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Lorgha's father said she had to marry a frog. That was the common way to refer to the ruler of the Anamorphic Realm, because no one knew his real shape. It was only in a cylindrical mirror that his reflection looked like a frog: a result of the curving surface distortion. His true form was not allowed for anyone to behold; Lorgha, as his wife, would be the only one among humans to do so.
At first, she wouldn't hear of it.
Her father waited for her sobs of protest to subside. "Haven't you learned anything from mathematics?" he said reproachfully. "Haven't you learned that what you need for ruling a kingdom is precision, logic, and the sense of inevitability that it instills? You have to marry that which will bring a stable alliance to our lands. Has mathematics not taught you to submit to rational reasoning?"
But that wasn't the only thing that Lorgha learned. Her favorite subject was geometry, and she was fascinated by how, with the same ease as it described our universe, it also described paradoxical realms: the ones where parallel lines converge, like on a sphere, and ones where they diverge, like on a hyperbolic surface.
The prince looked like a puffy frog in the cylindrical microcosm of the mirror, and Lorgha racked her brain as to what his real form was. Everything about his image was an artifact of mirror distortion. The downturned arch of a smile, the long, delicate fingers, thinner than a human's, which, if Lorgha was honest with yourself, set her secret places aflutter: she didn't know whether those nimble curves were an illusion of bending lightrays.
She didn't know how to find out. She didn't know enough geometry to transform his image into his real shape.
One day she found a clue, but not in mathematics. Stealing away from her watchful governesses, she discovered a book of fairy tales in the attic. It spoke about a princess who married a frog. She put two and two together.
"I found out what he really looks like. He's a man," Lorgha said to her tutor. "That's all there is! That's his whole secret, his mysterious, mind-twisting form. It's right under our noses."
"How did you arrive at this conclusion?" the baffled tutor asked.
"In the fairy tales the princess marries a frog that turns out to be a man."
The tutor shook her head. "But have you tried to examine that reflection from the geometric perspective? Did you subject each point in the reflection to a reverse spherical mirror transformation, so as to compute its location in reality?"
"I don't have enough knowledge for that yet," Lorgha said. And I don't need it, because I found a shortcut, she refrained from adding. Life is far different from what mathematics - and you - can teach me. In fairy tales, the simplest answer is often true, the secret is in plain view, overlooked by everyone.
"Then I'll do it for you. I hope you haven't told your father your decision yet," the tutor said, glancing at her student with concern. Lorgha's heart quivered. Flush with pride and confidence in having figured out the riddle, she had already told her father 'yes'. "Let's start with this. The cylindrical mirror creates distortion in which direction?"
"Horizontal," Lorgha said. It was obvious: the mirror curves horizontally, but is straight vertically.
"So if it does not create a vertical distortion, how is it possible that your prince is a man squished into a small, round animal shape?"
Lorgha's conviction developed crack lines. "Perhaps it's a man lying on his side," she said, hanging on to hope.
The tutor sighed. She proceeded to write out equations and draw figures. As her pencil flicked across the page, a figure materialized. Why is she drawing a piece of driftwood?, Lorgha thought at first. An old log with deep grooves, curved into an unnatural, circular shape? Like an artsy furniture item that imitates nature, meant more as a conversation piece than to sit on. But mathematics has taught her inexorable logic. The pencil's journey across paper showed her how the converging lines in the mirror translate into parallel lines in reality. It's too horrifying to admit it yet, but she knew she'd have to. That which was crystallizing before her eyes, the grooved log with an elongated vestige of a head - that was her future spouse, the co-ruler of the kingdom. That's his true shape.
Those skinny, lifeless twigs were the mirror-fingers that had set her heart aflutter. In her mind's eye, in her helpless rage, they snap off with a dry crack as she stomps on them with her heel.