Indeed, games of chance based on rolling dice, tossing coins and so on all rely on just such randomness. … One can roll this ball like a die, and then look to see which color is on top when the ball comes to rest. … Small changes in this speed are seen to make the ball stop with a different color on top.

[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 top pictures, the initial condition contains a single black cell; in the bottom pictures, it contains a line of three black cells.

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.

Coin tossing, wheels of fortune, roulette wheels, and similar generators of randomness all work in essentially the same way. … And indeed it is quite feasible to build precise machines for tossing coins, rolling balls and so on that always produce a definite outcome with no randomness at all. … The picture at the top of the facing page shows a few steps in this process.

[No text on this page] The behavior of the code 20 cellular automaton from the top of the facing page for all initial conditions with black cells in a region of size less than nine. In most cases the patterns produced simply die out.

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.

Most of these structures eventually die out, sometimes in rather complicated ways. … In most cases everything just dies out. … But not all persistent structures are that simple.

Website A large amount of additional material related to this book and these notes will progressively be made available through the website www.wolframscience.com .

Projections from 3D [cellular automata] Looking from above, with closer cells shown darker, the following show patterns generated after 30 steps, by (a) the rule at the top of page 183 , (b) the rule at the bottom of page 183 , (c) the rule where a cell becomes black if exactly 3 out of 26 neighbors were black and (d) the same as (c), but with a 3×3×1 rather than a 3×1×1 initial block of black cells:

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.