An automorphism is a map from a thing onto itself that preserves everything about it. Not a change. Not an improvement. The same object, carried onto the same object, with every relation intact.

Most of what gets sold as love is a different map. It takes you somewhere and calls the distance growth. You come out the other side described as better, and you can't check the claim, because the person doing the describing is also the one who moved you.

An automorphism doesn't do that. It moves you and leaves you provably yourself.

Every object has one for free. The identity — the map that does nothing, that sends each part of you to exactly where it already was. It counts. It is in the group. But nobody writes columns about it, because it asks nothing and costs nothing and you can have it alone in a room.

The interesting ones are the others. Something moves and nothing breaks. You get rearranged — a new apartment, new hours, somebody else's family at the table, a Tuesday that used to be yours — and every relation that was true of you before is still true after. You are not diminished into the arrangement. You are the same structure, sitting differently.

Here is the part that took me a while. The size of the group is the measure. A thing with a large automorphism group is a thing with a great deal of symmetry — many ways to be moved and still be itself. A thing with only the identity is rigid. Nothing can be done to it at all.

So the capacity to be loved is not a matter of being fixed in place. It is the opposite. It is how much can happen to you without costing you anything true.

I think about this on the walk along the river, where the same water goes past in a different order every day and the river is not a different river. Nobody would say the river compromised.

• • •

The test is not whether it feels like love. It is whether, afterward, everything that was true of you is still true.

Ask it about a person. Ask it about a job. Ask it about a city.

If the answer is no, that was not the map.