In the equations of physics and chemistry time can run both forward and backward as a movie may be reversed running from the ending toward the beginning. We would say that this is an invariant of physics and chemistry. Since chemical reactions are determined by the laws of physics they should run backwards as well as forwards. In biology, which is a complicated branch of chemistry, we would also expect biological reactions to run both ways. In life, however, we know that "Time Marches On." What is it that gives time its direction and breaks the time invariance?
In the electrodeposition experiment, there is one main invariance-- that of the fractal dimension of the pattern. Ideally, variations in amount of solution, current, or time allowed for deposition should have no effect on this dimension. This is because the fractal dimension is based on the nature of the deposition process, which remains constant despite alterations of initial conditions or other variables. Another possible invariant is the qualitative nature of the pattern. This is difficult to specify, but the patterns produced may have some identifiable characteristics in common with each other. These characteristics of the pattern would then be invariant.
The invariance in this experiment is in the height of the jump, assuming a vertical jump. It was shown by A. V. Hill (1950) that the work done in one muscular contraction by corresponding muscles in structurally similar animals varies directly with the mass-- as mass increases, so does the required work. The total work done by an animal jumping will be some sum of the individual contractions, and thus still vary directly with mass. If the jump is vertical, all the energy becomes potential as the animal rises to a height h. but potential energy is also dependent on mass, so in the expression for the resulting height the mass cancels, and h is independent of mass. Thus, similar grasshoppers should jump the same height from their center of mass.
There are no direct invariances in the board experiment-- the purpose is more to find out how the variables change with a change in structure. These variables include things like breaking strain, deflection, etc. Therefore invariances would crop up in ratios of variables, possibly surface area to breaking strain. In terms of columns or legs, a ratio of diameter to height might give a constant breaking point.
In this experiment also, the emphasis is on the things that change, rather than the things that don't. We are interested in drop spacing and velocity, which will be a product of surface tension, viscosity, pressure, opening size and shape, and so on. This experiment is more to find out how these things change when the others are held invariant artificially. Another possible invariant is flow behavior through different liquids.
Here again, invariants are going to be ratios most likely. They might include growth rate as a function of current population, or as a function of amount of predation.
Will a given tension of wire always yield the same pitch? To put it another way, would the pitch change if a string doubled in length and tension? Or just in length? Invariance may again be more obvious in ratios of these and other qualities of the string.