Symmetry, simplicity, and just what is required

Very often, when we look at something, we have an immediate, intuitive sense its rightness or wrongness. This sense of rightness or wrongness most often comes directly from the symmetries we see and our sense about these symmetries.
The essence of this rightness or wrongness hinges on the issue of necessity. There is an intimate and fundamental connection between arbitrariness, necessity, and symmetry, which says, in a nutshell, this. Everything in nature is symmetrical unless there is a reason for it not to be. When this law is violated, we feel that something is unnatural, and that is the way in which symmetry plays such a fundamental role.

Imagine you are looking at the sky one day. Suppose suddenly you were to see a cloud which is perfectly square. Without even thinking, you would know that is was not a natural cloud. You would know it must have been made by an airplane, or by some other non-natural process. You know this instantly, within the first tenth of a second. Why is this so?
It is because you have an immediate familiarity with clouds as symmetry structures. Although clouds are loose and asymmetrical, still their characteristic form, the quality which makes them clouds, is a definite symmetry structure of a certain type. If we were to see a square cloud, we would be seeing a different kind of symmetry structure, and we would know, at once, that it was artificial. We would know it cannot have come about as a result of the normal cloud-making process because the cloud process does not produce that kind of symmetry structure.
This example shows that the symmetry structures in the world are very close to us. We perceive them instantly and subconsciously, without even knowing it. This mode of perception gives us an intuitive sense of which symmetry structures are appropriate or not appropriate in various situations. When we see the square cloud, we instantly register that something is “wrong”. Our sense of what is right and what is wrong thus depends on subtle and detailed awareness of the kinds of symmetry structures which are appropriate and natural in various different situations.

Each thing in the world is subject to various influences. It has various degrees of similarity and difference compared with other things, according to its situation. And in itself also, it has various degrees of similarity and difference. This is what we call its symmetry structure. Symmetry is a precise way of talking about similarities.
We observe that in any thing, there must be just the right amount of similarity and difference. Its internal degrees of similarity and difference must match, exactly, the degrees of similarity and difference which it experiences in the world.
When we make something which is just right, we have hit the degree of similarities and differences — its internal symmetries — just right.
On the other hand, when we are wrong we can also always analyze the wrongness of what we have made in terms of symmetries. Either the symmetries are less than the situation requires. Or the symmetries are more than the situation requires.

In general, a harmonious structure — and the simplest structure — is one whose internal similarities and differences correspond exactly to the degrees of similarity and difference that exist in its conditions. That is the best definition of simplicity.
Consider the shape of a bubble. When we have a soap bubble floating in the air, it roughly has the shape of a sphere. Although we can give various sophisticated mathematical explanations for this fact, there is one very simple explanation, more fundamental than all the others. It is simply this. The air pressure on the inside of the bubble presses out with equal force in all directions. The same is true of the air pressure outside the bubble, pressing in. It presses with equal strength all over the bubble. Under these circumstances the bubble must take on the form of a sphere, because a sphere is the only volume-enclosing shape whose surface is the same at every point.
Suppose you saw a bubble in the shape of a cube. You would know, right away, that something was wrong because a cube has too many differences in it. Mainly, the corners of the cube are different from any other points and the edges are different from the middle of the sides. Such a structure could only come about under circumstances where the forces or processes also had a comparable level of complexity, where the pattern of forces somehow gave rise to eight points which were “special”. Since you know the forces in a bubble aren’t like that, you know the bubble can’t take on the form of a cube.

We can express this idea, in the most general way, by saying that things which are similar must be similar, and things which are different must be different.
Or I can put it more precisely: The degree of similarities which exist in a structure must also correspond to the degrees of difference in the conditions there.

This is a profound idea, which — I believe — no one has so far managed to express in a fully mathematical way. If it could be expressed precisely, it would be the rule from which everything, all form, derives.

A building which is perfectly made, and perfectly simple, is one in which the symmetries correspond exactly to what is required — neither more, nor less — just as we see in nature.

Modern buildings: too few internal symmetries Postmodern buildings: too many symmetries

Making a thing whose symmetries are exactly right is extraordinarily hard. It means, that we have to be so simple that all the necessities are in perfect balance. Simplicity is the state in which all structure is removed, except exactly that structure which is required.

#book/The Nature of Order/2 The process of creating life/17 Simplicity#