2D generalizations [of entropies] Above 1D no systematic method seems to exist for finding exact formulas for entropies (as expected from the discussion at the end of Chapter 5 ).

Manifold [properties and] undecidability Given a particular set of network substitution rules there is in general no finite way to decide whether any sequence of such rules exists that will transform particular networks into each other.

However, in an actual shell material can only be added on the outside of what already exists, and this can be represented by restricting θ to run over only part of the range - π to π . … As the pictures in the main text show, shells of actual molluscs (both current and fossil) exist throughout a large region of parameter space.

But above the line, except for reversible rules, there is no guarantee that any pattern satisfying the constraints can exist.

Sometimes physical ridges exist on shells in correspondence with their pigmentation patterns.

The problem was first solved in the early 1960s; the solution using 6 colors and a minimal number of steps shown on the right below was found in 1988 by Jacques Mazoyer , who also determined that no similar 4-color solutions exist.

Examples of undecidability Once universality exists in a system it is known from Gordon Rice 's 1953 theorem and its generalizations that most questions about ultimate behavior will be undecidable unless their answers are always trivially the same. … Also undecidable are many questions about whether strings exist that satisfy particular constraints (see below ).

The Riemann Hypothesis is also equivalent to the statement that a bound of order √ n Log[n] 2 exists on Abs[Log[Apply[LCM, Range[n]]] - n] . … In 1972 Sergei Voronin showed that Zeta[z + (3/4 +  t)] has a certain universality in that there always in principle exists some t (presumably in practice usually astronomically large) for which it can reproduce to any specified precision over say the region Abs[z] < 1/4 any analytic function without zeros.

And indeed this same issue also exists for processes other than growth.

Elaborate pigmentation patterns, for instance, typically exist just on an outer skin, and are made up of only a few types of cells.