Did You Know...
This page boldly goes where some remarkable discoverers have gone before, and tries to make sense of the amazing things that they found.
(Sorry. I always wanted to say that!)
A Winter's Tale: Hunting the Analemma
When this story began, I had never heard of the Analemma. If someone had asked me, I might have guessed wildly that it was a small animal on the endangered species list (which turns out not to be the case).
To start with what I thought I knew...
The winter solstice marks the shortest day of the year, right? (Right!)
It occurs (for the northern hemisphere) when the Earth's axis is tilted directly away from the Sun, so a particular place (London, say) spends the most time in shadow as the Earth rotates and spends the least time in the light. Right? (Right again!)
So the winter solstice is the time when sunrises start getting earlier, and sunsets start getting later... right? (Wrong!)
It was pointed out to me (by my mother-in-law, who knew a lot of interesting stuff) that the time of sunset starts getting later well before the solstice (early December, depending on latitude), while the time of sunrise doesn't start getting earlier until well after the solstice (early January, depending on latitude). Sure enough, when I looked up sunrise and sunset times (as you can do here) she was correct... but why?
My first thought was that maybe it took time for the Earth to pass through the solstice point, so to speak... but it doesn't take that long!
I didn't realise that I was starting down a trail that was to get more complicated (and more interesting) until I finally understood (I think) what the Analemma really is, and why it looks the way it does.
Before heading off down this trail yourself (supposing that you want to) here is a little thought experiment that might act as a kind of map:
Imagine you are in a large-ish room with no windows and black walls (representing the solar system). In the middle of the room is a lamp without a lamp-shade (representing the Sun). (What? OK, it's a lamp with a high-wattage bulb!)
Imagine you are standing with your back against the middle of one wall, carrying a globe of the Earth. It's one of those globes you see in schools (not too big to lift), with a stand and an angled spindle around which the Earth rotates. (You might find that it has a mysterious figure-of-eight pattern printed on it somewhere, but it doesn't matter if it doesn't.) You are standing facing the "Sun" lamp, and you are holding the globe so that the top of the angled spindle is tilted towards you. Are we OK so far?
Oh yes... and imagine that you are completely transparent, so that the light from the "Sun" will be able to go right through you when it needs to as if you weren't there. You don't want to cause an eclipse. (It's a thought experiment... anything goes.)
If all is well, you are holding the globe so that it represents the Earth in the winter solstice position. If that doesn't make sense, don't worry.
Now press the switch on the stand that I didn't mention, and imagine that the globe starts turning slowly, in an anti-clockwise direction (as seen from above). Pick a city in the northern hemisphere but south of the Arctic Circle, say San Francisco or New York or London, and keep your eye on it. Half of the globe will be in shadow (the side nearest your chest, since you are facing the "Sun"), and half the globe will be in the light of the "Sun". As your city of choice moves, it will spend more of its time in the dark side than the light side, because of the tilt of the globe. Your city of choice, in fact, is experiencing its shortest day of the year.
Now we will make things a little more realistic, because the "Earth" needs to move around the "Sun". This is a little tricky. You have to walk in an anti-clockwise circle around the room, but always facing in the same direction, i.e. towards the opposite wall (because the Earth's axis always faces the same way as it orbits the Sun). This means that you begin by shuffling mostly sideways (to your right), and then start moving forwards as well as sideways, so that you are tracing out a circle on the floor with the "Sun" in the centre. Eventually you will have to walk sideways and backwards, after you reach the other side of the room! How fast should you walk? Well, at a speed so that the globe you are holding rotates about 365 times (representing days of the year) while you make one complete circuit of the room (which represents one year).
Are we good so far? Now imagine that you are looking carefully at the globe as you walk around. You will see that the shadow on the globe is itself turning slowly, in an anti-clockwise direction (as seen from above). This is really important. The globe is turning, but so (much more slowly) is the Sun's shadow. This means that your city of choice has to make slightly more than one revolution from "noon" on one day (when it is facing the "Sun" most directly) to "noon" on the next day (when it is again facing the "Sun" most directly). I hope that's clear.
It turns out, therefore, that a solar day (which is what we live to, and is approximately 24 hours long according to our time-pieces) represents a bit more than one revolution of the Earth, whereas a sidereal day represents exactly one revolution of the Earth and is a little less than 24 hours according to our time-pieces.
When you have walked a quarter of the way around the room, watch your city of choice again. It should now be spending the same amount of time in the light as it does in the dark. Can you see why? This represents the Spring Equinox, where day and night are of equal length. And as you keep on moving round the room, the days will get longer, until you reach the Summer Solstice at the far side of the room, with your back to the "Sun" (you are completely transparent, remember, so the light is shining through you!). Your city of choice is now spending its maximum time in the light and its minimum time in the shadow.
If all of this makes sense so far, we are ready for the next bit.
Imagine you are in your city of choice at "midnight" on one day, and then again at "midnight" the next day. The Earth will have rotated one-and-a-small-bit times between those two "midnights", and so the stars you would see at each "midnight" will not be quite in the same position - they will appear to have drifted a little to the right. And so, each "midnight", we appear to have moved slightly right-to-left through the constellations. If we divide up a strip of all the constellations into twelve sections as the ancients did, and imagine pictures to go with those sections, what we have are the "signs of the zodiac", the zodiac being the strip of constellations that we appear to drift through from one "midnight" to the next "midnight".
We haven't quite reached the Analemma yet. We have to get a bit more realistic again. Before we do, we should understand that if the "solar day" (the time from "noon" to "noon", slightly more than one revolution of the Earth) were exactly 24 hours long and constant, which it isn't, then the puzzle we started with wouldn't exist. After the shortest day of the year, the sunrises would start getting earlier (by our time-pieces), and the sunsets would start getting later (by our time-pieces).
OK, now fasten your mental seat-belt... There are two different reasons (that we need to worry about) why the length of the "solar day" varies, and the first reason (believe it or not) is the easier to understand. Let's take that one first.
In our thought experiment, we walked around the "Sun" in a circle, at a constant speed (we were trying to, anyway), but that isn't quite what really happens. The Earth goes round the Sun in an ellipse, not a circle, and it is closest to the Sun some time in January (which is not what one might expect!).
Furthermore, the Earth travels faster the nearer it is to the Sun, according to Kepler's Second Law. So when walking around the imaginary room, you need to be a little closer to the "Sun" soon after you set off, and get a little further away from it as you head around to the opposite side of the room. But more importantly, because of the way objects move in an elliptical orbit, you need to be moving a little faster when your are closer to the "Sun", and a little slower as you get further away.
OK, off you go... Now, as you walk, look carefully at the shadow on the globe again. It is no longer turning around at a constant rate, but is going at its fastest some time in January, and at its slowest some time in July (the other side of the room). Of course, the difference isn't much, but it adds up. So when you set off (and later in late November / December, when you have gone all around the room and approach your original position), it takes a little longer than normal for the city of your choice to catch up with the turning shadow, and so sunsets (and noons, and sunrises) happen a little later each day according to our 24-hour time-pieces.
Can you see that as the yearly turning of the Earth's shadow speeds up, the effect is to slow down the apparent daily motion of the sun across our sky? That will turn out to be important later on.
As the days shorten towards the solstice, and the rate at which the yearly turning of the Earth's shadow speeds up, the "latening" of the sunsets gets a boost (which we see as an apparent lengthening of the afternoon before the solstice). And as the days lengthen after the solstice the "latening" of the sunrises is still happening, which delays the date on which the time of sunrise (according to our 24-hour time-pieces) starts getting earlier again. The effect we are talking about is only a few minutes difference, and once you have moved some distance around the room the "normal" day-lengthening effect swamps these small variations.
The Analemma is now in sight! We just need to take a look at the second reason why the length of the "solar day" varies. This one is a little tough, and if I don't make it clear, don't worry - there are movies at the Analemma web site which will do a better job than I can!
Back to our starting position in the room... Start walking in the peculiar way you did before, paying close attention to the slowly moving shadow and the path that your chosen city takes into and out of it. As you set off from the Winter Solstice position, mark somehow where your chosen city is at "noon", relative to the frame of the globe.
Keep walking, and when the globe has rotated once-and-a-small-bit, notice where your chosen city is at the next "noon". The difference between the two successive "noon" positions is a short imaginary line on the surface of the Earth that is more or less parallel to the floor.
Now walk around until you are near to the right hand wall, i.e. the Spring Equinox position, and do the same thought experiment again. This time that same short line between successive "noon" positions is tilted relative to the floor, and is shorter than the first line when you were at the Winter Solstice. (I said it was tough!). This is because the Sun's yearly relative motion is now partly in a south-to-north direction, and therefore only partly in a west-to-east direction. (Remember that as you walk around the room, the yearly shadow is turning anti-clockwise as seen from above.)
In other words, when the Earth is tilted at right angles to the Sun, instead of towards or away from it, the effect of the "noon shift" is at its mimimum.
And so we get two contributions to "noon shift" - one from the Earth not going around the Sun at uniform speed, being fastest in January and slowest in July - and one from the effects of the tilt of the Earth's axis, which is fastest around December 21st and June 21st (the solstices) and slowest around March 21st and September 21st (the equinoxes). If you add these two effects together (observing that one effect has a one-year cycle, and the other a half-year cycle) you get "the equation of time" which describes how the noon position of the Sun (12 o'clock solar time) shifts with respect to 12 o'clock as measured by our 24 hour time-pieces (which run to "mean solar time").
It may help (if you get confused later on) to remember that as the yearly turning of the Earth's shadow speeds up, the effect is to slow down the apparent daily motion of the sun across our sky. Just think back to those thought experiments we did!
Now... one last thought experiment, a different one, and you are there! Imagine having an accurate watch, setting up a camera on a tripod, and taking a picture of the Sun at some particular time at the Winter Solstice, then taking another picture a week later at the same time by your watch, then another one a week later... and so on... (not adjusting your watch for summer time, and in miraculously good weather with no clouds to get in the way) and then superimposing all those pictures on top of each other. What would you expect to see?
If the earth were not tilted, and was orbiting the Sun in a circle at constant speed (which it doesn't), what you would see is a single bright spot. Does that make sense?
If the Earth were not tilted, and was orbiting the Sun in an ellipse at varying speed (as it does), what you would see is a series of close spots in a horizontal straight line. Does that make sense?
But the Earth is tilted. What you actually would see is... the Analemma.
As you look at one of those great Analemma photographs, it might also be interesting to note that as the Earth turns, it takes about a minute for the Sun to move across its own diameter in the sky. This gives some indication of the variations in time that the different images of the Sun in the Analemma represent.
That's me done! The rest is up to you... I suggest you start at the Analemma Web Site ... and good hunting!
Magical Loops Part 1: Conway's Game of Life
Let's jump right in and play the game - I'll explain what it's about as we go along (in case you haven't met it before).
You can use a Go board (a chess board isn't really big enough), or anything with a large number of squares on it, (the more squares the better). You also need some markers (you could use Go stones) to go in those squares.
I'm going to use a spreadsheet, since it's more convenient and you can make it as big as you like. Here it is:
Notice that each cell (square) on the board has 8 neighbouring cells (except at the edges of the board, which we'll ignore for now).
We can now lay down markers in any pattern we like. I'll start with something simple:
What we are looking at can be called "Generation 1". We are going to create a new pattern ("Generation 2"), which is derived from "Generation 1" pattern according to some simple rules concerning deaths and births:
- Death: Any occupied cell in "Generation 1" that has no occupied neighbour cells, or only one occupied neighbour cell, dies from starvation or isolation, and won't appear in the "Generation 2" pattern.
- Death: Any occupied cell in "Generation 1" that has 4 or more occupied neighbour cells dies from overcrowding, and won't appear in the "Generation 2" pattern.
- Survival: Any occupied cell in "Generation 1" that has 2 or 3 occupied neighbour cells is OK, and will survive to appear in the same place in the "Generation 2" pattern.
- Birth: Any unoccupied cell in "Generation 1" that has 3 occupied neighbour cells is ready for a birth, and will become an occupied cell in the "Generation 2" pattern.
The same rules apply in each succeeding generation.
If we look at our simple "Generation 1" pattern, we can see that the two blobs at each end of the pattern will die, from rule 1, and the blob in the middle will survive, from rule 3. We can also see that a new birth will take place above and below the middle blob, from rule 4.
"Generation 2" for this pattern will therefore look like this:
So what will "Generation 3" look like? Right! It will look exactly like "Generation 1" and the generations in this case will repeat their forms endlessly - this kind of pattern is called an "oscillator".
If we start with something different:
The next generation will look like this:
And a few generations later we get this:
Where will this pattern end up? Right now I don't know, and there is no way (even with the most powerful computer) of calculating where it will end up - we just have to run the simulation and see what happens. The possible outcomes are:
- The whole population of blobs will eventually die.
- The pattern of blobs will become stable, unchanging from one generation to the next.
- The pattern of blobs will cycle through a number of generations (2 or more) and repeat this cycle endlessly.
- As a variation on 3, the same thing may happen but the pattern shifts as it goes along - a "glider".
- Various patterns will emerge in any combination of 2 to 4.
It is interesting that if the pattern was not initially symmetrical in either the left-right or up-down direction, then such a symmetry may develop during the evolution of generations - and once it does develop, it is never lost. As we watch the patterns evolve, we can understand how this development of symmetry comes about.
It is very rare that the population will grow indefinitely (with these particular rules), and it was not certain for some time whether such a pattern would ever be found. (I am aware of one such discovery - a pattern that repeatedly generates "gliders" - but there may now be others.) This is one of the aspects that John Conway was investigating when he invented his "game". The game actually has many variations (e.g. a board with hexagonal cells) and many possible rules.
Any real board that you play with has an edge, and we don't want an edge in this game. Using a computer program (which may or may not be a spreadsheet) you can get round this problem in two ways:
- Make the board big enough so that the pattern never reaches the edge, or
- Join the left hand edge to the right hand edge, and the top edge to the bottom edge, so that there are no actual edges. This makes a kind of finite, folded 2-dimensional universe - you can go as far as you like in any direction, but eventually you will be back where you started. This is a kind of "bodged job", so to speak, but then our own 3-dimensional universe may be folded in just such a way (I haven't caught up with the latest theory in that department!).
You will find many more examples of evolving patterns, animations, programs you can run and a fuller description of the Game of Life here.
- If you are comfortable using spreadsheets...
- As an alternative to the many programs you can run, you are welcome to have a copy of the simple Excel spreadsheet that I use - I will e-mail it to you.
- It has two grids (each of 30x30 cells, but you can change that) side by side, representing "Generation N" and "Generation N+1". Any pattern you put in the left hand grid (entering 0 or 1 in each cell - they are all 0 initially) will immediately show the next generation in the right hand grid. You can select and copy the right hand (Generation N+1) grid to the clip-board, then select the top left hand cell of the left hand (Generation N) grid, and click "Paste Special... Values" (not "Paste"). If you customize your toolbar to add the icon for "Paste Values" then you will advance the generations each time you click that icon.
Before finishing, it is worth mentioning that the Game of Life is an example of a whole family of investigations into how Nature works, in particular the fascinating study of Cellular Automata. From these investigations we see how even very simple rules, repeated many times, can lead to very complex (and sometimes very beautiful) results.
These investigations focus on models of behaviour that involve loops, often following the pattern shown in this template:
We will see a truly awesome example of this template in action, in the next article!
Magical Loops Part 2: The Amazing Mandelbrot Set
You probably already know about the Mandelbrot Set. At least you may have seen some of the computer-generated images - fractal patterns of incredible detail and beauty (some examples below).
However far one expands an image, one discovers still more detail and beauty... and all of this from a set of extremely simple rules, run in a loop.
The rules themselves are no more complicated than the rules for Conway's Game of Life, which we looked at above. What happens when the rules are run, and the results interpreted in a particular way, is far from simple!
Let's take a quick look, so that we can see where we are going!
The above image shows the complete set. By the end of this article, hopefully you will understand quite a few things about this picture, if you didn't already.
If you expand the view of the "valley" between the biggest blob and the next-biggest blob, you see this:
By the time you have zoomed in another seven times you see this in a tiny part of the original image:
And you can go on and on, deeper and deeper, uncovering still more wonders. I understand that some people, watching the computer program revealing the details, enter a kind of altered state of mind, and it isn't hard to see why.
The problem that I found when trying to understand what was behind these awesome images was that there seemed to be two kinds of explanation. One kind was full of fearsome looking mathematical equations. The other kind avoided the maths but missed out on what was really going on.
Rather foolishly, you might think, I am going to try to explain what I think is happening in a hopefully clear way, but without ignoring the maths.If you are a mathematician yourself (I am not bad at maths, but that does not make me a mathematician), you are probably way ahead of me, and any corrections will be gratefully received!
The Mandelbrot Set, you will quickly discover, is a set of complex numbers, and as soon as you hear that you might head for the nearest exit. Please don't!
It turns out that complex numbers are nothing more than points on a flat map, and tell you where you are on that map, in the same way that latitude and longitude tell you where you are on the surface of the earth. Latitude and longitude are much more complicated than complex numbers - trust me.
The next thing you discover is that you have to manipulate complex numbers using complex arithmetic, at which point you might again be heading for the nearest exit. Please don't!
It turns out that in order to understand the (simple) complex arithmetic used in the Mandelbrot Set, you just need to be able to imagine rotating a triangular shape, and then shrinking or expanding that shape in a way that keeps its proportions the same (it's still the same shape, but bigger or smaller).
This isn't kiddie-speak, by the way: I have just described to you what is really involved in multiplying complex numbers together. If you're comfortable with that, then we're good to go.
All the action takes place in a flat map, and that's the first thing we need to understand. It isn't hard at all - but we really do need to understand it. (By the way, I am using the word "map" to convey an important idea, not in any strictly mathematical sense. You probably won't find other people using this term in this context.)
How do we work out where we are on the map? Much like we do on any other map: how far East (or West) we are from some starting line, and how far North (or South) we are from starting line. In this case, it's easy because the map is flat. The two starting lines cross in the middle of the map at right angles to each other, much like the Equator and the Greenwich Meridian do on the surface of the Earth. Where these lines cross is called the "origin".
How do we measure distances on this map? Just with numbers, either whole numbers like 1, 2, 3, or numbers with fractional parts like 0.265, 1.592, or when we get in deep maybe something like 0.99998265739923. The units we are measuring in don't matter - you can imagine that we are talking about feet, miles, kilometres or light-years.
The map in which the Mandelbrot Set lives has a circular boundary. The circle is 4 units across, or 2 units from the centre to any edge. You can cross the boundary and keep going, but if you do then you are never coming back... we'll see later why this is so.
The last thing to know about the map is how we describe where we are on it. I am going to use distances East and North from the origin ( E and N for short). For example (E=0, N=0) describes the centre of the map. (E=2, N=0) describes the right hand edge of the map. I hope that that's clear.
I will use negative numbers if we are going in the opposite direction (West or South). For example, (E=0, N=-2) describes the bottom edge of the map (the south pole, so to speak). Unfortunately, the minus sign is sometimes hard to spot!
- If you're into maths, you will know that distances East and West represent the normal "real" numbers that we are used to, and the distances North and South represent the so-called "imaginary" numbers, but it isn't necessary to understand this in order to make sense of the Mandelbrot Set.
OK, please go take a look at the map! (There are also some explanatory words below the diagram, which you might not spot unless you scroll down.)
The next diagram is similar to one we have seen before, when we looked at Conway's Game of Life.
It shows the basic loop which is going to be used later in a special way.
An explanation of the loop appears below the diagram.
Now, the way that the Set gets generated is to pick a point on the map (called C), and run the loop we have just been looking at for a while, seeing where Z (the positions generated from the initial C) goes. One of two things will happen: either Z will bounce around inside the Map, staying there forever (in which case C is in the Set), or it will cross the boundary of the Map and zoom off, never be seen again (in which case C is not in the Set). At this stage of explanation, we have absolutely no clue about why this happens!
We then pick another starting position C, and keep going... with the help of a computer, naturally.
And this is where some explanations leave you, high and dry. We'll get there very soon... meanwhile, the following diagram summarizes what has just been said:
Well, we're ready to go. We are going to choose a starting place (C), one which I happen to know is part of the Mandelbrot Set. (I know that because it falls inside the black area on the picture of the Set, and also because I used a simple spreadsheet to track what happens through 2,000 generations - more of that later.)
The next diagram shows what happens on the first time through the loop, and the following diagram will show what happens on the second time through the loop.
You will remember that each time around the loop we are computing a new position Z which is equal to the old Z multiplied by itself (Z2), plus a positional offset equal to the value of C (that last bit will make sense when you explore the diagram).
If you take it slowly and work your way through the two diagrams and their explanations, you should come out the other side in the same way that I did - feeling that at last I was able to get some kind of handle on what's really going on.
This is an example of what happens the first time round the loop:
This is what happens the next time round the loop:
The examples we have looked at so far are where the starting position has been well inside the Mandelbrot Set. But the really interesting region of the Set is its infinitely detailed, fractal edge. What is going on there?
Remember that the edge of the Set is the region where the tiniest change in the starting position makes a huge change to the outcome.
An analogy with a simpler situation might help. Supposing that we were running a loop that just computed the square of a plain number (X2 rather than Z2, say), and fed the result back in as an input.
If X starts out as a number greater than 1, then X2 will always greater than X. If we start out with an initial X = 2, then successive loops quickly zoom off into the distance (2, 4, 8, 16, 32...). If X is greater than 1 by the tiniest amount, then X2 is still greater than X, and the outcome is still the same: successive loops will eventually zoom off into the distance, although at first the increase will be slow. If we start out with an initial X = 1.001, the initial sequence would be 1.001, 1.002001, 1.004006004001... There is no doubt about the outcome, eventually we will be over the horizon and far away.
If X is exactly equal to 1, then X2 is the same as X, however often we run the loops. And if X is less than 1, then the sequence will generate numbers that get smaller and smaller (a half of a half is a quarter, a quarter of a quarter is a sixteenth, and so on).
If we look again at the beautiful coloured images we started with, the black colour indicated the region inside the Set. In this simple analogy, those would be regions where X started out being less than or equal to 1. The other colours represent how fast we zoom away, with the deepest blue corresponding to initial values of X such as 2, where we zoom away very quickly.
Let's look at a similar thing happening with the Mandelbrot Set.
The following diagram shows part of a loop that might, or might not, be going to zoom out of the map.
In the following (and last!) diagram, there is no doubt what will happen next.
Hopefully what you have seen in all this is what the loops actually do when a test is being made to determine whether a starting position C is inside or outside the Mandelbrot Set.
We also have some idea of how, when a starting position C turns out to be very close to the edge of the Set, a tiny change in C in some direction (however small) will have a drastic effect on the outcome. It is these tiny changes that give us the amazing detail we see at the edge of the Set - but surely nobody could have predicted the wonderful complexity of that detail.
The Mandelbrot Set (like Conway's Game of Life) gives us all some insight into how complexity develops in the world around us.
And that's the limit of my understanding right now! If I become further enlightened then I will update this article (in which case I will say so on the Latest Updates page).
You will find plenty more information about the Mandelbrot Set here.
- If you are comfortable using spreadsheets...
- I have a small Excel spreadsheet, which I would be happy to e-mail to you, that computes successive versions of Z from an intial value C (expressed as E and N coordinates, in other words real and imaginary values). The spreadsheet has some 2,000 steps in it, but of course you can make it as long as you like. The screen is split half way down, so that I can see the end of the sequence at the same time as the beginning of the sequence, and it is immediately obvious whether a change to the starting value C has caused the loops to zoom out of control.
- It's a great tool for gaining some understanding of what is going on, so long as you don't try to push the level of accuracy too far for Excel to deal with.
- The actual calculation of a new Z from an old Z (since you're here) is quite simple:
The new E component of Z2 = (Old E component of Z)2 - (Old N component of Z)2.
The new N component of Z2 = 2 x (Old E component of Z) x (Old N component of Z).
To add the value of C, you just add C's E component to the E component of Z2, and you add C's N component to the N component of Z2.
Nature by Numbers: a dreamlike animation of the Fibonacci sequence, the Golden Ratio, and more
In "Nature by Numbers," filmmaker Cristobal Vila presents a series of breathtaking animations illustrating various mathematic principles, and how they apply to Nature.
He begins with the Fibonacci sequence, followed by the Golden and Angle Ratios, the Delaunay Triangulation and Voronoi Tessellations.
This brilliant animation is worth seeing even if the mathematics sound scary. It is a fascinating insight into the mystery of how mathematics, seemingly so abstract, is embodied in the natural world around us.
I have written an article in my Categorian blog which presents the video and provides some notes and links to follow up (if you need any) on the mathematics. I wrote them to help me learn more about what was going on in the animations, but they might be useful for others also.
Click the image above, or go here,
to read my article and to play the video.