Another way to state the Einstein equations—already discussed by David Hilbert in 1915—is as the constraint that the integral of RicciScalar Sqrt[Det[g]] (the so-called Einstein–Hilbert action) be an extremum. … In their usual formulation, the Einstein equations are thought of as defining constraints on the structure of 4D spacetime. But at some level they can also be viewed as defining how 3D space evolves with time.

And indeed experiments have been done which try to enforce this by choosing the angles for the polarizers only just before the photons reach them—and too close in time for a light signal to get from one to the other. … When set up for pairs of particles, Bell's inequalities tend just to provide numerical constraints on probabilities. But for triples of particles, it was noticed in the late 1980s that they can give constraints that force probabilities to be 0 or 1, implying that with the assumptions made, certain configurations of measurement results are simply impossible.

Note (c) for Systems Based on Constraints…Tiling [problems] The constraints discussed here are similar to those encountered in covering the plane with tiles of various shapes. … For some time it was believed that any set of tiles that could cover the plane could be arranged to do so repetitively.

Sometimes the rules serve mainly as a constraint. … Simple progressions and various forms of repetition have presumably been used in music since at least the time of Pythagoras . … Usually they are based on constraints, and occasionally they can be thought of as providing evidence that simple constraints can have complicated solutions.

And with the constraint of reversibility, it turns out that it is impossible to get a non-trivial phase transition in any 1D system with the kind of short-range interactions that exist in a cellular automaton. … The discrete nature of phase transitions was at one time often explained as a consequence of changes in the symmetry of a system.

But emphasis on evolutionary rather than mechanistic explanations for a long time caused little further work to be done along these lines. … There were quite a few attempts—often misguided in my opinion—to use traditional ideas from physics and engineering to derive forms of biological organisms from constraints of mechanical or other optimality. … The idea of comparing systems in biology and engineering dates back to antiquity, but for a long time it was mainly thought of just as an inspiration for engineering.

The proposed content and medium of the messages has however steadily changed, usually reflecting what seemed to be the most significant human achievements of the time—yet often seeming quaint within just a few decades. … Schemes that might however get at least some distance include sending: • waveforms made of simple underlying elements; • long complicated sequences that repeat precisely; • a diversity of kinds of sequences; • something complicated that satisfies simple constraints. … (If cases could be found where the sequences as a whole were forced not to have any obvious regularities, then pattern-avoiding sequences might perhaps be good since they have constraints that are locally fairly easy to recognize.)

Note (f) for Systems Based on Constraints…But if one insists that the variables are whole numbers, then the problem is more analogous to the discrete constraints in the main text, and becomes much more difficult. And in fact, even though such so-called Diophantine equations have been studied since well before the time of Diophantus around perhaps 250 AD, only limited results about them are known.

In 1967 they observed so-called long-time tails not expected from existing calculations, and although it was realized that these were a consequence of fluid-like behavior not readily accounted for in purely microscopic approximations, it did not seem plausible that large-scale fluid phenomena could be investigated with molecular dynamics. … In the 1960s there was also interest in so-called lattice gases in which—by analogy with spin systems like the Ising model—discrete particles were placed in all possible configurations on a lattice subject to certain local constraints, and average equilibrium properties were computed.

Most use the same basic scheme: to look for signals that show a narrow band of frequencies—say only 1 Hz wide—perhaps changing in time. … But although there are now practical constraints associated with the fact that on Earth only a few frequency regions have been left clear for radio astronomy I consider this to be a remarkable example of reliance on details of human intellectual development.