Coin tossing, wheels of fortune, roulette wheels, and similar generators of randomness all work in essentially the same way. … The bottom pictures on the facing page show what happens to two sets of points that start very close together. The most obvious effect is that these points diverge rapidly on successive steps.

As one example, consider a large number of circular coins pushed together on a table. … For identical coins this constraint is satisfied by the simple repetitive pattern shown on the right. … In two dimensions similar issues arise as soon as one has coins of more than one size.

For dice and coins there are some additional detailed effects associated with the shapes of these objects and the way they bounce. … Note that in practice a coin tossed in the air will typically turn over between ten and twenty times while a die rolled on a table will turn over a few tens of times. A coin spun on a table can rotate several hundred times before falling over and coming to rest.

Fluttering If one releases a stationary piece of paper in air, then unlike a coin, it does not typically maintain the same orientation as it falls. … A similar phenomenon can be seen if one drops a coin in water.

Indeed, games of chance based on rolling dice, tossing coins and so on all rely on just such randomness. … It is such sensitivity to randomness in the initial conditions that makes processes such as rolling dice or tossing coins yield seemingly random output.

Two examples of what can happen when the kneading process above is applied to nearby collections of points. In both cases the points initially diverge exponentially, as implied by chaos theory. … What differs between the two cases is the detailed digit sequences of the positions of the points: in the first case these digit sequences are quite random, while in the second case they have a simple repetitive form.

The last two pictures in each row above give the distribution of points whose coordinates in two and three dimensions are obtained by taking successive numbers from the linear congruential generator. If the output from the generator was perfectly random, then in each case these points would be uniformly distributed.

In the third picture, however, the points where the curve crosses the axis come in two regularly spaced families. And as the pictures on the facing page indicate, for any curve like Sin[x] + Sin[ α x] the relative arrangements of these crossing points turn out to be related to the output of a generalized substitution system in which the rule at each step is obtained from a term in the continued fraction representation of ( α – 1)/( α + 1) .

Alkane properties The picture on the facing page shows melting points measured for alkanes. … Unbranched alkanes yield melting points that increase smoothly for n even and for n odd. Highly symmetrical branched alkanes tend to have high melting points, presumably because they pack well in space.

The successive steps in the evolution of each substitution system are obtained at the points indicated by arrows.