How high would it scale?

[MasterIsander]

How high would it scale?

The space they were in was not a room. It was a bundle of potentiality, a soft darkness ready to take any form. Alice stood in the center, and in front of her was a Swan. He was no ordinary bird, a creature of lines and light, an abstract conductor whose name was simply a symbol of purity and cold, mathematical beauty.

—Look,— said the Swan, and his voice sounded like the soft tinkle of crystal. He waved a single wing ray in the air, and a fiery square was born out of the darkness, hovering in front of Alice.
— He has an order. The harmony of repetition. This is symmetry.

He touched the square, and it rotated ninety degrees on its axis. "Click." The square coincided with itself.
"Alone." A quarter turn.
Another click, one hundred and eighty degrees.
"two. Half a turn."
Click — two hundred and seventy.
"Three."
And finally, a full circle — three hundred and sixty. The same as zero degrees.
"Four." These are rotational symmetries.

Then the Swan passed the ray vertically through the center of the square. One half mirrored the other, merged.
"Five." The vertical axis.
The horizontal line is six.
And then two diagonal lines — seven and eight.
— Eight transformations, after which the square remains itself. Eight ways to say, "It's still me."

Alice nodded, her eyes alight with interest. Then the Swan erased the square. A new figure appeared in his place. It looked like three arcs glued together into something sharp and streamlined at the same time— the Reuleau triangle. Swan crumpled it so that all sides and angles became different from each other.
"And this?" Alice asked.
"It's almost an asymmetry. Try to turn it."

Alice mentally tried to combine the figure. It was rotating, but only after a full revolution did it coincide with itself. The axis of symmetry? No axis divided it into mirrored halves.
"Doesn't it have symmetry?" — the girl asked, disappointed.
"Almost." But no. — Swan made the figure make a complete turn. — There is one transformation that leaves any thing unchanged. Even the most uneven and twisted ones. Even you, when you wake up in the morning with your hair in a tangle.
"What's that?"
— The identity. Do nothing. Just say, "It's you." And it counts. So everything in the world has at least one symmetry. One sacred coincidence with oneself.

Alice frowned. She stared at the rotating triangle, at its stubborn, unwilling arcs.
"Is there... anything that doesn't even have this one?" At all? So that it... wouldn't even be itself?

The swan froze. His light lines wavered.
— No, Alice. This is impossible by definition. The object must be identical to itself. This is the law.
— But you said that laws are just agreements in our heads! — the girl did not give up. "Imagine that! Please!"

The swan swayed slowly, as if hesitating. Then he sighed, a sound like wind in the void.
— Alright. But we will think beyond the limit. Not about things, but about... principles.

He erased all the shapes. Now there was only a dot in front of them.
— Let's say it's Non-Existence. Not emptiness, but active denial. The transformation "identity" wants to say, "You are you." But Non-existence responds, "There is no "you." There is no subject for your action." It cannot be fixed to convert.

"Too clever,— Alice snorted. "It's like teasing a cat with the ghost of a mouse." Show me something… which you can almost see.

Swan thought about it. And he created a door. Simple, rectangular, hanging in the dark.
"Is that the door?"
— This is a transformation that leads out of symmetry. — The swan went up to it and opened it. There was no darkness or light outside the door. There was nothing, and this nothing was not empty, it was as thick as rejection. — Imagine an object that is always different. Not just changing, but fundamentally different at every moment. At time t, he is A. At the moment t+0, he is no longer A. And so it always is. If you apply the identity to him — "remain yourself" — it will no longer be applied to what is. It's always late. It is always false.

— Like the Heraclitus River, which cannot be entered twice?
"Worse." Which you can't enter even once. Because the moment you touch the water, it's no longer the river you started entering. And your leg is not that leg anymore. There is no starting point. There is no "self".

Alice walked to the edge of the door and peered into the dense nothingness. She felt uneasy, but she didn't back down.
"So it's something."… No uniform at all? Infinitely fluid?
"No," Swan suddenly perked up. — Fluidity is also a property. This is also something that can be called "identical to fluidity." I think... it must be something that breaks the logic itself. Imagine... — he created a small, bright crystal — X. — Here is object X. And it obeys the rule: X ≠ X. It is not its equal. In your world, this is nonsense. In a world where real contradictions are possible, this is its essence. How do I apply the identity to it? The operation id(X) = X assumes that the result is the same X. But if initially X is not equal to X, then even the identity breaks it. It does not preserve him — it imposes on him a law that is not in him.

Alice was silent, thinking. Then her face lit up.
"I get it!" It's like... a silence that's louder than any sound. But not real silence, but one that negates the very idea of sound. Or like a gaze that sees, but it has nothing to look at because it cancels out what it's looking at. So it's an absolutely asymmetrical thing... it's not even a thing. This is the rule of self-destruction of any rule.

The swan looked at her, and its abstract shape shone with a quiet, respectful light.
"Almost." We've come back to where we started. To Oblivion. But not to a passive one, but to one that acts. Which is the principle of difference before every thing. It is this door itself. A transformation that does not save, but erases. It also erases itself as a transformation. Such a "thing" has no symmetries, because for symmetry you need something stable that can be preserved. And there's nothing to save here. Only a pure, impersonal, asymmetrical difference.

He closed the door and it opened.
"Have we found it?" Alice whispered.
"We found its trail. An idea. You can't hold it, just like you can't hold water in your hands. But you can understand that it is possible. Beyond logic, beyond form. — The swan created a small square again and started it spinning in front of Alice. — This is our world. Symmetrical, recognizable, identical to itself. And then..." he waved his wing, and the square disappeared without a flash, as if it had never existed, —then there is reality without mirrors. Where even the echo forgets its own name.

Alice stared at the place where the square had disappeared for a long time. She didn't see God or meaning. She saw... an opportunity. The possibility of being so different that this otherness destroys the very concept of "being."
"Thank you, Swan," she said at last. "That was the best story about nothing I've ever heard."
"It's not a story," Swan corrected her, and for the first time there was a hint of sadness in his voice. "It's just a door that we opened a crack. And now it will always be with you.