Dimensionality in the Annealing Metaphor

(~960 words)

The concept of annealing has escaped the materials science context where it originated. In the material context, it is straightforward to see how the characteristics of physical space might constrain annealing. For example, the 3-dimensional “kissing number” (with a value of 12) constrains the number of neighbors a 3-dimensional atom can have and thereby the possibility of movement, diffusion, and change within a 3-dimensional material. If the kissing number was greater than 12, a given annealing treatment regime would be more effective. The kissing number is merely one measure of the abundance and profile of interatomic associations in a material—associations which carry stress-energy, and which stress-energy is reduced in annealing via the shifting of those associations. In many systems, among them metals, societies, brains, each “atom” or basic unit has a number of associations, which in turn have their own associations, and the shape of the association network can be reckoned as living in a space with a certain dimensionality and geometry. Analyzing such spaces helps to enrich the annealing metaphor, pinpointing similarities and contrasts with material annealing in its 3-dimensional Euclidean space.

Let’s start where annealing began. A piece of metal essentially never has pure crystal structure throughout. On the one hand, the casting process usually proceeds in an uneven way, with multiple centers of solid crystalline order (“nucleation sites”) growing inside the molten metal as the entire piece solidifies, leaving irregular “grain boundaries” between the multiple “grains” that grew. Metal pieces that solidify quickly have many small grains, and pieces that solidify slowly have few but large grains. On the other hand, nothing is perfect, and even a mostly pure crystalline grain will have “point defects” like a vacancy where there should be an atom, or an interstitial atom wedged where there should be none. In a perfect FCC or HCP metal crystal, every atom would have 12 neighbors (following the 3-dimensional kissing number), but in any actual piece of metal, however, many atoms will not have 12, and their vacant or interstitial neighbors present opportunities. Metals anneal when vacancies and interstitial atoms march through the material (often along grain boundaries), readjusting grains to relieve stress or even creating entirely new grains within.

Annealing is well known to be affected by the preexisting abundance of point defects and grain boundaries and the treatment temperature, but the theoretical angle to armchair about here is how different dimensionalities (and kissing numbers) would affect annealing. Given an abundance of point defects (say, 0.1% of lattice sites being irregular), how many of these any given atom has as a neighbor will depend on the kissing number. If this number is very small, then any atom will rarely have a defective neighbor, but if this number is very large, then any atom likely may. The more atoms with defective neighbors, the more possibilities for vacancy or interstitial diffusion, and the greater efficiency at minimizing stress. So the hypothesis is that increasing the dimensionality of a metal makes it easier to anneal. Unfortunately, our Universe furnishes us with zero tools to change the dimensionality of the space that metals are in, so this must remain a hypothesis.

Instead of a metal, one can imagine a society where every individual has 12 friends. Or instead of 12, maybe 6, or perhaps just 2. These three numbers are the kissing numbers in 3, 2, and 1 dimensions, respectively, and they are totally conceivable even for real humans in our ordinary World, albeit rather drearily. The point is that human society, which it ostensibly embedded in the 3-dimensional World, can have a structure that belongs to a different dimensionality. If everyone had 24 friends, then human society would be effectively 4-dimensional. These numbers are of course totally crude, but there is an actual fact of the matter as to how humans are associated in society. I have spoken to f people in the past week, paid g people in the past month, touched h people in the past year. If we had an appropriate dataset, it would be possible to reconstruct the effective dimensionalities of human societies, down to their local fluctuations across communities and time. The dimensionality of urban areas is certainly higher, for instance, than the dimensionality of rural areas.

Some societies are easier to anneal than others. Many people have too many associations or too few for their local social lattice, but in societies with high dimensionality, it is easy for people to diffuse and find their balance. Because of the parallel association of dimensionality and “heat” in annealing, to heat a society is to increase the number of interpersonal associations, and to decrease interpersonal associations (as in pandemic lock-downs) is to cool it.

The hallmark of brains is that neural tissue is not populated by oval-ish cells like everywhere else in the body, but instead by extraordinarily spindly cells that branch and reach and associate directly with myriads of other neurons, clearly many more than 12 of them. Axons and dendrites are the transcendence of neurons in neural tissue beyond the 3-dimensional kissing number. The brain is 3-dimensional but its neural network is not. The exact mechanism of annealing in brain is unknown but it certainly involves neurons changing which other neurons they are connected to and how strongly, and because of neural tissue’s higher-than-3-dimensionality, it is uniquely capable of doing so among all human tissues.

There are many more systems whose dimensionality and annealing properties could be analyzed. The fundamental picture is that, despite the 3-dimensionality of space in our Universe, systems inside of it can adopt effective dimensionalities that are quite different via various mechanisms, and because there is a keen relationship between spatial associations and annealing, annealing in these systems will vary in methodical ways from annealing in materials.

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