"Through a line and a point not in the line there is exactly one plane."
She's that point, she knows. Shuji and AKIRA are the line, and she's the point outside it. Nobuta doesn't mind; it's the only way a triangle can be formed, they can't all be collinear, and this geometry, the unlikely 180 sum of corners pulling at different angles, is the secret to her smile.
You can see the building his dad works as president in from the tofu man's house. A building that tall needs a strong base to support it. Triangles are stable, more stable than squares. Akira knows all this because he does listen to people sometimes, even when he doesn't really have to.
"One of these days," he says to his hand, "people are going to build castles over us, kon!" And he dissolves into giggles at the mental image.
Because haven't they already built monuments over them? The horror booth, the self-expressive paint-on-uniform trend, the business. Most of all there were the changes, big changes, small changes, inside and outside and between them. And there were the things that wouldn't change. There's Shuji and Akira and Nobuta, ever changing and never changing at the same time, the only kind of ever-after he wants in this not-fairy-tale.
("I'm not gonna be a dime on the street when I grow up," he tells Shuji the following morning, arms flapping in excitement. "I'm going to be part of a castle. Shuuuji and Nobuta too!"
"I'll actually be happy for you if you just grew up," Shuji retorts.)
It's very simple, really. Him, Nobuta, Akira - because they are only three, it's simple. If it were different number - if they were four, like Aoi had tried to make them, or five, or ten, it never would have worked out. It would have been a tangle of complications and expectations, which Shuji has always hated. Hell, even if they were just two, if it were just Shuji and Akira or Shuji and Nobuta, it wouldn't have worked out either; the shortest distance between two points is a line, and Shuji isn't the type of person you can force into that intimacy.
But as things are, they're only three. It's all so simple, basic, in fact, but sometimes Shuji has a hard time believing that this friendship (the word still feels awkward in his tongue and in his mind, he wonders how Akira can say it so easily, but then again it's Akira) is real.
The miracle here is in the math. Shuji figures he can believe in it.