Self-assembly means symmetry

08 Jul 2017

I may have spent rather a lot of time playing with small spherical magnets, zen magnets.

icosahedron made of zen magnets

A chain, connected to a blob, will draw more chain out of the blob with only a little help - you connect the chain to the blob and then pull on it. A helix, connected to a chain, will slurp up the chain, creating more helix of that same diameter, powered by a gentle swirling motion.

This is called self-assembly, which is annoying, because the word doesn’t explain anything. It’s like asking “why do birds tend to fly elsewhere for the winter?” and getting the answer “instinct”. That’s not an explanation, it’s just a label.

The mechanism is the symmetry of the chain, or the helix. They’re not accidental, they are crucially necessary for the self-assembly to occur.

A three-quarters finished dodecahedron, lying on a table, doesn’t necessarily get completed. You could imagine it calling out for completion, asking the world “please finish me!”. But it might be in an unsympathetic, hard-hearted part of the world, and not find a receptive ear. But if you find a three-quarters finished dodecahedron, and want to heed its imaginary request, then you can read the details from one part of the dodecahedron, and via symmetry, apply it to the unfinished part of the dodecahedron, in order to finish it. Not all shapes allow even the most receptive environment to read them and finish them like that.

Viral capsules are geometric for the purpose of self-assembling. If the temperature or the pH of the surrounding water is right, even surface tension or other thoughtless forces are receptive enough to finish the shape.

(This image is not actually of viruses, these are much larger, but still microscopic, radiolarians drawn by Haeckel. The point is that symmetry occurrs naturally.)

Radiolarians drawn by Haeckel

Normally we think of active processes like wind or waves acting on passive substances such as stone or clay. In the case of self-assembly, we flip the subject / object role. The usually-passive substance is recruiting, maybe even compelling, the dynamics of the surrounding water to complete itself.

Normally, we think of computer programs acting on data. A sorting procedure sorts the records. However, some computer programs are not simply slightly deflected from their normal course by particulars of the data they are operating on (the sorting procedure takes longer on bigger files), they are wildly, intensely influenced by the particulars of the data they are operating on. In those cases, it is more reasonable to say that the data controls the program, inverting the way we speak about them. Those computer programs are interpreters. The Python interpreter obeys the text of the Python source code.

To say that the Platonic icosahedron uses the force of magnetism, the loose magnets, and the hands, eyes, and brain of an obliging passers-by, to cause a version of itself to appear in the real world is only somewhat unreasonable.

Circles and spheres, parabolas and catenary curves, arguably owe their frequency to the fact that, due to their symmetry, they can be found as fixed points of more or less mindless dynamics. If you want to make something, or find something, and the thing that you want to make is symmetrical, then you may be able to lean on the action of the symmetry to help you make it or find it. For example, a potter’s wheel rotates, which is the action of the symmetry of the bowl which the potter’s wheel makes. You could make a wire by creating a very long thin mould and then casting it, but the usual way to do it, wire-drawing, incorporates the linear motion which is the action of the linear symmetry of the goal wire. Rorsach ink-blots are made symmetrical via the folding action (180 degree rotation of half of the paper), and so on.

Maybe the procedure that Feynman described for manufacturing flat surfaces: “We can make flats by rubbing unflat surfaces in triplicates together—in three pairs—and the flats then become flatter than the thing you started with.” could have been invented in the first place by trying to figure out an action which has flatness as its only fixed point.

There are procedures, techniques, practices that relate to themselves in a self-consistent, lawful way. Deciding things via using “one person one vote”, and laws that forbid particular things to everyone, are symmetrical with respect to a permutation of people (Rawls’s veil). Kant’s Categorical Imperative is an attempt to “bootstrap” ethical truths from this kind of symmetry, similar, as far as I can tell, to the bootstrap method in physics. The use of an abacus or counting with pebbles on a counting-board has lots of internal symmetries among the procedures for counting, adding, and multiplying, as well as the correspondence between signifier and and signified.

(I am not an expert, but) I think pushing pebbles around a board, beads in grooves or wires, tends to explode into fully-formed existence in history without a long historical record of how it was developed. I think this is because a single person, working with a partial abacus procedure, can act like the receptive environment (the water of the right temperature and pH, or the python interpreter), and finish the procedure according to symmetries inside the partial procedure. Flipping the sense of the verb, the partly finished abacus-math procedure controls the person who is inventing it, using their receptive mind to finish inventing itself.

This leads to recognizability. The chinese counting-board user would recognize and understand what the incan yupana-board user is doing.

I think Scott Alexander’s “How the West Was Won” is too breezy and casual, handing out certificates of universality and ahistoricity left and right. When Scott Alexander describes a summoner summoning a demon, and then the demon wearing their skin, what they mean, in boring non-supernatural language, is that someone:

  1. created or encountered a fragment of a highly-symmetrical idea, and then
  2. the idea self-catalyzed to its full, recognizable state inside their mind and then
  3. influenced the inventor (or the inventor’s culture) in a characteristic way, “possessed” by this fascinating, highly-symmetrical idea.

The “wearing their skin” aspect corresponds (straightforwardly) to the idea not changing the inventor’s name, or the inventor’s culture’s name.

How could an idea, however fascinating, control someone? One possibility is it might exploit the fractures between multiple people.

A focal point a.k.a. Shelling point is some point that people will tend to use in the absence of communication, because it seems natural, special, or relevant to them.

In a lab experiment, if you are asked to point to one of four squares, and you are rewarded if you successfully coordinate with someone else playing the same game, and one of the four squares is a different color, then you would likely point to the square of distinguished color.

By creating expectations that everyone else will behave selfishly, the idea of capitalism recruits people who might otherwise have behaved unselfishly to behave selfishly.

The idea that “all decent people X” (for example, X might be “attend a Christian church” in the American Midwest), recruits people who might not otherwise X, in order to signal decency to everyone else, who “of course” are familiar with the idea that “all decent people X”.

After a particular course has particular well-known shape, academics teaching that course may conform very narrowly to that shape:

If something big and difficult, requiring many people working together, building a very large bridge for example, was accomplished for the first time via a particular method. Then someone who comes along later and wants to do something similar via a slightly-different method. Every potential supporter might ask “Why are you doing it this way? Everyone knows that this prior art did it this other way. It’s risky enough attempting to build a bridge. Don’t stray from what works! You won’t be able to recruit supporters!”. That is, the second mover may have a harder time recruiting supporters unless they imitate the first mover’s method quite closely.

That is, the first time something difficult is accomplished, it carves a groove in the space of how things are accomplished and the groove is likely to be exactly followed the second time, which makes the groove deeper, and so on.

Some moderately complicated and symmetrical social structure ideas, such as the idea of the recursive patron/client relationship, the idea of the Magna Carta, the idea of the joint-stock corporation are candidates to be this kind of Alexandrian demon.