Seeing things


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I sometimes suspect that I return to the same topics time and again. Not too often I hope, because that will put people off reading this blog (in case anyone does!) This is possibly a topic which I may have already addressed, but hopefully this post will be interesting anyway.

It seems obvious to me that we all see things differently, and I’m talking about vision here, not “seeing” as a philosophical point of view. Some are short sighted, some long sighted, and others have impaired vision. I see a colour as a shade of blue, while my wife sees it as a shade of green.

Toyota Celica 2.0 GT (ST202) shown in Bright T...
Toyota Celica 2.0 GT (ST202) shown in Bright Turquoise Pearl (colour code 756). (Photo credit: Wikipedia)

One could argue that the difference is merely where the line is drawn, but I think that it is more than that. Apart from the physical differences in the lenses of our our eyes, we may have differences in the physical structure of the rest of our eyes, perhaps in the rods and the cones, and it is highly likely that the physical structures of our brains are different, and our minds (which I think of as the software that runs of the hardware of the brain) are definitely different.

It’s no surprise then that my wife and I disagree on whether a colour is a shade of blue or of green. (Actually we disagree about a lot of things. I believe that it goes with being married for 40+ years!)

Plymouth Valiant 100 of some 40 years ago seen...
Plymouth Valiant 100 of some 40 years ago seen on street in New Orleans (Photo credit: Wikipedia)

In Googling around as I write this post I found an article about the brain’s colour processor. Interestingly it has a section entitled “Color is Personal” which is a part of my theme for this post. This section, however, is not really relevant to my theme as the author then discusses Achromatopsia, where damage to the colour processor causes all sensation of colour to disappear.

It seems that even in our own brains and thinking processes the idea of colour is not fixed. I read another article which describes our own personal perception of colours as “malleable”. The implication of this is that a person might describe a colour as “a shade of green” one day, and “a shade of blue” on another day. Is there no hope of a definitive answer?

Newton's color circle, showing the colors corr...
Newton’s color circle, showing the colors correlated with musical notes and symbols for the planets (Photo credit: Wikipedia)

A physicist could help us out, couldn’t he/she? He/she could measure the frequency of the light and say, definitively, that the colour is blue, or it is green, couldn’t he/she? Well, sort of. This would work for very simple colours, but real world colours are rarely made up of just one colour. The scientist’s scope would likely show a range of frequencies resembling a mountain range. That blue/green colour might have traces or red or violet, and is fairly certain to have more than one peak in the blue/green range.

Albert Einstein showed us that if a scientist was moving at a high speed relative to us, he/she would measure the frequencies in the colour differently from a scientist whose spectroscope was alongside us and not moving or moving at the same speed as us.

General Relativity
General Relativity (Photo credit: Wikipedia)

The ambient light has an effect on the colours that we perceive. A red object in red light doesn’t look red. Other objects of different colours look different in a red light. Similarly, it is difficult to determine the colours of cars and other objects under the yellow/orange sodium lights. According to Wikipedia, the colour of a street light has effects other than simple colour perception – it appears to affect peripheral vision.  New LED technology may be able to remove some of these deficiencies.

There are innumerable effects which affect or perception of colour. The most recently famous illusion is the dress which appears to people to be either black and blue or white and gold, but there are many such illusions. One which I came across a long time ago is the chessboard illusion. In this illusion, two square appear to be different colours, but are in fact the same colour. This illusion is usually shown in monochrome, but the illusion works in colour too, and depends on the shadow of the cylinder to produce the effect.


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One brain is very like any other brain. When a scientist shows someone a colour on a card, the same areas of the brain show activity in all individuals, if we exclude some cases where brain function is abnormal for some reason. We can’t delve very much deeper into this issue as we don’t know what this activity signifies, beyond the bare fact that the person was shown a card with a colour on it. We certainly can’t tell if they see it as a shade of blue or a shade of green, and we can’t tell what their subjective experience is when the brain activity occurs.

In some individuals a number or letter may invoke a sensation of colour. Such people might have the sensation of seeing something green when they think of or read the number 6. I don’t know if this imprinted behaviour because the person was presented with a green symbol when first learning their numbers or whether or not it was merely a chance association that arose at a different time, or indeed if it was because of some neurological happening or trauma that has allowed the association to happen.

English: A graph or how the brain interprets color
English: A graph or how the brain interprets color (Photo credit: Wikipedia)

Anyhow, when we see something, there are many stages to the process that  starts with light leaving the object, reaching our eyes, being refracted by the lens of the eye to form an image on the retina at the back of the eye, being sensed by the rods and cone cells in the retina, and sending signals to the brain, which then processes the data.

The amazing thing here is that the image sent to the brain is pretty messy. The eye is not a perfect sphere, the retina is curved in three dimensions and the resolution is pretty rubbish. The retina has at least one major gap in it, rods and cones are not evenly distributed across the retina. Our perception however, is smooth and break free. We have our image processing hardware and software in the brain to that for that.

Retinoblastoma retina scan before and after ch...
Retinoblastoma retina scan before and after chemotherapy (Photo credit: Wikipedia)

It means we can watch a soccer match, and we can see the black and white panels or the ball rotating as it spins across the television screen, when the unprocessed image that reaches our eyes may be quite blurred. Seeing is believing!


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Spring is in the Air

English: Graeme Crosby at Pukekohe race track ...
English: Graeme Crosby at Pukekohe race track in New Zealand on 17 April 2011. (Photo credit: Wikipedia)

On the race track of the year, spring is the bendy bit just before the long fast straight of summer. You have left the tedious long drag of the back straight of winter and you are looking forward to being able to blithely put the foot down before, eventually braking for the twists and turn of autumn.

Spring promises a lot, daffodils, apple and cherry blossom, even the first flush of grass that requires you to dig the mower out of the shed or garage. It carefully doesn’t promise sudden drops to what feels like arctic temperatures and gales that knock trees over! One term for spring weather is “changeable”. It doesn’t promise equinoctial gales, though apparently there is no real evidence for any such thing.

English: Beautiful yellow Hibiscus flower 'Gol...
English: Beautiful yellow Hibiscus flower ‘Gold Blush’? (Photo credit: Wikipedia)

When we pass the magic date of September 1, it becomes officially spring, but no one told the weather. We try and persuade ourselves that the weather this week is better than the weather last week, and sometimes it is and sometimes it isn’t.

As spring wears on the signs of the approach of summer start to appear. The length of the day increases, and spring flowers start to blossom. I like to watch the deciduous trees come into leaf. First the buds swell, and then they burst into vivid green clusters of leaves, firstly quite pale as the leaf’s internal factories start manufacturing chlorophyll, and then darker green as the chlorophyll builds up in the leaves.


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Some trees, like the cherry and apple trees seem to favour blossom over leaves at the start of the blossoming season. The little new leaves are there, on the apple trees, but they are overshadowed by the mass of blossom. Cherry trees appear to be all blossom. They are not the only ones – magnolias trees blossom before bearing leaves. The rather showy magnolia flowers appear before the leaves and quickly fall apart to carpet the ground with rapidly btowning petals.

Daffodils spring up, again promising better weather to come, and temperatures do start to slowly rise, but not without a few cooler spells as a nod back to winter. On the warmer days, I’ve seen Monarch butterflies tempted out of hibernation, but it’s a dangerous time for Monarchs. I seen those that failed because they are caught by a cold snap or because they have exhausted the reserve built up in the autumn, lying dead on the grass.

English: Ice in the dunes At the end of a bitt...
English: Ice in the dunes At the end of a bitterly cold snap the pools above the tideline were all frozen. (Photo credit: Wikipedia)

Other birds and animals become more active, like the Tuis brawling in the trees like a bunch of youths fuelled by testosterone. Moreporks become more active in the night, though their calls never really stop the whole year round. All birds ramp up their activities.

Spring is a time for lambs. Usually they will have been born for a month or more when spring starts, but late comers will still be found and the early comers are still quite small, leaving them very vulnerable to a late cold snap.

English: Spring Lambs Signs of Spring in the f...
English: Spring Lambs Signs of Spring in the fields (Photo credit: Wikipedia)

If I’ve made Spring appear bleak, it isn’t really. The lengthening of the days as well as the increases in temperature make the nicer days very pleasant, and one can slowly, layer by later, reduce the stifling amounts of winter clothing, with the prospect, round the corner, of the much lighter styles of summer.

With food being shipped around the world these days, the so-called summer vegetables can be purchased the year round, though in the depths of winter some vegetables attract premium prices, and out of season produce can be somewhat lacking in flavour. But soon enough more local vegetables come into season and prices fall and flavour improves.


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I just noticed that the Vernal (spring) Equinox is nearly here. It will be on September 23 at 8:22pm NZST. This of course means that the day and the night will be approximately the same length, 12 hours, on that day, and from then on until the Autumnal Equinox in March days will be longer than the nights. Nights will shrink to 8 hours and 50 minutes at the Summer Solstice.

We move to daylight saving time (NZDT) at the end of the month. The clocks go forward so we lose an hour in bed! Never mind, it’s another sign that summer is on its way. The dog will enjoy getting his food an hour earlier too.

English: Raw feeding: Golden Retriever eating ...
English: Raw feeding: Golden Retriever eating raw pig foot (Photo credit: Wikipedia)

In the autumn we expected the temperatures to start to fall, but if they don’t for a day or two, and summer temperatures persist, we sometimes call this an “Indian Summer”. I don’t think that there is a similar term for a late cold period except perhaps a “cold snap”. Of course, there may well be an unseasonably warm period in early spring, which causes plants to bloom or animals to come out of hibernation early. Hopefully this would not be followed by deep cold, which would spell disaster for the animals and plants.

English: Indian Summer
English: Indian Summer (Photo credit: Wikipedia)

Weather forecasting is not capable of predicting the onset of cold or warm periods, but long term trends are able to be predicted, though only in terms of probability. One such prediction involves something called the El Niño Southern Oscillation, and the effect leads to changes to the patterns of  weather over the country. The western areas tend to get more rainfall and the eastern areas tend get less, to the extent that drought conditions ensue. It is likely according to Australia’s Bureau of Meteorology that the current El Niño effect may continue into next year.

While this promises a golden summer in the east of the country and a wetter time in the west, this is not good news for dairy farmers, who have been struggling as a result of the global downturn in dairy prices. Farmers who plant crops will also be hit, as they will need to provide water from other sources, such as bores, irrigation schemes, or they may even have to truck water in. Lack of water will also affect power generation, and also tap water.


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It seems strange writing that, since at the moment we are experiencing a period of wet weather over much of the country as a low pressure weather system has stalled over the country, becoming disconnected from the systems that move lows and highs across the country. So we are stuck with wet weather for the next few days.

Never mind. This may, hopefully, be winter’s last gasp. We are probably tracking through the first chicane of spring, and the rest of the season will see a general improvement until we turn the final corner and accelerate into summer.


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Trains, boats and planes

Refugees arrive in Travnik, central Bosnia, du...
Refugees arrive in Travnik, central Bosnia, during the Yugoslav wars, 1993. (Photo credit: Wikipedia)

There’s a horrifying refugee crisis going on in Europe where floods of people from the Middle East are trying to get into the richer and stabler countries like Germany and the UK. They are fleeing wars and persecutions in their own countries, and are paying ruthless individuals to transport them mainly in overloaded boats from Asia to Europe.

Tragically, people are being killed in this process, as people are stifled in trucks and drowned falling from boats or suffering similar misfortunes. I haven’t heard of cases, but it would not surprise me to learn that unscrupulous have been killing refugees and taking whatever small possessions that they have.

Children of the United Kingdom's Children's Mi...
Children of the United Kingdom’s Children’s Migrant Programme (Photo credit: Wikipedia)

Whether a person is a refugee or merely a migrant, they are leaving one country for another because they believe that life will be better in a new country. Such movements are older than the human race itself. It is believed that the human race evolved in Africa and relatively quickly spread though much of the then accessible world. Members of the “Homo” family of species at that time were widely spread in Eurasia as well as the home continent of Africa.

The Homo family of species spread through much of Europe and Asia probably as a result of their intelligence and their high rate of breeding. Being hunters and gatherers and increasing population would put pressure on scarce resources, forcing families and groups to travel further for food and resulting in migrations in search of food.

This is a recreated vector image in SVG. The o...
This is a recreated vector image in SVG. The original “Human_evolution_scheme.png” was made by José-Manuel Benitos. The following was stated by the original author: “Simplified scheme of human evolution, it does not try to be trustworthy, but a symbol of this process” (Photo credit: Wikipedia)

Of course there would have been many other factors, but I’ll not go into that as I don’t know much about early human migrations. One effect of the rise and spread of humanity was the decline of the other Homo species. I assume that there is a link between the two phenomenon as they happened, apparently, at about the same time.

Maybe a cleverer Homo Sapiens stole the resources that the other Homo species needed, or maybe the other Homo species succumbed to some influence that did not affect Homo Sapiens, such as a disease or a climate change. Maybe our ancestors destroyed the other species in a pre-stone age holocaust. I’ve not studied the literature on the subject, so I’m ignorant of what was the likely cause of the decline of the other Homo species.

English: Human evolution splitter view
English: Human evolution splitter view (Photo credit: Wikipedia)

Whatever happened in those early days appears to have left the human race the urge to keep moving on. This urge has prompted us to send people to the moon and to send spacecraft to all (local) parts of the universe. Of course refugees in general don’t have much choice in the matter. They need to move or they are dead.

There is probably a spectrum stretching from migrant to refugee that covers all people who change countries or even regions. At the one end you have the forced movement of people between countries, by the authorities or an invader, through people fleeing war or persecution, to those who flee unpopular regimes which won’t actually kill or persucute them, to those who choose to migrate for political, cultural, reasons, right through to those who like to experience a different living environment.


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I’ve changed countries myself, and though my migration was voluntary and for a better life, it was a huge upheaval to move countries. You have to leave friends and relatives, all the things that you have known, much of which you may miss, to pack up your life and relocate it to a new country, where the culture is different if not in type then in detail, and you do not know how you will cope.

Of course, voluntary migrants have it easy in comparison to the refugees. They often cannot bring any possession with them, and they may not like the culture (which may espouse a different religion of course) and the likelihood of them returning to their original homes is remote. As refugees they will almost certainly miss their countries more than a person further up the migrant-refugee spectrum would.

Remains of an Orthodox church in the city cent...
Remains of an Orthodox church in the city center. The church was destroyed during the war but has since been reconstructed. (Photo credit: Wikipedia)

Nevertheless, some voluntary migrants suffer at some stage from home sickness. Perhaps when an elderly relative dies and they cannot return for the funeral, or when a sibling who has remained in the “homeland” has a child. I’ve seen home sickness triggered by a simple treat brought by a visitor from the “homeland” that is unavailable in the new country.

I’ve not suffered very much from the syndrome myself, but I’ve known people who have and it is not a trivial thing. Home sickness can make a person physically ill, and if they are frail, it can even kill them. It can seriously disturb a person’s mental health, especially if they are prone to depression or similar mental illnesses. I’d say, however, that almost every single person who leaves one country for another suffers from it, except perhaps those of a persistently roving disposition.

"Homesickness Can Be Cured" - NARA -...
“Homesickness Can Be Cured” – NARA – 514527 (Photo credit: Wikipedia)

Some voluntary migrants cannot cope with living in a new country. These are the ones that pack up and go back “home”. It may be culture, it may be relatives, it may be what you can buy in the shops, but these people make the decision to return from whence they came. I used to wonder why they did it, until I went back for a couple of years, and found that I hated it and couldn’t wait to get back to the new country. After that I had more sympathy for those returning migrants.

This contrasts strongly with refugees, who, although they see their target countries as being better than their homelands, are going to face a hugely different culture, possible religious and racial intolerance, all without the safety net of being able to return to their homelands. Even if they are able, at some time in the future, to return, it is likely that their homelands would have become strange and alien. Likely other people will be living their, with new customs and even religions.

We have a phrase which describes the process of settling in a new land, or adapting to local customs, to making friends and watching children forming bonds with others in the new country. It’s called “putting down roots”. Let’s hope that the refugees all find a place where they can join happily with the local society and put down some roots. Not to forget their homelands totally, but to rejoice in their new homeland. Those of us who are voluntary migrants should welcome these “involuntary migrants”.


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A Miscellany


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I’m going to try something different in this post. I won’t try to stick to a single theme, but will try to create a miscellany of short themes and see how it goes.

Firstly, a friend of mine is a keen photographer who keeps a blog and for five years he has posted one or more photos to his blog every day. It’s a fantastic achievement, and since I have trouble posting once a week, I can only imagine the persistence, application and diligence needed to post once every day.


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I’ve mentioned before that I have started blogs at various times in the past and have been unable to keep them going. This time around, for reasons that I can’t really fathom, I have managed to keep going for coming up to 150 posts now. If Brian succeeds in reaching 5 years of posting he will have posted over 1,870 times. Which is mind-boggling!!

One of the reasons that he has given for dropping his self-imposed regime of daily posting is that he feels that the quality of his posted pictures is possibly suffering from the requirement to post something every day and that the temptation to post a merely adequate (from his point of view) picture just to keep the chain going is strong.

English: Locomotive in KiwiRail livery (not a ...
English: Locomotive in KiwiRail livery (not a particularly good photo, but perhaps adequate until a better is found) (Photo credit: Wikipedia)

I must say that I have not seen any deterioration in his pictures, but I don’t look at them with his eyes. So I will continue to look forward to his posts, even though he will not be posting daily pictures once he reaches the 5 year target.

Of course, this has made me think about this blog and how long I intend to keep it going. I’ve posted nearly 150 times so far which represents a bit under three years. I’d like to reach at least 5 years too, which will be around 250 posts. That’s the current target, so let’s see if I can reach it.

English: 250 West Pratt Street in Baltimore
English: 250 West Pratt Street in Baltimore (Photo credit: Wikipedia)

Next topic. If I was able to ask God a question, I would say “Quantum Physics. What were you thinking!” We like to think that the universe is logical and consistent. If it wasn’t then there would be no guarantee that the sun would not blink out 10 seconds from now. 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0! OK we are fine this time.

I’m told, and I have an inkling about it from my reading and thinking, that Quantum Physics is logical and consistent. However the various popularizations of it appear counter intuitive and paradoxical. How can Schroedinger’s poor cat be both alive and dead? What exactly is a superposition of states? What (if anything) is the “collapse of the wave form”?

Omega-Point-Multiverse
Omega-Point-Multiverse (Photo credit: Wikipedia)

General and Special Relativity were considered mysterious and paradoxical when Einstein first published his papers. At the time someone claimed that only three people in the world truly understood it, but it didn’t take long for it to be taught down to college and undergraduate levels. While strange and challenging, it was soon accepted as true by the majority of people who had come across it, although people still create web sites where they try to prove that Einstein was wrong.

Quantum Physics is also taught at undergraduate levels I believe but it remains (or so I get the impression) as a work in progress. The famous Copenhagen Interpretation was formulated around 90 years ago by Niels Bohr and Werner Heisenburg and there is still no accessible standard interpretation that is accepted by the majority.

English: Experiment suggested by Heisenberg, p...
English: Experiment suggested by Heisenberg, part 1: Wide hole in barrier screen gives only a very general idea of location of photon as it moves toward detection screen. Photons will arrive at a small spot on the detection screen. (Photo credit: Wikipedia)

Niels Bohr once said “Anyone not shocked by quantum mechanics has not yet understood it.” Richard Feynman said “Nobody understands quantum mechanics.” So, God, Quantum Physics. What were you thinking about?

Flags and things. There is a big debate in this country at the moment about whether or not to change the flag. People often refer to the Canadian flag as a case where a new flag was adopted after public discussion. It’s not a good case study though as the actual adoption of the winning flag involved a farcical mistake : “Through a six-week period of study with political manoeuvring, the committee took a vote on the two finalists: the Pearson Pennant (Beddoe’s design) and the current design. Believing the Liberal members would vote for the Prime Minister’s preference, the Conservatives voted for the single leaf design. The Liberals, though, all voted for the same, giving a unanimous, 14 to 0 vote for [it]”. (From Wikipedia).

Flag of Spanish Vexilology Society (Asociación...
Flag of Spanish Vexilology Society (Asociación Española de Vexilología) (Photo credit: Wikipedia)

The process for our new flag was decided on early. Firstly all submissions would be reviewed, and a “top 40” would be selected by a panel composed of, basically, a bunch of celebrities and other. No disrespect to them, but they were not flag experts, and if they were chosen to sort of represent the man/woman in the street that is what they achieved.

Of the top 40, four were “approved” by the ruling National cabinet. It’s not too clear how this was done, but only conspiracy theorists would contend that three out of the four contained the fern emblem that the Prime Minister favoured and that he somehow influenced the selection.

Stjórnarráðið in Reykjavik, the seat of the ex...
Stjórnarráðið in Reykjavik, the seat of the executive branch of Iceland’s government (Photo credit: Wikipedia)

Now we have the top four we are supposed to vote in a referendum to pick one to go up against the current flag in a second referendum. Interestingly a group has been formed to promote a fifth design (originally in the top 40) over the top four. We will see where if anywhere that this movement goes. I’d guess it will eventually fail.

One hundred words to go. Interest is high on the attempt of Jarryd Hayne, a former Australian rugby league player who has secured a spot in San Francisco 49ers NFL team. (That’s what is called “American Football” everywhere except the USA). Good luck to him, I say. I expect to see a surge of popularity for the sport in this part of the world.

RLWC - Fiji v Ireland, Gold Coast 2008. Fijian...
RLWC – Fiji v Ireland, Gold Coast 2008. Fijian fullback Jarryd Hayne. (Photo credit: Wikipedia)

He seemed to bring something new to the American game, but time will tell if opposition coaches find ways to nullify his effect or whether other players will adopt some of his style, which to my naive eye seems to be a more fluid running game. Or maybe he really is an exceptional player. Time will tell.

Well, I quite enjoyed jumping around through various topics but I don’t think that I will do it every time. I’ll just take it as it comes.

A show jumping course
A show jumping course (Photo credit: Wikipedia)

 

Cycling through life

English: cycle that rotates on its axis Españo...
English: cycle that rotates on its axis Español: ciclo que gira sobre su propio eje (Photo credit: Wikipedia)

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.

English: SINE and COSINE-Graph of the sine- an...
English: SINE and COSINE-Graph of the sine- and cosine-functions sin(x) and cos(x). One period from 0 to 2π is drawn. x- and y-axis have the same units. All labels are embedded in “Computer Modern” font. The x-scale is in appropriate units of pi. Deutsch: SINUS und COSINUS-Graph der Funktionen sin(x) und cos(x). Eine Periode von 0 bis 2π ist dargestellt. Die x-Achse ist in π-Anteilen skaliert entsprechend 0 bis 2π bzw. 0° bis 360° (Photo credit: Wikipedia)

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.

Graphing process of y = sin x (where x is the ...
Graphing process of y = sin x (where x is the angle in radians) using a unit circle. (Photo credit: Wikipedia)

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.

English: Section of ellipse showing eccentric ...
English: Section of ellipse showing eccentric and true anaomaly (Photo credit: Wikipedia)

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.

Illustration of Gauss-Newton applied to a curv...
Illustration of Gauss-Newton applied to a curve-fitting problem with noisy data. What is plotted is the best fit curve versus the data with the fitting parameters obtained via Gauss-Newton. (Photo credit: Wikipedia)

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.

Claudius Ptolemäus, Picture of 16th century bo...
Claudius Ptolemäus, Picture of 16th century book frontispiece (Photo credit: Wikipedia)