What is Symmetry?

I would like to explain what symmetries are. So, what is symmetry? Because symmetries are everywhere. They appear in daily life, they appear in art, they appear in mathematics, and they also appear – and this is my main point – they appear and they are fundamental in theoretical physics. So, what is symmetry?

I should start by showing this ball. A sphere has infinitely many symmetries. For example, if you turn it around like this the sphere does not change. In the same way, if I take a mirror and I put a mirror in the middle, the sphere would not change, it would look the same in the mirror. And so on, and so forth. So, this is an example of symmetry of an object.

What would be a way to define symmetry? A symmetry of an object in space is a transformation of space, like a rotation, for instance, or a mirror. It is a transformation of space, but it does not transform the object. That is an important thing: symmetry is something that could transform something, for example, a sphere, but does not transform it. There is some tension between something that is transformed and something that is not.

There are many examples. For instance, if you have a set of cards and you take one, you see a painting like this. A priori it has no symmetry. No left, no right symmetry. Then, if you take another side, there is still almost no symmetry. A heart here could have left and right symmetry. But then there is this and this that breaks this symmetry. So, this object still has no symmetry. If I take other examples, like this one, it has rotation symmetry. This is what we call symmetry of order two in mathematics. Because if you turn it twice, you do a complete turn, so the space itself will be invariant. We are back to nothing, to the identity. These were examples of symmetries of order two, like mirror symmetry, left and right, like a rotation of a half turn. But there are, of course, other possibilities.

The other possibility is that if you take a square, you have left and right symmetry. Mirror symmetry with respect to the middle. But you also have up and down symmetry with respect to a mirror that will cut this square here. If I take a three-dimensional object, I can have even more symmetries: I have left and right symmetry like this, and again left and right for each of these faces. So this object, as we can see, has many more symmetries. Each of them is an order two symmetry.

Now lets go back to the square, and I will give new examples, which are not order two, but order four now. If I do just a quarter of a turn, than this is also invariant. Because a square is an order four symmetry. We can do 4 turns, and the space will be back to itself. And the object itself is invariant on each fourth turn of a complete turn, 90 degrees.

Till now I gave examples of symmetries, which were of order two or order four – a finite order. These symmetries are called by mathematicians “discreet symmetries”, in the sense they are not continuous, just for the finite steps. If you take a rotation of a sphere, you have infinite symmetry. Any rotation of whatever angle along the axis that goes through the center of a sphere is symmetry. Of course, there is some notion of cyclicity, in the sense that if I make a rotation of 360 degrees, the whole space itself will also left invariant.

The central point of symmetry is that it is a transformation, but the most important is what is not being transformed. So, it is a bit paradoxical property of symmetry. In more technical terms, what is important is the invariance. It is very fundamental both in mathematics and in physics.

Till now I discussed a notion of symmetry of an object. It was relatively concrete, it was an object in space, something we can imagine, something we can see. But there is also more abstract notion of symmetry, which is also very fundamental in modern mathematics. This is the idea of symmetry of a space itself. Not just an object in space, but the space itself. So, it is the symmetry of properties of the various spaces that we can study in mathematics. It is such a deep notion, that there is the idea of mathematician Klein, who proposed two centuries ago to define geometry via symmetry. So, the other way around: not to start with geometry and then to notice that there is symmetry; but to define geometry from properties.

It is a bit too vague, so I should give an example. Consider a three-dimensional space that we are in. The first important symmetry is what we call homogeneity. Homogeneity means, like the democracy principle, that all points are equal. So, if you are in an empty space, every point is, obviously, the same. It is about mathematics. But it is also important in cosmology. If you consider the universe at cosmological scale, even galaxies would be like a grain of dust – infinitesimally small. At this scale, the whole universe is homogeneous. Of course, there are some tiny deviations, like the Earth. But generally you can model universe very well like homogeneous system. This was one property.

Then there is a second property, which is that all directions in space are equal. Whatever direction you look at the sky still would be equal, if you are at cosmological scales or in the empty mathematical universe. If you try to characterize these properties mathematically, than you would say, that property of homogeneity means that the universe is invariant on translations by some distance in some direction. So, homogeneity is nothing but the universe invariant in translation.

Till now I spoke about the homogeneity of space, so all points are equal. But there is a second property of space, which is very important. In technical terms, it is called isotropy. Isotropy in Greek means that all directions are essentially the same. It is again the democracy principle – all directions are equal. In practice, it means that in any direction the universe looks the same, since it is completely empty. If you consider the universe at cosmological scales, all directions will appear exactly the same.

This is the property of isotropy. And these two properties of symmetry are related to transformations that are called translations, because we can translate the whole universe by a finite distance in a given direction. Invariance of the universe and translation is homogeneity, while isotropy is invariance of space and rotations. And as I said what is fundamental in symmetry is what is invariant. And what is invariant? What is preserved by rotations and translations? These are precisely length and angles. Each of them knows about relativity in the sense that the distances between the points are in respect to two points, and angles are always with respect to two directions. So, in symmetry there is always a distinction between relative and invariant, absolute. It would be a good conclusion to say that symmetry is about what does not change, but what could change, and about what could have been relative, but is actually absolute.