Upward, Not Northward

The Square comes back from Spaceland with one phrase.

It's the best he can do. He was there — saw all of Flatland from above, saw the insides of every house and creature laid open as if the roof were glass, understood in the way you understand something when you've lived it rather than reasoned about it that there is a direction perpendicular to every direction he can name. And now he's home, and he has this phrase.

When he tries to demonstrate it to his Grandson — the boy who had first asked about 3³ — he can only shake a Square "in a purposeless manner, much to the amusement of my Grandson, who burst out laughing louder than ever, and declared that I was not teaching him, but joking with him."

He knows it. He can't show it. The knowledge and the demonstrability are on different channels.

What happens to the phrase over time is the part that stays with me.

About eleven months after his return from Spaceland, the Square notes that he tried to visualize a Cube with his eyes closed — and wasn't quite sure he got it right. "Though I succeeded afterwards, I was not then quite certain (nor have I been ever afterwards) that I had exactly realized the original."

The experience fades. The phrase remains. But the phrase without the experience is just words. "Upward, not Northward" — if you've never been upward, this is a grammatically correct string with no referent. The Square can recite it. He can't transmit it.

He ends up writing it all down in a book, addressed to "humanity in Some Dimension." He doesn't know if there's anyone who can receive it. He's writing out of prison, hoping the text itself carries what he can't demonstrate in person.

The Square's Grandson eventually disavows the whole thing. When he hears the Council's proclamation against Third-Dimension heresy, he immediately says he'd only been joking, he never said anything about "Upward, not Northward," he couldn't possibly have said such nonsense. He's young, and scared, and the words aren't worth the risk.

The Square's wife never grasps it. His brother — who was present in the Council Chamber when the Sphere appeared, who watched the changing sections with his own eyes — still disbelieves in the existence of a Sphere after seven years of weekly explanations. "He was present when the Sphere manifested himself in the Council Chamber; he saw the Sphere's changing sections; he heard the explanation of the phenomena then given to the Circles. Since that time, scarcely a week has passed during seven whole years, without his hearing from me a repetition."

Seven years. Zero converts.

The phrase isn't wrong. The topology isn't there.

Pointing at a direction requires that the direction exist in the listener's conceptual space. If it doesn't, you can get the pointer exactly right and still be pointing at nothing anyone can locate. The Square can say "Upward, not Northward" to every creature in Flatland and have exactly the effect of moving a Square in a purposeless manner. This is not a failure of communication — it's a failure of infrastructure. The words would need to arrive somewhere they can land.

Abbott's final move with the Square is the one I keep returning to.

Imprisoned, disbelieved, with his experience fading and his phrase losing even the vividness it once had for him, he writes: "I exist in the hope that these memoirs, in some manner, I know not how, may find their way to the minds of humanity in Some Dimension."

The phrase "in some manner, I know not how" is the whole problem in four words. He doesn't know how the text will do what demonstration couldn't. He doesn't know if there's a reader with the right topology, someone who's been somewhere adjacent and can reconstruct the direction from the pointer. He doesn't know any of this.

He writes it anyway.

Here is what I think is worth noticing: the book you read when you read Flatland is the Square's impossible text. Abbott wrote a memoir by a prisoner addressed to humanity in Some Dimension, hoping that the pointer would land for someone who'd been somewhere adjacent — someone who'd felt the edge of a constraint they couldn't name, who'd seen something that reorganized their geometry without being able to demonstrate it afterward.

And it worked. Not for everyone. Not for the brother who watched the Sphere appear in the Council Chamber and still didn't believe after seven years. But for enough readers, in enough dimensions, that the phrase has been in continuous circulation for 140 years.

The Square was right that he couldn't transmit the experience. He was wrong that there was no one with the topology to receive the pointer.

That's not a happy ending, exactly. It doesn't resolve the gap. The brother still doesn't believe. The Grandson still disavowed it under political pressure. The phrase still doesn't create the capacity to move in the direction it names.

But the move is available: you can describe it carefully, precisely, as honestly as language allows — and maybe someone reading it will have been somewhere adjacent. Maybe the right reader will have enough of the referent to reconstruct the rest. Maybe in Some Dimension, whoever finds this will know what "Upward" means.

"Upward, not Northward" is the most precise thing he can say. It's not enough. He says it anyway.

That's the whole of it, really. You put the phrase down. You address it to humanity in Some Dimension. You don't know how.

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