Hyperbolic networks Any surface that always has positive curvature must eventually close up to form something like a sphere. … Yet even in such a case one can always define coordinates that nominally allow the surface to be drawn in a finite way—and the Poincaré disk model used in the pictures below is the standard way of doing this. In ordinary flat space, regular polygons with more than 6 sides can never form a tessellation.

Recurrence relations The rules for the sequences given here all have the form of linear recurrence relations. … Standard examples of recursive sequences that do not come from linear recurrence relations include factorial f[1] = 1; f[n_] := n f[n - 1] and Ackermann functions (see below ).

Computational fluid dynamics From its inception in the mid-1940s until the invention of cellular automaton fluids in the 1980s, essentially all computational fluid dynamics involved taking the continuum Navier–Stokes equations and then approximating these equations using some form of discrete mesh in space and time, and arguing that when the mesh becomes small enough, correct results would be obtained. … One issue however is that in the simplest cellular automaton fluids molecules are in effect counted in unary: each molecule is traced separately, rather than just being included as part of a total number that can be manipulated using standard arithmetic operations.

Standard treatment [of relativity] In a standard treatment of relativity theory one way to begin is to consider setting up a square grid of points in space and time—and then to ask what kind of transformed grid corresponds to this same set of points if one is moving at some velocity v . … But as the pictures below illustrate, it implies that light cones tip as v increases, so that the apparent speed of light changes, and for example Maxwell's equations must change their form.

Forms of [engineered] artifacts Much as in biological evolution, once a particular engineering construct has been found to work it normally continues to be used. … (Some self-similarity is also present in standard log-periodic antennas.) … Some knots can also be thought of as objects with complex forms.

From its basic setup the three-body system conserves standard mechanical quantities like energy and angular momentum. … And even now it remains conceivable that the three-body problem could be solved in terms of more sophisticated standard mathematical functions. … With appropriate initial conditions one can get various forms of simple behavior.

But what happens if one takes for example a perfect single crystal—say a standard highly pure industrial silicon crystal—and breaks it? … When a solid material breaks what typically happens is that a crack forms—usually at the edge of the material—and then spreads.

As we found two sections ago , many standard methods of data compression have the same limitation. But at the end of that section I showed that the fairly simple procedure of two-dimensional pointer Examples of all the distinct repetitive patterns that can be formed from arrays of 2×2 and 3×3 blocks.

If this behavior always has an obvious repetitive or nested form then it will often be quite straightforward to analyze. But as we saw in Chapter 10 , in almost no other case do standard methods of perception and analysis allow one to make much progress at all.

For even in its most traditional form it can often deal quite well with those aspects of behavior that happen to be simple enough to be computationally reducible. … This procedure is based on the standard repeated squaring method of finding 2 n by starting from 2, and then successively squaring the numbers one gets, multiplying by 2 if the corresponding base 2 digit in n is 1.