laptop, connected to the Internet. When you request a webpage, your request is sent to another node on the network, which then sends it to another node, and that forwards it on to yet another node and eventually the request arrives at the destination.

The clever part is that you might think that every “node” on the Internet needs to know where all the other nodes are, but in fact all it needs to know is where to send the request next.

It seems almost magical. Your computer doesn’t know where wordpress.com is, though it does look up its unique address (known as an IP address). It still doesn’t know where wordpress.com is, so it sends the request and the IP address to your ISP. Your ISP looks at the IP address and sees that it isn’t one local to the ISP, so it passes it on.

As noted above the message is passed on and on until it reaches its destination and then more magic happens as the remote machine responds to the request and sends the response all the way back. It may even travel back by a different route.



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The magic is that some of the nodes know around 200,000 addresses on the Internet and where the next step should go. These addresses are in the most part partial in that the address will be like the street address, without the building number.

So although the Internet is a complex network with many many connections between nodes, the basic principle by which it works is simple, based on an address lookup system (DNS) and a simple unique address for each device on the Internet.



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(OK there’s more to it than that, but the complexities are mainly at the “edges” of the Internet and mainly spring from the need for security and for organisations to have a “gateway” or single address on the Internet).

When we plan a journey over the road network, we generally have some idea of where we are going or we get out a map. We then scan it for the start and end of our journey and work out what direction we need to travel and the intermediate towns.

But if we travelled like a message travels on the Internet we would first travel to the nearest town and ask someone where we need to go to get to our destination. He or she would point us to the next town to which he or she believes we should go. We would then travel to the next destination and ask again.

It would seem that such a process could result in us going round and round in circles, but eventually we will reach a place where the traffic director knows a large part of the roading network and is able to redirect us to another city which is known to be closer to our destination. Once we are on the right road, the process will eventually result in us reaching our destination.

Another network is the network formed by people we know and the people that they know, and the people that they know and so on. There is a theory that to from you, to someone you know to someone they know and so on, it takes six or less steps to reach any person on the planet. This is referred to as “six degrees of separation“.

Similar numbers can be calculated for smaller sets of people. The Kevin Bacon Number relates movie stars through films that they have starred in with other people. Number higher than 4 are rare. The Erdős Number relates people by the number of scientific papers that they have co-authored.

head of Paul Erdös, Budapest fall 1992 (Photo credit: Wikipedia)

These somewhat whimsical numbers do demonstrate how closely linked the human race is. So far as I know no study has been done of the importance of bridging individuals is. I’m talking about those who perhaps emigrate to a country, thereby directly linking together two populations that may be less loosely connected, increasing the connectivity and reducing the number of degrees of separation.

Why Pi?

If you measure the ratio of the circumference to the diameter of any circular object you get the number Pi (π). Everyone who has done any maths or physics at all knows this. Some people who have gone on to do more maths knows that Pi is an irrational number, which is, looked at one way, merely the category into which Pi falls.

There are other irrational numbers, for example the square root of the number 2, which are almost as well known as Pi, and others, such as the number e or Euler’s number, which are less well known.

Illustration of the Exponential function (Photo credit: Wikipedia)

Anyone who has travelled further along the mathematical road will be aware that there is more to Pi than mere circles and that there are many fascinating things about this number to keep amateur and professional mathematicians interested for a long time.

Pi has been known for millennia, and this has given rise to many rules of thumb and approximation for the use of the number in all sorts of calculations. For instance, I once read that the ratio of the height to base length of the pyramids is pretty much a ratio of Pi. The reason why this is so leads to many theories and a great deal of discussion, some of which are thoughtful and measured and others very much more dubious.

Ancient and not so ancient civilisations have produced mathematicians who have directly or indirectly interacted with the number Pi. One example of this is the attempts over the centuries to “square the circle“. Briefly squaring the circle means creating a square with the same area as the circle by using the usual geometric construction methods and tools – compass and straight edge.

This has been proved to be impossible, as the above reference mentions. The attempts to “trisect the angle” and “double the cube” also failed and for very similar reasons. It has been proved that all three constructions are impossible.

English: Drawing of an square inscribed in a c... — English: Drawing of an square inscribed in a circle showing animated strightedge and compass Italiano: Disegno di un quadrato inscritto in una circonferenza, con animazione di riga e compasso (Photo credit: Wikipedia)

Well, actually they are not possible in a finite number of steps, but it is “possible” in a sense for these objectives to be achieved in an infinite number of steps. This is a pointer to irrational numbers being involved. Operations which involve rational numbers finish in a finite time or a finite number of steps. (OK, I’m not entirely sure about this one – any corrections will be welcomed).

OK, so that tells us something about Pi and irrational numbers, but my title says “Why Pi?”, and my question is not about the character of Pi as an irrational number, but as the basic number of circular geometry. If you google the phrase “Why Pi?”, you will get about a quarter of a million hits.

Animation of the act of unrolling a circle's c... — Animation of the act of unrolling a circle’s circumference, illustrating the ratio π. (Photo credit: Wikipedia)

Most of these (I’ve only looked at a few!) seem to be discussions of the mathematics of Pi, not the philosophy of Pi, which I think that the question implies. So I searched for articles on the Philosophy of Pi.

Hmm, not much there on the actual philosophy of Pi, but heaps on the philosophy of the film “Life of Pi“. What I’m interested in is not the fact that Pi is irrational or that somewhere in its length is encoded my birthday and the US Declaration of Independence (not to mention copies of the US Declaration of Independence with various spelling and grammatical mistakes).

What I’m interested in is why this particular irrational number is the ratio between the circumference and the diameter. Why 3.1415….? Why not 3.1416….?

Part the answer may lie in a relation called “Euler’s Identity“.

$e^{i \pi} + 1 = 0$

This relates two irrational numbers, ‘e’ and ‘π’ in an elegantly simple equation. As in the XKCD link, any mathematician who comes across this equation can’t help but be gob-smacked by it.

The mathematical symbols and operation in this equation make it the most concise expression of mathematics that we know of. It is considered an example of mathematical beauty.



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The interesting thing about Pi is that it was an experimental value in the first place. Ancient geometers were not interested much in theory, but they measured round things. They lived purely in the physical world and their maths was utilitarian. They were measuring the world.

However they discovered something that has deep mathematical significance, or to put it another way is intimately involved in some beautiful deep mathematics.

English: Bubble-Universe's-graphic-visualby pa... — English: Bubble-Universe’s-graphic-visualby paul b. toman (Photo credit: Wikipedia)

This argues for a deep and fundamental relationship between mathematics and physics. Mathematics describes physics and the physical universe has a certain shape, for want of a better word. If Pi had a different value, that would imply that the universe had a different shape.

In our universe one could consider that Euler’s Relation describes the shape of the universe at least in part. Possibly a major part of the shape of the universe is encoded in it. It doesn’t seem however to encode the quantum universe at least directly.

English: Acrylic paint on canvas. Theme quantu... — English: Acrylic paint on canvas. Theme quantum physics. Français : Peinture acrylique sur toile. Thématique physique quantique. (Photo credit: Wikipedia)

I haven’t been trained in Quantum Physics so I can only go on the little that I know about the subject and I don’t know if there is any similar relationship that determines the “shape” of Quantum Physics as Euler’s Relation does for at least some aspects of Newtonian physics.

Maybe the closest relationship that I can think of is the Heisenberg Uncertainty Principle. Roughly speaking, (sorry physicists!) it states that for certain pairs of physical variables there is a physical limit to the accuracy with which they can be known. More specifically the product of the standard deviations of the two variables is greater than Plank’s constant divided by two.

In other words, if we accurately know the position of something, we only have a vague notion of its momentum. If we accurately know its velocity we only have a vague idea of its position. This “vagueness” is quantified by the Uncertainty Principle. It shows exactly how fuzzy Quantum Physics.

The mathematical discipline of statistics underlay the Uncertainty Principle. In a sense the Principle defines Quantum Physics as a statistically based discipline and the “shape” of statistics determines or describes the science. At least, that is my guess and suggestion.



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To return to my original question, “why Pi?”. For that matter, “why statistics?”. My answer is a guess and a suggestion as above. The answer is that it is because that is the shape of the universe. The Universe has statistical elements and shape elements and possibly other elements and the maths describe the shapes and the shapes determine the maths.

This is rather circular I know, but one can conceive of Universes where the maths is different and so is the physics and of course the physics matches the maths and vice versa. We can only guess what a universe would be like where Pi is a different irrational number (or even, bizarrely a rational number) and where the fuzziness of the universe at small scales is less or more or physically related values are related in more complicated ways.



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The reason for “Why Pi” then comes down the anthropological answer, “Because we measure it that way”. Our Universe just happens to have that shape. If it had another shape we would either measure it differently, or we wouldn’t exist.



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