1. Using Internet try to find out all you can about the land of Tor'Bled-Nam.

2. What is the very essence of mathematical visualization? Key-words: magnification, abstract mathematics, complex numbers, miracles of mathematics.

We are now in a position to see how the Mandelbrot set is defined. Let be some arbitrarily chosen complex number. Whatever this complex number is, it will be represented as some point on the Argand plane. Now consider the mapping whereby is replaced by a new complex number, given by

,

where с is another fixed (i.e. given) complex number. The number will be represented by some new point in the Argand plane. For example, if happened to be given as the number , then would be mapped according to

so that, in particular, 3 would be replaced by

and would be replaced by

When such numbers get complicated, the calculations are best carried out by an electronic computer.

Now, whatever с may be, the particular number 0 is replaced, under this scheme, by the actual given number . What about itself? This must be replaced by the number Suppose we continue this process and apply the replacement to the number ; then we obtain

.

Let us iterate the replacement again, applying it next to the above number to obtain

and then again to this number, and so on. We obtain a sequence of complex numbers, starting with 0:

Now if we do this with certain choices of the given complex number c, the sequence of numbers that we get in this way never wanders very far from the origin in the Argand plane; more precisely, the sequence remains bounded for such choices of с which is to say that every member of the sequence lies within some fixed circle centred at the origin (see Fig. 1). A good example where this occurs is the case , since

Fig. 1. A sequence of points in the Argand plane is bounded if there is some fixed circle that contains all the points. (This particular iteration starts with zero and has ).

in this case, every member of the sequence is in fact 0. Another example of bounded behaviour occurs with , for then the sequence is: ; and yet another example occurs with , the sequence being . However, for various other complex numbers с the sequence wanders farther and farther from the origin to indefinite distance; i.e. the sequence is unbounded, and cannot be contained within any fixed circle. An example of this latter behaviour occurs when , for then the sequence is ; this also happens when , the sequence being ; and also when , the sequence being

.

The Mandelbrot set, that is to say, the black region of our world of Tor'Bled-Nam, is precisely that region of the Argand plane consisting of points for which the sequence remains bounded. The white region consists of thoses points с for which the sequence is unbounded. The detailed pictures that we saw earlier were all drawn from the outputs of computers. The computer would systematically run through possible choices of the complex number c, where for each choice of с it would work out the sequence and decide, according to some appropriate criterion, whether the sequence is remaining bounded or not. If it is bounded, then the computer would arrange that a black spot appear on the screen at the point corresponding to с. If it is unbounded, then the computer would arrange for a white spot. Eventually, for every pixel in the range under consideration, the decision would be made by the computer as to whether the point would be coloured white or black.

The complexity of the Mandelbrot set is very remarkable, particularly in view of the fact that the definition of this set is, as mathematical definitions go, a strikingly simple one. It is also the case that the general structure of this set is not very sensitive to the precise algebraic form of the mapping that we have chosen. Many other iterated complex mappings (e.g. ) will give extraordinarily similar structures (provided that we choose an appropriate number to start with – perhaps not 0, but a number whose value is characterized by a clear mathematical rule for each appropriate choice of mapping). There is, indeed, a kind of universal or absolute character to these 'Mandelbrot' structures, with regard to iterated complex maps. The study of such structures is a subject on its own, within mathema­tics, which is referred to as complex dynamical systems.

Text 4. Cosmology and the Big Bang

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Get information from the Internet to confirm or refuse the following statement:

Now, it follows from the equations of Einstein's general relativity that this positively closed universe cannot continue to expand forever. After it reaches a stage of maximum expansion, it collapses back in on itself, finally to reach zero size again in a kind of big bang in reverse.

As far as we can tell from using our most powerful telescopes – both optical and radio – the universe, on a very large scale, appears to be rather uniform; but, more remarkably, it is expanding. The farther away that we look, the more rapidly the distant galaxies (and even more distant quasars) appear to be receding from us. It is as though the universe itself was created in one gigantic explosion – an event referred to as the big bang, which occurred some ten thousand million years ago.¹ Impressive further support for this uniformity, and for the actual existence of the big bang, comes from what is known as the black-body background radiation. This is thermal radiation – photons moving around randomly, without discernible source – corresponding to a temperature of about 2.7° absolute (2.7 K), i.e. -270.3° Celsius, or 454.5° below zero Farenheit. This may seem like a very cold temperature – as indeed it is! – but it appears to be the leftover of the flash of the big bang itself! Because the universe has expanded by such a huge factor since the time of the big bang, this initial fireball has dispersed by an absolutely enormous factor. The temperatures in the big bang far exceeded any temperatures that can occur at the present time, but owing to this expansion, that temperature has cooled to the tiny value that the black-body background has now. The presence of this background was predicted by the Russian-American physicist and astronomer George Gamow in 1948 on the basis of the now-standard big-bang picture. It was first observed (accidentally) by Penzias and Wilson in 1965.

I should address a question that often puzzles people. If the distant galaxies in the universe are all receding from us, does that not mean that we ourselves are occupying some very special central location? No it does not! The same recession of distant galaxies would be seen wherever we might be located in the universe. The expansion is uniform on a large scale, and no particular location is preferred over any other. This is often pictured in terms of a balloon being blown up (Fig. 1). Suppose that there are spots on the balloon to represent the different galaxies, and take the two-dimensional surface of the balloon itself to represent the entire three-dimensional spatial universe. It is clear that from each point on the balloon, all the other points are receding from it. No point on the balloon is to be preferred, in this respect, over any other point. Likewise, as seen from the vantage point of each galaxy in the universe, all other galaxies appear to be receding from it, equally in all directions.

Fig. 1. The expansion of the universe can be likened to the surface of a balloon being blown up. The galaxies all recede from one another.

This expanding balloon provides quite a good picture of one of the three standard so-called Friedmann-Robertson-Walker (FRW) models of the uni­verse – namely the spatially closed positively curved FRW-model. In the other two FRW-models (zero or negative curvature), the universe expands in the same sort of way, but instead of having a spatially finite universe, as the surface of the balloon indicates, we have an infinite universe with an infinite number of galaxies.

In the easier to comprehend of these two infinite models, the spatial geometry is Euclidean, i.e. it has zero curvature. Think of an ordinary flat plane as representing the entire spatial universe, where there are points marked on the plane to represent galaxies. As the universe evolves with time, these galaxies recede from one another in a uniform way. Let us think of this in space-time terms. Accordingly, we have a different Euclidean plane for each 'moment of time', and all these planes are imagined as being stacked one above the other, so that we have a picture of the entire space-time all at once (Fig. 2). The galaxies are now represented as curves – the world-lines of the galaxies' histories – and these curves move away from each other into the future direction. Again no particular galaxy world-line is preferred.

Fig. 2. Space-time picture of an ex­panding universe with Euclidean spatial sections (two space dimensions de­picted).	Fig. 3. Space-time picture of an ex­panding universe with Lobachevskian spatial sections (two space dimensions depicted).

For the one remaining FRW-model, the «negative-curvature model, the spatial geometry is the non-Euclidean Lobachevskian. For the space-time description, we need one of these Lobachevsky spaces for each 'instant of time', and we stack these all on top of one another to give a picture of the entire space-time (Fig. 3). Again the galaxy world-lines are curves moving away from each other in the future direction, and no galaxy is preferred.

Of course, in all these descriptions we have suppressed one of the three spatial dimensions to give a more visualizable three-dimensional space-time than would be necessary for the complete four-dimensional space-time picture. Even so, it is hard to visualize the positive-curvature space-time without discarding yet another spatial dimension! Let us do so, and represent the positively curved closed spatial universe by a circle (one-dimensional), rather than the sphere (two-dimensional) which had been the balloon's surface. As the universe expands, this circle grows in size, and we can represent the space-time by stacking these circles (one circle for each 'moment of time') above one another to obtain a kind of curved cone (Fig. 4). Now, it follows from the equations of Einstein's general relativity that this positively closed universe cannot continue to expand forever. After it reaches a stage of maximum expansion, it collapses back in on itself, finally to reach zero size again in a kind of big bang in reverse (Fig. 4). This time-reversed big bang is sometimes referred to as the big crunch. The negatively curved and zero-curved (infinite) universe FRW-models do not recollapse in this way. Instead of reaching a big crunch, they continue to expand forever.

At least this is true for standard general relativity in which the so-called cosmological constant is taken to be zero. With suitable non-zero values of this cosmological constant, it is possible to have spatially infinite universe models which recollapse to a big crunch, or finite positively curved models which expand indefinitely. The presence of a non-zero cosmological constant would complicate the discussion slightly, but not in any significant way for our purposes. For simplicity, I shall take the cosmological constant to be zero.¹ At the time of writing, the cosmological constant is known observationally to be very small, and the data are consistent with its being zero. (For further information on cosmological models, see Rindler 1977.)

Fig. 4. (a) Space-time picture of an expanding universe with spherical spatial sections (only one space dimension depicted), (b) Eventually this universe re-collapses to a final big crunch.

Unfortunately the data are not yet good enough to point clearly to one or other of the proposed cosmological models (nor to determine whether or not the presence of a tiny cosmological constant might have a significant overall effect). On the face of it, it would appear that the data indicate that the universe is spatially negatively curved (with Lobachevsky geometry on the large scale) and that it will continue to expand indefinitely. This is largely based on observations of the amount of actual matter that seems to be present in visible form. However, there may be huge amounts of invisible matter spread throughout space, in which case the universe could be positively curved, and it could finally recollapse to a big crunch – though only on a time-scale far larger than the years, or so, for which the universe has been in existence. For this recollapse to be possible there would have to be some thirty times as much matter permeating space in this invisible form – the postulated so-called 'dark matter' – than can be directly discerned through telescopes. There is some good indirect evidence that a substantial amount of dark matter is indeed present, but whether there is enough of it 'to close the universe' (or make it spatially flat) – and recollapse it – is very much an open question.

Text 5. Black Holes

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