In general the density for an arrangement of white squares with offsets v is given in s dimensions by (no simple closed formula seems to exist except for the 1 × 1 case) Product[With[{p = Prime[n]}, 1 - Length[Union[Mod[v, p]]]/p s ], {n, ∞ }] White squares correspond to lattice points that are directly visible from the origin at the top left of the picture, so that lines to them do not pass through any other integer points.

Most common is to assume that within each phoneme-length chunk of a few tens of milliseconds the vocal tract acts like a linear filter excited either by pure tones or randomness.

Block cellular automata With a rule of the form {{1, 1}  {1, 1}, {1, 0}  {1, 0}, {0, 1}  {0, 0}, {0, 0}  {0, 1}} the evolution of a block cellular automaton with blocks of size n can be implemented using BCAEvolveList[{n_Integer, rule_}, init_, t_] := FoldList[BCAStep[{n, rule}, #1, #2]&, init, Range[t]] /; Mod[Length[init], n]  0 BCAStep[{n_, rule_}, a_, d_] := RotateRight[ Flatten[Partition[RotateLeft[a, d], n]/.rule], d] Starting with a single black cell, none of the k = 2 , n = 2 block cellular automata generate anything beyond simple nested patterns.

Properties [of number theoretic sequences] (a) The number of divisors of n is given by DivisorSigma[0, n] , equal to Length[Divisors[n]] . … The number of ways of writing an integer n as a sum of two primes can be calculated explicitly as Length[Select[n - Table[Prime[i], {i, PrimePi[n]}], PrimeQ]] .

And while there are various triangulation methods that for example avoid triangles with small angles, no standard method yields networks analogous to the ones I consider in which all triangle edges are effectively the same length. … But while explicit coordinates and lengths are not usually discussed, it is still imagined that one knows more information than in the networks I consider: not only how vertices are connected by edges, but also how edges are arranged around faces, faces around volumes, and so on.

History [of computational complexity theory] Ideas of characterizing problems by growth rates in the computational resources needed to solve them were discussed in the 1950s, notably in the context of operation counts for numerical calculations, sizes of circuits for switching and other applications, and theoretical lengths of proofs.

A typical example is finding a placement of components in a 2D circuit so that the total length of wire necessary to connect these components is minimized (related to the so-called travelling salesman problem).

In a soap foam, the geometrical layout of this network is determined by surface tension forces—with connections meeting at 120° at each node, though being slightly curved and of different lengths.

(The maximum length of any chain is sometimes called the dimension of a poset, but this is unrelated to the notions of dimension I consider.)

Polymers whose lengths differ by more than one or two repeating units often seem to smell different, and it is conceivable that elaborate general features of shapes of molecules can be perceived.