But this discovery does not appear to have been followed up, and systems equivalent to simple 2D Turing machines were reinvented again, largely independently, several times in the mid-1980s: by Christopher Langton in 1985 under the name "vants"; by Rudy Rucker in 1987 under the name "turmites"; and by Allen Brady in 1987 under the name "turning machines".

But at intermediate times one will see all sorts of potentially dramatic gullies that reflect the pattern of drainage, and the formation of a whole tree of streams and rivers.

In cases where an infinite sequence of networks is allowed, there are typically particular subnetworks that can occur any number of times, making the sizes of allowed networks form arithmetic progressions.

What practical computers always basically do is to repeat millions of times a second a simple cycle, in which the processor fetches an instruction from memory, then executes the instruction. … Once compiled, a program can be executed any number of times.

Diagrammatic and mechanical methods for minimizing simple logic expressions have existed since at least medieval times.

Indeed, in many respects, what is called mathematics today can be seen as a direct extension of the particular notions of arithmetic and geometry that apparently arose in Babylonian times.

CAEvolveList applies CAStep t times.

In quantum field theory particles of any mass can always in principle exist for short times in virtual form.

Quantum effects Over the years, many suggested effects have been thought to be characteristic of quantum systems: • Basic quantization (1913): mechanical properties of particles in effectively bounded systems are discrete; • Wave-particle duality (1923): objects like electrons and photons can be described as either waves or particles; • Spin (1925): particles can have intrinsic angular momentum even if they are of zero size; • Non-commuting measurements (1926): one can get different results doing measurements in different orders; • Complex amplitudes (1926): processes are described by complex probability amplitudes; • Probabilism (1926): outcomes are random, though probabilities for them can be computed; • Amplitude superposition (1926): there is a linear superposition principle for probability amplitudes; • State superposition (1926): quantum systems can occur in superpositions of measurable states; • Exclusion principle (1926): amplitudes cancel for fermions like electrons to go in the same state; • Interference (1927): probability amplitudes for particles can interfere, potentially destructively; • Uncertainty principle (1927): quantities like position and momenta have related measurement uncertainties; • Hilbert space (1927): states of systems are represented by vectors of amplitudes rather than individual variables; • Field quantization (1927): only discrete numbers of any particular kind of particle can in effect ever exist; • Quantum tunnelling (1928): particles have amplitudes to go where no classical motion would take them; • Virtual particles (1932): particles can occur for short times without their usual energy-momentum relation; • Spinors (1930s): fermions show rotational invariance under SU(2) rather than SO(3); • Entanglement (1935): separated parts of a system often inevitably behave in irreducibly correlated ways; • Quantum logic (1936): relations between events do not follow ordinary laws of logic; • Path integrals (1941): probabilities for behavior are obtained by summing contributions from many paths; • Imaginary time (1947): statistical mechanics is like quantum mechanics in imaginary time; • Vacuum fluctuations (1948): there are continual random field fluctuations even in the vacuum; • Aharonov–Bohm effect (1959): magnetic fields can affect particles even in regions where they have zero strength; • Bell's inequalities (1964): correlations between events can be larger than in any ordinary probabilistic system; • Anomalies (1969): virtual particles can have effects that violate the original symmetries of a system; • Delayed choice experiments (1978): whether particle or wave features are seen can be determined after an event; • Quantum computing (1980s): there is the potential for fundamental parallelism in computations. … And indeed in recent times consequences of this—such as violations of Bell's inequalities—are what have probably most often been quoted as the most unique features of quantum systems.

The h[q] have been introduced in almost identical form several times, notably by Alfréd Rényi in the 1950s as information measures for probability distributions, in the 1970s as part of the thermodynamic formalism for dynamical systems, and in the 1980s as generalized dimensions for multifractals.