In the pictures below, the n th point has position ( √ n {Sin[#], Cos[#]} &)[2 π n GoldenRatio] , and in such pictures regular spirals or parastichies emanating from the center are seen whenever points whose numbers differ by Fibonacci[m] are joined.

Feynman diagrams [and networks] In the standard approach to particle physics, possible interaction processes are represented by networks in which each node corresponds to an elementary interaction, and the nodes are joined by connections which correspond to the propagation of particles in spacetime.

Sequential substitution systems [emulating cellular automata] Given the rules for an elementary cellular automaton in the form used on page 867 , the following will construct a sequential substitution system which emulates it: CAToSSS[rules_] := Join[rules /.

The sequence {1, 2, 2, 1, 1, 2, …} defined by the property list  Map[Length, Split[list]] was suggested as a mathematical puzzle by William Kolakoski in 1965 and is equivalent to Join[{1, 2}, Map[First, CTEvolveList[{{1}, {2}}, {2}, t]]] It is known that this sequence does not repeat, contains no more than two identical consecutive blocks, and has at least very close to equal numbers of 1's and 2's.

Runs of digits [in numbers] One can consider any base 2 digit sequence as consisting of successive runs of 0's and 1's, constructed from the list of run lengths by Fold[Join[#1, Table[1 - Last[#1], {#2}]] &, {0}, list] This representation is related to so-called surreal numbers (though with the first few digits different).

The outer walls of pollen grains are often covered with a certain density of tiny columns that can form spikes, or can have plates on top that can form cross-linkages and can join together to appear as patches.

Going beyond ordinary networks, one can consider hypernetworks in which connections join not just pairs of nodes, but larger numbers of nodes.

Standard axioms for lattice theory are ( ∧ is usually called meet, and ∨ join) {a ∧ b  b ∧ a, a ∨ b  b ∨ a, (a ∧ b) ∧ c  a ∧ (b ∧ c), (a ∨ b) ∨ c  a ∨ (b ∨ c), a ∧ (a ∨ b)  a, a ∨ a ∧ b  a} Boolean algebra (basic logic) is a special case of lattice theory, as is the theory of partially ordered sets (of which the causal networks in Chapter 9 are an example).

One possible such ordering for numbers with a total of m digits is GrayCode[m_] := Nest[Join[#, Length[#] + Reverse[#]] &, {0}, m] The succession of sizes and digit sequences of numbers ordered in this way are shown below.

And in fact in general finding such packings is an NP-complete problem: it is equivalent to the problem of finding the maximum clique (completely connected set) in the graph whose vertices are joined whenever they correspond to grid points on which non-overlapping circles could be centered.