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. … And it turns out that in this particular case this pattern is quickly produced if one actually pushes coins together on a table. … In two dimensions similar issues arise as soon as one has coins of more than one size.

[No text on this page] Further examples of three-dimensional cellular automata, but now with rules that depend on all 26 neighbors that share either a face or a corner with a particular cell. In the top pictures, the rule specifies that a cell should become black when exactly one of its 26 neighbors was black on the step before. In the bottom pictures, the rule specifies that a cell should become black only when exactly two of its 26 neighbors were black on the step before.

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. … Things appear somewhat simpler with boiling points, and as noticed by Harry Wiener in 1947 (and increasingly discussed since the 1970s) these tend to be well fit as being linearly proportional to the so-called topological index given by the sum of the smallest numbers of connections visited in getting between all pairs of carbon atoms in an alkane molecule.

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. … And if one does this in practice, what one will typically find is that the outcome seems quite random. … It is such sensitivity to randomness in the initial conditions that makes processes such as rolling dice or tossing coins yield seemingly random output.

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.

In each case, all cells are initially white, and one of the rules given on the left is applied for the specified number of steps. Note that in the later cases shown, the head often visits the same position on the grid many times.

• Is there a path shorter than some given length that visits all of some set of points in the plane? … One can get an idea of the threshold of NP completeness by looking at seemingly similar problems where one is NP-complete but the other is in P. Examples include: • Finding a Hamiltonian circuit that visits once every node in a given network is NP-complete, but finding an Euler circuit that visits once every connection is in P. • Finding the longest path between two nodes in a network is NP-complete, but finding the shortest path is in P. • Determining satisfiability for a Boolean expression with 3 variables in each clause is NP-complete, but for one with 2 variables is in P.

This basic mechanism may well be the main one at work in many biological systems: each cell or each organism becomes separated from others, and while it survives, it can exhibit organized behavior. … It is an inevitable feature of having a closed system of limited size that in the end the behavior one gets must repeat itself. … And instead, starting from any particular initial condition, the system will only ever visit a tiny fraction of all possible states.