Some plants have leaves with simple smooth boundaries that one might imagine could be described by traditional mathematical functions.

Note that isotropy can also be characterized using analogs of multipole moments, obtained in 2D by summing r i Exp[  n θ i ] , and in higher dimensions by summing appropriate SphericalHarmonicY or GegenbauerC functions. … (Sums of squares of moments of given order in general provide rotationally invariant measures of anisotropy—equal to pair correlations weighted with LegendreP or GegenbauerC functions.)

And even in present-day situations, if we are exposed to objects or activities outside the areas of human endeavor with which we happen to be familiar, it can be very hard for us to tell which features are immediately purposeful, and which are unintentional—or have, say, primarily ornamental or ceremonial functions.

This function is shown below.

In Mathematica functions like TrueQ and IntegerQ are set up always to yield True or False —but just by looking at the explicit structure of a symbolic expression.

Around the same time Michael Barnsley also used so-called iterated function systems to make pictures of ferns—but he appears to have viewed these more as a curiosity than a contribution to botany.

Note, however, that in predicate logic the expressions that appear on each side of any rule are required to be so-called well-formed formulas (WFFs) consisting of variables (such as a ) and constants (such as 0 or ∅ ) inside any number of layers of functions (such as + , · , or Δ ) inside a layer of predicates (such as  or ∈ ) inside any number of layers of logical connectives (such as ∧ or ⇒ ) or quantifiers (such as ∀ or ∃ ). (This setup is reflected in the grammar of the Mathematica language, where the operator precedences for functions are higher than for predicates, which are in turn higher than for quantifiers and logical connectives—thus yielding for example few parentheses in the presentation of axiom systems here.)

(Examples include evaluating standard mathematical functions and simulating the evolution of cellular automata and Turing machines.) • NP (non-deterministic polynomial time): solutions can be checked in polynomial time.

And certainly pattern matching with __ in Mathematica, as well as polynomial manipulation functions like GroebnerBasis , routinely deal with problems that are formally NP-complete.

A still better approximation is obtained by subtracting Sum[LogIntegral[n r i ], {i, - ∞ , ∞ }] where the r i are the complex zeros of the Riemann zeta function Zeta[s] , discussed on page 918 .