Science – Page 9 – Me on the net

Growing up, down and sideways, and George Clooney.

Often a plant seed will end up in a place that is not particularly suitable for it. In particular it may end up in a gravel path or similar where there is little real soil. Or it may end up in sand which drains quickly and may, if near the sea, contain high amounts of salt.

In such an environment it may grow stunted or may be deformed. For instance Bonsai trees are kept in a small container and kept relative short of nutrients so that stay small and become gnarled and twisted. They may even have their roots trimmed.

One can imagine a society of concerned individuals fighting against the sustained torture of the trees treated in such a manner, but strangely, I’ve never heard of one. Maybe it is because trees can’t scream?

All members of a species have the same genetic make up, the same genotype. All individuals grow in much the same way, to produce similar adult individuals. This is termed the phenotype.

There may be sexual dimorphism, where the female of the species differs from the male of the species, but in most ways, all members of one sex are pretty similar to one another. I am not too dissimilar from George Clooney. My wife is much like Angelina Jolie.

Of course individuals are not identical. I’m slightly taller than George, for instance. This difference can be genetic, or it may be environmental. My genes may be the cause of the difference, or maybe the environment when we were growing up has slightly affected our growth. Our good looks are almost certainly genetic.

Sometimes a plant grows in a particular way in one environment will look completely different in another environment. Also a young specimen of a plant may look different from a mature specimen of the same species. Lanceword (Pseudopanax crassifolius) has a juvenile form so different from the mature form that it was initially thought to be two different species.

The environmental effect on the phenotype or expressed shape can be seen in genetically identical twins. One would expect their phenotypes to be identical at all ages, however, while “identical twins” look very very similar there are detectable differences.

English: Comparison of typical zygote developm... — English: Comparison of typical zygote development in monozygotic identical and dizygotic twins. (Photo credit: Wikipedia)

For instance if one twin had suffered a serious illness at a critical stage of growth, then their adult sizes may be significantly different. If one twin had a rich diet and the other twin a restricted diet that also might affect their sizes and expectations of longevity. Scientists can tell a lot about the processes of growth and development by studying genetically identical twins.

A more subtle variation in the phenotype can be seen when populations are considered rather than individuals. A population of moths that lives on darker surfaces may tend to be darker in colour than the same species that lives on lighter surfaces. Since this effect happens slowly, over many generations, it appears to be a genetic change or shift. However this change to the genotype is at a lower level than the species as the lighter moths and the darker ones can interbreed.

Some plants look completely different if grown in different environments. The weed that grows in gravel may look completely different from the weed that grows a metre away in a more favourable environment. It’s as if a switch has been thrown which turns on a totally different way of “building” the plant, as it may well be something like that.

If the genome of the organisation is a “program” to “build” the plant, it is perfectly feasible that a lack of resources at an early stage in the plant’s life might well kick in a different path in the developmental process from the path that it would take if resources were abundant.

if (abundant_resources == true)

then build_good_version

else build_poor_version

This is a simple branching process in a computer program, but the process is almost certainly a lot more complex in real life. However the principle is sound, I believe.

A simple iterative process can be used to generate complex shapes that look a lot like real plants. Minor changes to the process can cause significant changes to the end results. Tall thin shapes can morph into shorter bushier ones with a few changes to fixed numbers (constants) in the iterative process.

The phenotype of a plant of a particular species will be similar in all individuals. If an individual has leaves, stem, flowers of a particular sort then the phenotype of an individual in a different (eg poorer) environment, will most likely have similar parts, though some differences will be obvious.

Maybe the stem into of being long and flexible, it may be a lot shorter and stiffer. Maybe the leaves will be a lot thicker and fleshier in the poor environment plant and may therefore be able to retain water which will be scarcer in the poor environment. Perhaps the flower will be more robust in the harsher environment.

One would expect such variations of phenotype, the poor and the rich, to be implicit in the genome if the wider environment is patchy, with areas of rich soil mixed up with areas where the soil is poor. Otherwise, the ability of the genome to be expressed in multiple ways would likely be bred out of the population, as nature always goes for the simpler rather than the more complex.

The flexibility of the genome is something that the organism benefits from its whole life. For example there are some fish which live in groups of one male and several females. If the single male is killed by another fish, an octopus or a human being, one of the females will change sex and become male, taking the place of the missing male fish.

I’ll not speculate on how that happens in detail, but it seems that it must be implicit in the genome. The trigger is the absence of the male fish, but how the “genetic program” detects this, I don’t know, but once is does it transforms the largest female into a male, presumably by triggering changes in the genetic organs. That’s bound to be a complex process.

The central idea in this post is that the genome is much like a computer program and that the environmental influences are like the parameters to such a program. This is probably an over simplification in many ways, but by considering it as a program can explain why the same genome can produce such different individuals.

A computer program can be controlled by inputs while it is running, and similarly the environment can shape an organism while it is growing and after it has reached maturity. The idea of organism as computer controlled machine is not new, but I like to bring it out and have another look at it now and again.



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The Solstice Again

Today is the summer solstice in the Southern Hemisphere, the time when the sun is furthest south in the sky and hence at its highest. From here on in, the days get shorter as we slide back towards winter.

In the Northern Hemisphere, it is of course the winter solstice, and those living there can expect the days to lengthen, as they move towards summer. Today is the Northern Hemisphere’s shortest day.

Seasonal lag means that we can look forward to the warmest months of the year after the solstice, and those unfortunate enough to live in the Northern Hemisphere can look forward to a couple of their coldest months before things start to warm up.

I read somewhere that winter months are the months when people tend to put on weight and this was attributed to the fact that in winter, in the coldest weather people tend to exercise less and eat more. The reduced exercise is attributed to the tendency to stay home in the warm, by the fireside to avoid the often hostile weather.

And the eating more is because, well, what else is there to do but eat, when you are trapped by the weather. Our ancestors used to use up all the reserves that they had laid up for just this occasion, the hams and preserves, dried fruit and root vegetables and so on.

When the summer solstice happens, the weather is warmer and better, so people can get out an exercise, and, for our ancestors at least, agriculture kept them on the move, and the aim was to replenish the stores for the winter months, hence an emphasis on growing rather than eating. Besides, most crops would not be ready for harvesting.



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The winter solstice is one candidate for the start of the year. It marks a definite point in the cycle of the year. It’s after the solstice (a few months after the solstice) that things start growing again. The summer solstice is probably not a good choice as things are humming along then, ploughing and planting, growing and nurturing so it doesn’t really fit as the start of the year.

The spring or vernal equinox falls in March, around the 21st in the Northern Hemisphere. This is also a candidate for the start of the year, but to my mind, it is too late. Winter is tailing off at that time, things are starting to grow and because of the seasonal lag, it’s the start of spring. The year, are I see it, is already under way.



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Interestingly our fiscal year ends on 31st March. This is the date used by individuals to account for tax obligations. In many countries using the Gregorian calendar, the fiscal year ends on 31st December and almost aligns with the (winter) solstice based year. Other countries which use other calendars have fiscal years which relate to the local calendar.

As I have said the summer solstice in the Southern Hemisphere falls on 21st December (in most years). It is an astronomical point in time, not a whole day and can happen on 20th December. In the decade from 2010 to 2020 it falls on the 20th on three occasions.



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The summer solstice, Christmas Day, and the official 1st January New Year Day all fall within just over a week of each other. There is good reason to suspect historical links between these days, and there is much debate on the actual historical relationship between these events.

It is often said that early Christians adopted the winter solstice in the Northern Hemisphere, to squeeze out or replace a pagan celebration at that time. This may or may not be the case (or it may be partially true), but what is evident is that many cultures outside of the Tropics of Cancer and Capricorn celebrate a festival at around the time of the solstice.

Between the two Tropics the sun is overhead twice in a year while the sun reaches a southerly point at the time of the southern solstice (winter in the north and summer in the south) and a northerly point at the time of the northern solstice, the hottest time occurs when the sun is overhead. This divides the year into unequal parts in these latitudes.

The climate of these regions is dependant on local conditions, such as whether or not the region is close to an ocean or is in the middle of a continent, and many tropical areas have wet and dry seasons, typically of unequal extents. One example know to many people outside the tropics is the monsoon season when a regions rainfall may predominantly happen.



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On the Arctic and Antarctic circle, at the solstices the sun just grazes the horizon at the summer solstice and the day lasts 24 hours. At the winter solstices the sun just barely reaches the horizon and the night lasts 24 hours. Closer to the poles the number of sunless days or days with the sum always above the horizon increase. At the poles the sun is below the horizon for three months and above it for three months. (I hope this is correct. I did research this a little, but I am not 100% sure).

Interestingly, I learnt recently that the sunset will continue to become later for the next few weeks. The reason for this according to the linked article is because we have tied our clocks to 24 hours exactly and the day is not exactly 24 hours long. Not only is it not exactly 24 hours, but its length varies during the year. In Wellington the sunset goes out to around 3 minutes to 9 and doesn’t dip below that time until 7th January 2015.

English: Sunrise at Winter Solstice (December ... — English: Sunrise at Winter Solstice (December 21, 2006 at 8 a.m.) as viewed through the doorway half way up Maiden Tower (Photo credit: Wikipedia)

(December data here, January data here).

While looking up these numbers I noticed that the day length in Auckland is nearly half an hour shorter up there. Also sunset is about a quarter of an hour later in Wellington meaning that when the summer weather finally arrives we will have an extra 14 minutes to enjoy the balmy evenings. That’s yet another reason to prefer Wellington over Auckland! We have more time to celebrate the solstice.

[Darn! I completed this on Monday but forgot to publish it. Better late than never, I guess!]

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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How do I get from A to B?

We had reason to visit another suburb today. It wasn’t until I was sitting waiting for some traffic lights to change that I thought about how I was navigating from home to destination.

We just got into the car and drove there. I didn’t consider the route in advance, and it seemed that I just pointed the car and we got there. Obviously I knew the way, as we had been there or through there a number of times in the past. But I didn’t have the destination in mind from start to finish, at least not consciously. I’m not sure that I had it in the forefront of my mind at all.

I knew that it was in that direction though, and that did not leave a lot of route options. I did have a general feeling that I should go south, in this instance and that really only leaves two options, the back road, or the motorway. The back road is a lot prettier!

I made the choice to take the back road but it was not, as I said, at the forefront of my mind, as I was doing other things at the time, like finding my keys, my phone, my wallet and these things occupied the forefront, while the decision about which route to take was more background.

So the route was chosen more of less in the background, but not subconsciously. Much the same process happened on the way there, and at each junction or turning point, I didn’t have to consider at the front of my mind which direction I should drive. I just did it. Some part of my mind knew that to get to our destination I had to turn right, or go straight on or whatever.

This is good because the front of my mind was doing the driving, keeping the car on line, signalling, accelerating or braking, keeping us safe on the road. Except that it wasn’t right at the front mind, since I was also talking to my wife about various things. Christmas things from memory.

English: Two motorcycle trailing off the brake... — English: Two motorcycle trailing off the brakes through Tooele Turn at Miller Motorsports Park. Rider on the white bike is Warren Rose, Rider of the green bike is Dave Palazzolo (Photo credit: Wikipedia)

I’ve been driving for many years and I’m confident that if needed the driving part of my mind can instantly oust the things currently in my mind should the unexpected happen. Many year ago, when I used to smoke, I was driving with a friend and an emergency happened. When it was over I realised that I was no longer holding my cigarette. Meantime my friend was scrabbling between his legs where my cigarette had ended up when the driving part of my mind grabbed precedence and the cigarette holding part was temporarily ousted.

The route planning part of my mind would not suddenly get control like that, fortunately. That would be highly dangerous. I could if I had wanted have brought the route planning part of my mind to the front, but it wouldn’t say much except “turn left at the next junction”.

Turn Left, Turn Right (Photo credit: Wikipedia)

I have on occasion made a navigating mistake. I’m going to A and the route to B is the same in part. Suddenly I realise that I have missed a junction and will have to backtrack. It seems that the route finding part of my mind spends much of the time dozing and checks in only infrequently, sometimes missing the turning point or ritually following a more usual route.

It also seems that the information it keeps is like an instruction to take an action at each decision point rather than the whole route from home to destination as well as a general direction, less well specified. GPS guidance systems seem to work this way too in that they instruct you to take an action at each junction without setting out the whole route each time.



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The model of the mind that I’ve used above, of various parts of the mind at various levels of “forefrontness” or consciousness is nothing new. The need to make a part of the mind the one at the top of the conscious levels, suddenly as a result of a danger, or selectively by choice, as in route following reminds me of the way that computer programs

Computers have several methods for navigating through programs and reacting to things that happen when they are running. One big part is called “handling errors”. Dividing by zero is an error and if the computer reaches a point where it has to divide by zero something needs to be done. The program can report the error and gracefully stop, or it can take some action to fix the error and then carry on.

Computers handle error by means of “interrupts”. Whether the errors is software (eg divide by zero) or caused by hardware connected to it (eg input/output errors) the computer stops what it is doing and runs a bit of program that handle the errors by sending a message or fixing things up. The bits of program that were running are suspended and after the error is handled the bits that were running may be given back control.

The mind seems to work in a similar way. When an emergency arises the current part of the mind that is at the forefront gets suspended and the emergency is handled by another part of the mind. A pedestrian steps into the road and you react by standing on the brakes “before you know it” as the saying goes. As soon as the emergency is over, the conscious mind takes over again.

a short .gif of the Taiwanese animated pedestr... — a short .gif of the Taiwanese animated pedestrian road crossing sign (Photo credit: Wikipedia)

You do indeed react “before you know it”, one might say instinctively. But humans have not been driving cars for much more than one hundred years, so it appears that the reaction is not instinctive in itself, but is an instinctive reaction to a danger that has been learned. We seem to have this fast reaction to events which is instinctively based but can be applied to learned situations, which is much more flexible than hard-wired instincts would be.

So, pondering on how I get from A to B has led me to conjecture that there are parts of the mind which are forefront in our minds and other parts which are not directly in the forefront but which can be brought to the forefront in an instant, when an event happens. It is evident that these parts are only partially backgrounded as the mind as a whole has some aware of the location at the time, but they do act semi-autonomously, that is until the pedestrian steps out onto the roadway.



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Evidently there are parts of the mind that are less foregrounded and more backgrounded. When the part of the mind that is concerned with driving wants to signal or change gear, another part of the mind which controls the arms and legs wakes up and make the limbs move as needed.

I’ve spoken above as if all these different levels are discrete states, but I think it more likely that is a continuum from the foreground of the mind to the background or a least the series of levels of consciousness are close enough togerther to appear so. The mind is a complex and wonderful thing.



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[Comment: After finishing this post I went looking for other discussions of the same topic. I first found this Wikipedia article which has the issues mentioned in the article’s header. Interestingly the implication in the article is that there is a single level of consciousness at any one time. This I do not agree with. Another article I found was a little better, I feel, but only because it acknowledges that several levels may be active at the same time, but divides them into three levels with well defined scopes. I feel that it is a lot more complex than that, with all sorts of sections of the mind at all sorts of levels being active at the same time. Neither article deals with the issue of one section of the mind apparently seizing the highest level when an external event triggers it.]

Dis-Continuum

Where ever one looks, things mostly seem to be in lumps or clumps of matter. We live on a lump of matter, one of a number of lumps of matter orbiting an even bigger lump of matter. We look into the sky when the bigger lump of matter is conveniently on the other side of our lump of matter and we see evidence of other lumps of matter similar to the lump of matter that our lump of matter orbits.

We see stars, in short, which poetically speaking float in a void empty of matter. We can see that these stars are not evenly distributed and that they gather together in clumps which we call galaxies. Actually stars seem to clump together in smaller clumps such as the Local Cluster of a dozen or so stars, and most galaxies have arms or other features that show structure at all levels.

The galaxies, which we can see between the much closer stars of our own galaxy, also appear to be clustered together in clumps, and the clumps seem to be clumped together. Of course, the ultimate clump is the Universe itself, but at all levels the Universe appears to have structure, to be organised, to be formed of lumps and clumps, variously shaped into loops, whorls, sheets, arms, rings, bubbles, and so on.

OK, but in the other direction, towards the smaller rather than the larger, our planet has various systems, weather, orogenic, natural, social and evolutionary. All sorts of systems at all levels, from global scope to the scope of the smallest element.



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In other personal worlds, below the level our interactions with our families, we have all the systems that make up our own bodies. The system that circulates our blood, the system that processes our food, the system that maintains our multiple systems in a state homeostasis.

That is, not a steady state, but a state where all the individual systems self-adjust so that the larger system does not descend into a state of chaos, leading to a disruption of the larger whole. Death.

By and large most systems in our environment are made up of molecules, which are in turn made up of atoms. Atoms are a convenient stopping point on the scale from very large to very small. They are pretty “well defined”, in that they are a very strong concept.

Atoms are rarely found solo. They are sociable critters. They form relationships with other atoms, but some atoms are more sociable than others, forming multiple bonds with other atoms. Some are more promiscuous than others, changing partners frequently.



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These relationships are called molecules, and range from simple to complex, containing from two or three atoms, to millions of atoms. The really large molecules can be broken down to smaller sub-molecules which are linked repeatedly to make up the complex molecules.

To rise higher up the scale for a moment, these molecules, large and small are organised into cells, which are essentially factories for making identical or nearly identical copies of themselves. The differences are necessary to make cells into muscles or organs and other functional features, and cells that make bones and sinews and other structural parts of a body.

A section of DNA; the sequence of the plate-li... — A section of DNA; the sequence of the plate-like units (nucleotides) in the center carries information. (Photo credit: Wikipedia)

As I said, atoms are a convenient stopping point. Every atom of an element is identical at least in its base state. It may lose or gain electrons in a “relationship” or molecule, but basically it is the same as any other element of the same sort.

Each atom consists of a nucleus and surrounding electrons, a model which some people liken to a solar system. There are similarities, but there are also differences (which I won’t go into in this post). The nucleus consists a mix of protons and neutrons. While the number neutrons may vary, they don’t significantly affect the chemical properties of the atom, which makes all atoms of an element effectively the same.

An early, outdated representation of an atom, ... — An early, outdated representation of an atom, with nucleus and electrons described as well-localized particles on well-localized orbits. (Photo credit: Wikipedia)

Each component of an atom is made up of smaller particles called “elementary” particles, although they may not be fundamentally elementary. At this level we reach the blurry level of quantum physics where a particle has an imprecise definition and an imprecise location in macroscopic terms.

Having travelled from the largest to the smallest, I’m now going to talk mathematics. I’ll link back to physics at the end.

We are all familiar with counting. One, two, three and so on. These concepts are the atoms of the mathematical world. They can be built up into complex structures, much like atoms can be built into molecules, organelles, cells, tissues and organs. (The analogy is far from perfect. I can think of several ways that it breaks down).

Below the “atomic” level of the integers is the “elementary” level of the rational numbers, what most people would recognise as fractions. Interestingly between any two rational numbers, you can find other rational numbers. These are very roughly equivalent to the elementary particles. Very roughly.

One might think that these would exhaust the list of types of numbers, but below (in a sense) the rational numbers is the level of the real numbers. While many of the real numbers are also rational numbers, the majority of the real numbers ate not rational numbers.

The level of the real numbers is also known as the level of the continuum. A continuum implies a line has no gaps, as in a line drawn with a pencil. If the line is made up of dots, no matter how small, it doesn’t represent a continuum.

A line made up of atoms is not a continuum, nor is a line of elementary particles. While scientists have found ever more fundamental particles, the line has apparently ended with quarks. Quantum physics seems to indicate that nature, at the lowest level, is discrete, or, to loop back to the start of this post, lumpy. There doesn’t seem to be a level of the continuum in nature.

That leaves us with two options. Either there is no level of the continuum in nature and nature is fundamentally lumpy, or the apparent indication of quantum physics that nature is lumpy is wrong.

Pineapple Lumps (240g size) (Photo credit: Wikipedia)

It’s hard to believe that a lumpy universe would permit the concept of the continuum. If the nature of things is discrete, it’s hard to see how one could consider a smooth continuous thing. It’s like considering chess, which fundamentally defines a discontinuous world, where a playing piece is in a particular square and a square contains a playing piece or not.

It’s a weak argument, but the fact that we can conceive the concept of a continuum hints that the universe may be fundamentally continuous, in spite of quantum physics’ indications that it is not continuous.



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Fractals

Now and then I fire up one of those programs that displays a fractal on the screen. These programs use mathematical programs to display patterns on the screen. Basically the program picks the coordinates of a pixel on the screen and feeds the resulting numbers to the program. Out pop two more numbers. These are fed back to the program and the process is repeated.

There are three possible outcomes from this process.

Firstly, the situation could be reached where the numbers being input to the program also pop out of the program. Once this situation is reached it is said that the program has converged.

Secondly, the numbers coming out of the program can increase rapidly and without bounds. the program can be said to be diverging.

Thirdly, the results of the calculation could meander around without ever diverging or converging.

English: The Markov chain for the drunkard’s walk (a type of random walk) on the real line starting at 0 with a range of two in both directions. (Photo credit: Wikipedia)

A point where the program converges can then be coloured white. Where it diverges, the point or pixel can be coloured black. A point where the program seems to neither converge nor diverge can then be coloured grey. A pattern will then appear in the three colours which is defined by the equation used.

Anyone who has seen fractals and fractal programs will realise that a three colour fractal is pretty boring as compared to other published fractal images. Indeed the process that I have described is pretty basic. A better image could be drawn by colouring points differently depending on how fast the program converges to a limit. This obviously requires a definition of what constitutes convergence to a limit.

Convergence is a tricky concept which I’m not going to go into, but to compute it to say in a computer program you have to take into account the errors and rounding introduced by the way that a computer works. In particular the computer has a largest number which it can physically hold, and a smallest number. Various mathematical techniques can be used to extend this, but the extra processing required means that the program slows down.

I’m not going to explain how this difficulty is circumvented, since I don’t know! However the fact is that the computer generated fractals are fascinating. Most will allow you to continually zoom in on a small area, revealing fantastic “landscapes” which demonstrate similar features at all the descending levels. Similar, but not the same.

fractal landscape (Photo credit: Wikipedia)

The above far from rigorous description describes one type of fractal of which there are various sorts. Others are described on the Wikipedia page on the subject.

Another interesting fractal is created on the number line. Take a fixed part of the number line, say from 0 to 1, and divide it into three parts. Rub out the middle one third. This leaves two smaller lines, from 0 to 1/3 and from 2/3 to 1. Divide these lines into three parts and perform the same process. Soon, all that is left is practically nothing. This residue is known as the Cantor set, after the mathematician Georg Cantor.

English: A Cantor set Deutsch: Eine Cantor-Menge Svenska: Cantordamm i sju iterationer, en fraktal (Photo credit: Wikipedia)

This particular fractal can be generalised to two, three, or even higher dimensions. The two dimensional version is called the Sierpinski curve and the three dimensional version is called the Menger sponge.

One of the fractal curves that I was interested in was the Feigenbaum function. This fractal shows a “period doubling cascade” as shown in the first diagram in the above link. If you see some versions of this diagram the doubling points (from which the constant is determined) often look sharply defined.

I was surprised the doubling points were not in fact sharply defined. You can see what I mean if you look closely at the first doubling point in the Wolfram Mathworld link above. Nevertheless, the doubling constant is a real constant.

English: Bifurcation diagram Česky: Bifurkační... — English: Bifurcation diagram Česky: Bifurkační diagram Polski: Zbieżność bifurkacji (Photo credit: Wikipedia)

Another sort of fractal produces tree and other diagrams that look, well, natural. A few simple rules, a few iterations and the computer draws a realistic looking skeleton tree. A few tweaks to the program and a different sort of tree is drawn. The trees are so realistic looking that it seems reasonable to conclude that there is some similarity between the underlying biological process and the underlying mathematical process. That is the biological tree is the result of an iterative process, like the mathematical trees.

Русский: Ещё одно фрактальное дерево. Фракталь... — Русский: Ещё одно фрактальное дерево. Фрактальное дерево. (Photo credit: Wikipedia)

I’ve mentioned natural objects, trees, which show fractal characteristics. Many other natural objects show such characteristics, the typical example which is usually given is that of the coastline of a country. On a large scale the coastline of a country is usually pretty convoluted, but if one zooms in the art of the coastline that one zooms in on stays pretty much as convoluted as the large scale view.

This process can be repeated right down to the point where one can see the waves. If you can imagine the waves to be frozen, then one can take the process even further, but at some point the individual water molecules become visible and the process (apparently) reaches an end.

If you want a three dimensional example, clouds, at least clouds of the same type, probably fit the bill. Basically what makes the clouds fractal is the fact that one cannot easily tell the size of a cloud if one is simple given a photograph of a cloud. It could be a huge cloud seen from a distance or a smaller cloud seen close up. Of course if one gets too close to a cloud it becomes hazy, indistinct, so one can use those clues to guess the size of a cloud.



http://www.gettyimages.com/detail/165590047

Fractals were popularised by the mathematician Benoit Mandlebrot, who wrote about and studied the so-called Mandlebrot set, wrote about it in his book, “The Fractal Geometry of Nature”. I’ve read this fascinating book.

While I was searching for links to the Mandlebrot Set I came across the diagram which shows the correspondence of the period doubling cascade mentioned above and the Mandlebrot set. This correspondence, which I did not know about before, demonstrates the interlinked nature of fractals, and how simple mathematics can often have hidden depths. Almost always has hidden depths.

Electric Cars

There is a perception that electric cars are greener than petrol-driven cars. While I would not like to give the impression that I am against electric cars, as I am actually in favour of them, long term, in the short term I see some issues with them.

Firstly, consider the auto mobile. There are 250 million of them in the United States alone. That require a huge infrastructure which we don’t often consider. Firstly the crude oil is extracted from the ground using huge drills. While the technology is fairly basic, a lot of planning goes into a well before the hole is drilled, and then the drilling rig, and the workers are brought in and eventually crude oil flows.

It flows, ultimately to refineries where, apart from fuel oil, many other oil based products are extracted. The fuel is then trucked around the country or to other countries and ultimately to petrol supply stations (or gas stations as they are referred to in North America). Special equipment, the bowsers, are used to load the fuel into the cars.

The cars also require lubricating oil, which can be purchased in the petrol supply stations. More often the lubrication oil is supplied at special workshops set up to cater for the auto mobile users. These have special equipment to attend to and repair internal combustion engine. Replacement parts are manufactured and distributed to these workshops.

In contrast the fledgling electric car industry is small. There are few recharging stations, and the repair stations for electric cars are currently few and far between. Technicians who can work safely on electric cars are rare.

For electric cars to compete directly with petrol engined cars the infrastructure for electrical cars needs to match the current infrastructure for the petrol cars and that will require significant investments from someone. New electrical charging stations will need to be created or petrol supply stations will have to give up some space to electrical charging stations.

While charging stations are being created, there are less than 10,000 world-wide and a few thousand in the US. In the US there are approximately 3 charging points per station, so there are relatively few places to charge electric cars.

Charging an electric car at an outlet takes a minimum of 10 minutes and to do it this fast requires special equipment, for which special expertise is required. To provide this expertise requires special training, comparable to the expertise required to deal with petrol bowsers. Cross-training of petrol bowser experts in electrical outlets is of course possible, but the expertise is sufficiently different that a whole new pool of experts will need to be built up.

Bowser at Ariah Park, New South Wales, Australia. (Photo credit: Wikipedia)

When an electrical car requires repair for any reason, it will need to be taken to a mechanic who knows how to deal with one. It’s likely that some repair locations will switch to electric rather than petrol, since the equipment is so different, though these days the petrol repair locations already use sophistic electronics in the diagnosis and repair of petrol engined cars.

So, in summary, electric car facilities will have to replace petrol car facilities as electrical cars become more common. This will not happen quickly and easily as the industry supporting petrol cars will no doubt resist. The electric car industry will have an expensive fight on its hands as all new equipment will have to be provided and a fledgling industry wont have a lot of financial backing.

What is needed is for the costs of the petrol car industry to climb significantly, and that will cause other significant societal problems. Then it will make sense to invest in the electrical car industry.

Another issue as regards electric cars is related to the charging of them. It takes significantly longer to charge an electric car as opposed to filling up a car’s tank with petrol. In a fast charging station, with special equipment in the charging “bowser” and special connections in the car, it could take anything from 10 minutes upwards.

Cars can be charged at home, and from standard electrical connections, but this would normally have to happen overnight when there is less other usage of electricity in the home. However, if you charge a car from a standard electrical connection, it will take a long time, up to eight hours or more. So those who charge their cars at home can expect not to use the car in the evening, and a flat battery is more of an issue than a flat battery in a petrol car.

The cables both in the house and in the supply connections needs to be robust because of the inevitable heating from the continual high current, and if you be chance draw too much current, either the car charging or the house will be temporarily cut off. If you were to have medical equipment in the house then this could be life threatening.

A CH-46E Sea Knight helicopter's downwash kick... — A CH-46E Sea Knight helicopter’s downwash kicks up a dust cloud resulting in brownout (Photo credit: Wikipedia)

Of course there are mechanisms that can be ensured to reduce the impact of these problems, but that means that the wiring infrastructure in the house needs to be upgraded. It’s not a big problem until you multiply it by the number of houses that would need to be upgraded.

A bigger issue is that the electricity infrastructure is built for, really, quite light usage. If everyone in the street were to get an electric car, then the local infrastructure would come under stress. There are already “brown-outs” and “black-outs” of the infrastructure in the US at times of heavy demand. Add onto that the charging of numerous electric cars and one wonders if the infrastructure could be upgraded in a reasonable time or whether blackouts and flat batteries would become common.

This problem goes all the way back to generation, which currently depends mostly on fossil fuels in many parts of the world. It’s not much good if reducing fossil fuel usage at the consumer end results in increased fossil fuel usage at the generation end.

So while electric cars and fossil free generation should eventuate, at the moment there are high barriers to widespread adoption of electric cars and reduction of dependence on fossil fuels.



Embed from Getty Images

The Law of the Land and of Nature

(This should be posted early as I will not be in a position to write a post next weekend. If I don’t make it, it may be late!)

Laws. When we think of laws we think of speeding fines, fines for drunken driving and so on, but there are a myriad of other laws that we don’t often think about. Ones about wills and probate. Laws to protect the young, the elderly, and other vulnerable people. Laws to deter people from stealing and cheating. Laws for almost everything.

Laws are odd, when you think about it. They prevent or deter people from doing things. They basically act against personal freedom. They set limits and they are restrictive. But they are usually created on public demand or at least with public acquiescence, if not approval.

A whole section, a powerful section, of our societies has grown up to maintain and enforce our laws. The police are tasked with ensuring that laws are followed and are given the power to arrest those who appear to be breaking them, although they do not decide whether or not those persons have in fact broken the laws.

The judiciary are the people who decide whether or not the laws have been broken, based on the evidence provided. Sometimes the judges or magistrates act alone, and at other times they use the judgement of a randomly selected group of citizens formed into a jury.

If the person is found guilty there are several types of punishments, fines, bans, even imprisonment and in the past, execution. A whole group of people is needed to oversee the sentence.

From the time when the person is charged with a crime until and often after the verdict he or she will rely on the services of a specialist in laws, a lawyer or attorney. These people dedicate their lives to helping prove the innocence or guilt of persons charged with a crime.

Another huge section of society is responsible for making and amending laws. One of the functions of Government is to do this, but how do they decide how to add, change, or remove laws? I find that an interesting question. It seems to me that the representatives bring their own biasses and beliefs to the party.

They would be bringing a need hopefully communicated to them by the people who they represent, either for a change in the law or a new law or for the removal of a redundant law.

There’s a huge part of human endeavours devoted to making and administrating laws. Why do we spend such a large effort in stopping people doing things that they basically want to do?

Well, in most cases laws are there to prevent people doing things that cause harm to others, either intentionally (eg stealing) or inadvertently (eg drunken driving). Some laws are intended to prevent people harming themselves (eg drug laws).

Interestingly I heard about an experiment where a bunch of students were engaged to play a game, but the game had no rules. After a while the students started to make rules, even rules about how to make rules. So far as I know they didn’t make rules about enforcing rules or have penalties for breaking the rules! I’ve subsequently tried to find references on the Internet to this game, but I’ve been unable to find any, but it does seem as if human like rules or laws.

There is another sort of law – one which cannot be broken, the so-called law (or laws) of nature. Mankind has been obsessed with these laws for a long time, longer even than they have known that there were laws.

It’s not difficult to spot that day follows night follows day. That is, I guess, a law in itself. The underlying causes for it are not apparent. All sorts of theories were tried, including Sun Gods and flaming chariots and things like that. It was realised fairly early on that the movement of the sun across the followed some fairly simple rules.

For example the sun is higher in the sky in summer, and lower in winter, and the day is longer in summer than in winter and its height depends on how far north or south one is. One of the better explanations involved celestial sphere rotating about the earth. By this time it was becoming clear that the earth was round.

As the power of the scientific method became apparent, the likes of Newton and his contemporaries really applied its power. The idea that everything was explainable in terms of a set of known laws became really prominent. I think that what was missed was that all descriptions, even back to the Sun Gods and similar, were just that, descriptions.

Some trajectories of a particle in a box accor... — Some trajectories of a particle in a box according to Newton’s laws of classical mechanics (A), and according to the Schrödinger equation of quantum mechanics (B-F). In (B-F), the horizontal axis is position, and the vertical axis is the real part (blue) or imaginary part (red) of the wavefunction. The states (B,C,D) are energy eigenstates, but (E,F) are not. (Photo credit: Wikipedia)

Newton’s laws of gravitational attraction were descriptions of how masses appeared to move under gravitational forces. Every massive object was described as being gravitationally attracted to any other massive object and Newton provided equations to model this interaction.

When one asks what a “massive object” is, one finds oneself in a spot. A massive object is one which has mass and which is attracted to other massive objects. It is evident that this is a circular definition. OK, given the concepts of massive objects and attraction, why should they have particular a particular mathematical relationship, rather than any other?

Newton’s law of gravitational attraction have been hugely successful, notably being used to calculate the orbital period of Halley’s Comet. But in the early 1900s Einstein showed that Newton’s laws of motion and gravitation are not absolutely accurate if the massive objects. Einstein’s formulations of the laws is more accurate under these extreme conditions.

The question comes to mind – is Einstein’s formulation the ultimate explanation? Firstly I’d argue that it is a description and not an explanation at all. As such, some different description may be needed under even more extreme conditions or maybe to merge the scientific description of moving objects at high speeds and the description of quantum level events.

There may be no ultimate explanation or mathematical laws accessible to humans. A photon behaves as it does because that is the way that a photon behaves, if it is correct to label a particular physical phenomenon as a photon.

(This is the post for next Sunday/Monday, since I won’t be able to do a post then. The post after that should be back on schedule.)

A equals B

The whole universe is full of inequality. No two galaxies are exactly alike, no two planets are exactly alike, no two grains of sand are exactly alike, no two atoms of silicon are exactly alike. Wait a minute, is that last one correct?

Well, in one sense each atom of silicon is alike. Every silicon atom has 14 protons in its nucleus, and, usually, 14 neutrons. However it could have one or two neutrons extra if it is a stable atom, or even more if it is a radioactive atom. Alternatively it could have less neutrons and again it would be radioactive.

Monocrystalline silicon ingot grown by the Czo... — Monocrystalline silicon ingot grown by the Czochralski process (Photo credit: Wikipedia)

So two silicon atoms with the same number of neutrons in the nucleus are “equal” right? Well, of course a single atom by itself is seldom if ever found in nature, and two isolated similar atoms are very unlikely. But suppose.

An atom of silicon is said to have electron shells with 14 electrons in them. Without going into unnecessary details these electrons can be in a base (lowest) state or in an excited state. With multiple excitation levels and multiple electrons the probability of two isolated atoms of silicon with all electrons in the same excitation state is extremely low.

Atom Structure (Photo credit: Wikipedia)

In practise of course, you would not find isolated atoms of silicon at all. You would find masses of silicon atoms, perhaps in a random conformation, or maybe in organised rows and columns. One of the tricks of semi-conductors is that the silicon atoms are organised into an array, with an occasional atom of another element interspersed.

Atoms according cubical atom model (Photo credit: Wikipedia)

This has the effect of either providing an extra electron or one fewer in parts of the array. Under certain conditions this allows the silicon atoms and the doping element to pass the extra electron, or the lack of an electron (known as a hole) along the array in an organised manner, a phenomenon known in the macroscopic world as an electric current.

English: Drawing of a 4 He + -ion, with labell... — English: Drawing of a 4 He + -ion, with labelled electron hole. (Photo credit: Wikipedia)

So, while two atoms of silicon may in some theoretical physical and chemical sense be equal, in practice, they will be in different states, in different situations. What can be said about two silicon atoms is that fit an ideal pattern of a silicon atom, in that the nucleus of the atom has 14 protons. Some of the properties and states of the two atoms will be different.

At the very least the two atoms will be in different locations, moving with different velocities and with different amounts of energy. They can never be “equal as such. The best that you could probably say is that two atoms of the same isotope of silicon have the same number of neutrons and protons in their nuclei.

When we talk about numbers we stray into the field of mathematics, and in maths “equal” has several shades of meaning. When we say that one integer equals another integer we are essentially saying that they are the same thing. So 2 + 1 = 3 is a bit more than a simple equality and in fact that expression can be referred to as an identity.

Algebraic proofs are all about changing the left hand side of an expression or the right hand side of the expression or both and still retaining that identity between the two sides.

Mnemosyne with a mathematical formula. (Photo credit: Wikipedia)

In the real world we use mathematics to calculate things, such a velocities, masses, energy levels, in fact anything that can be calculated. Issues arise because we cannot measure real distances and times with absolute accuracy. We measure the length of something and we know that the length that we measure is not the same as the actual length of the object that we are measuring.

Lengths are conceptually not represented by integers but by ‘real numbers’. Real numbers are represented by two strings of digits separated by a period or full stop. Both strings can be infinite in length though the both strings are usually represented as being finite in length.

1 Infinite Loop, Cupertino, California. Home o... — 1 Infinite Loop, Cupertino, California. Home of Apple Inc. and one of Silicon Valley’s best known streets. (Photo credit: Wikipedia)

If we measure a distance with a ruler or tape measure, the real distance will usually fall between two marks on the ruler or measure. So we can say that the length is, say, between 1.13 and 1.14 units of measurement. If use a micrometer we might squeeze and extra couple of decimal places, and say that the length is between 1.1324 and 1.1325. With a laser measuring tool we can estimate the length more accurately still.

You can see what is happening, I hope. The more accurately we measure a distance, the more decimal places we need. To measure something with absolute accuracy we would need an infinite number of decimal places. So when we say that the distance from A to B equals 1.345 miles, we are not being exact, but are approximating to the level of accuracy that we need. Hence A is not really equal to B.

A particularly interesting case of A not being equal to B is in the mathematical case where one is trying to determine the roots of an equation. There are various method of doing this and there is a class of methods which can be designated as iterative.

One first makes a guess as to the correct value, puts that into the equation which generates a new value which is, if the iterative method chosen is appropriate, closer to the correct value. This process is repeated getting ever closer to the correct answer.

Plot of x^3 - 2x + 2, including tangent lines ... — Plot of x^3 – 2x + 2, including tangent lines at x = 0 and x = 1. Illustrates why Newton’s method doesn’t always converge for this function. (Photo credit: Wikipedia)

Of course this process never finishes, so we specify some rule to terminate the process, possibly some number of decimal places, at which to stop. More technically this is called a limit.

To prove convergence, in other words to prove that the process will generate the root if the process is taken to infinity, has proved mathematically difficult. I’m not going to attempt the proof here, but after several attempts from the time of Isaac Newton, this was achieved last century, with the introduction of the concept of limits.

One can then say, roughly, that the end result of an infinite sequence of steps in a process (A) is equal to a required value (B), even though the result no particular step is actually equal to B. You have to creep up on it, as it were.

I’ll briefly mention equality in computer programs and social equality/inequality, if only to say that I might come back to those topics some time.

The Negative Universe

Dandelion(negative) (Photo credit: tanakawho)

If cosmologists are to be believed the Universe came from nothing and is likely to return to nothing. This is odd as there seems to be an awful lot of it! There are between ten to the power 78 and ten to the power 82 atoms in the observable universe, according to some estimates. There’s also a huge amount of energy out there in the universe, and as Einstein said, this is equivalent to matter, according to his famous equation.

Maxwell's Equations — Maxwell’s Equations (Photo credit: DJOtaku)

It is likely that the enormous amount of matter and energy that we see out there is balanced by an equivalent amount of “negative” matter and energy somehow. “Negative” is in scare quotes because it may not describe what is actually going on. Anyway, the negative matter and energy may be incorporated into this universe somehow, which means that on average half the universe is this sort of energy. We can’t see it anywhere so far as I know, so it is a bit of a puzzle.

Large Format Doha Panorama Portra 400 (Photo credit: Doha Sam)

We can see evidence everywhere for “normal” matter and energy, and we should be able to see evidence of “negative” matter if it is anywhere near us. As I understand it, “negative” matter would behave differently to “normal” in various ways and should be detectable. I’m not sure in what ways it would be different – I can guess that there could be a gravitational attraction between particles of “negative” matter, just as there is between particles of “normal” matter, but there could be a gravitational repulsion between “negative” matter and “normal” matter for example.

Galaxy NGC 720 (NASA, Chandra, 10/22/02) (Photo credit: NASA’s Marshall Space Flight Center)

But my ignorance is almost total. I do believe it is true that “negative” matter should be detectable.) Since we can’t see or detect “negative” matter within our locality (ie “the observable universe“) it may be grouped elsewhere in the universe. If so, it may not have any observable effect in our neck of the woods, but inevitable it will have an effect at some time in the astronomical future.

Español: es la misma imagen que aparece en el ... — Español: es la misma imagen que aparece en el articulo en ingles: http://en.wikipedia.org/wiki/Sloan_Great_Wall (Photo credit: Wikipedia)

The reason that I say this is that the universe doesn’t seem to be expanding faster than the speed of light so any effect such as (possibly) gravity which does appear to have a “speed of light” effect will eventually affect our corner of the universe. But the situation is complex, and as the Wikipedia article says,

Due to the non-intuitive nature of the subject and what has been described by some as “careless” choices of wording, certain descriptions of the metric expansion of space and the misconceptions to which such descriptions can lead are an ongoing subject of discussion in the realm of pedagogy and communication of scientific concepts.

In other words, there are many misconceptions and misinterpretations around this topic. However any effect of the possible existence of “negative” matter on our little neck of the universe is likely to be felt a long time in the future, even on a cosmological time scale, I feel. “Negative” matter could have created a negative universe, I guess, which mirrors this universe.

In a negative universe at least one dimension would be reversed but all other dimension would have the same polarity as our universe. If an odd number of dimensions were reversed, would all but one cancel out? I’m not sure but a cursory mathematical examination would indicate that this would not be so, but I lack the time to explore the concept in depth.

In our universe things tend to a state of disorder. If one partition of a closed system contains all the matter (in the simplest case, as a gas) and the partition is removed then eventually the matter is eventually dispersed through the whole system. In a negative universe, possibly the opposite would apply – gas dispersed through a system could tend to bunch up in one part of the system. Maxwell’s demon could watch benignly without lifting a finger.

Our view of this is that it is extremely unlikely – would a glass spontaneously rise from the floor, gathering the scattered wine and land on the table complete with wine? Perhaps this is a parochial view, only true in our universe. In some alternate universe, this may be the normality – entropy may tend to decrease, order may tend to increase. Such an entropy twin may simple be the time reversed twin of the original universe. Or the original universe perceived from a time reversed perspective.

The Grand Canyon Time-Zero Project (Photo credit: futurowoman)

If the universe sprung out of nothing, then the sum of the universe is zero. Any object has its anti-object. Any event has its anti-event. Maybe the universe has a partner, an anti-universe if you like, or even a mirror universe. Time in our universe runs from the zero moment into the positive future. In a mirror universe would presumably (and debatably) run in the opposite direction from the zero moment and all spacial dimensions would be reversed.

Plus-Reversed,-1960 (Photo credit: Wikipedia)

This would correspond to a point reflection in time and space, which may or may not be the same as a rotation in time and space. I’m not sure. Some of complexities can be seen in this Wikipedia article on “parity”. In particular some interactions of elementary particles may display chirality, which means that they come in left and right handed versions, like gloves or shoes. All of the above means that if a person were to be point reflected into an anti-universe and all the elementary particles of his body were switched with their anti-particles, there may be no way for the person to tell that the switch had occurred.

different flowers from same plant (Photo credit: ghedo)

Sure, time would be reversed, but so would literally everything else, so a left-handed glove would appear, in the point reflected world, to still be a left-handed glove even though, if we could see the glove it would appear to us, from our point of view to be right-handed. Of course I’ve assumed for much of the above that the reflection that transforms a universe to its anti-universe is a point reflection.

In mathematical terms that means that all variables are reversed. That is x is replaced by -x, y by -y, z by -z and so on. It may be that the reflection may be in a line and the x dimension stays the same while the others are negated. Or it may be a reflection in a plane (a mirror reflection) where 2 dimensions are unchanged. Or it may be a reflection in a higher number of dimensions.

As you can see, the subject is complex and I’ve not got my head around it (obviously!), but I believe that if we were switched into the anti-universe (including all out particles) it would not look any different from this universe. In fact we would probably find ourselves discussing our anti-universe, which would be our original universe. In fact it would not matter which universe we called “the original” because they both would have come into existence at the same time and there would be no meaning to the term “original”.

A face.. (the original OMG Wall) (Photo credit: eworm)