Chapter I

I slept soundly, falling into a dream, a dream of a strange land. Looking about, it was hard to see much but lines on lines, as if I had stroked with softer and harder leads on a fixed T-square, or chosen to view the Z axis in a cad plan.

I realized I had read about this place - I thought, "I must be dreaming about Flatland." It could have been worse - I could have been dreaming about reality television. I wondered if I would run into A Square. I wondered if I *was* a square. I had a square meal before sleeping - pork chops and potato pancakes. I cut both into little squares and forked them together with the round carrots. A geometric repast.

In Flatland, one had to fear impalement on acute angles, so they preferred obtuse structures - pentagonal at least. Isosceles triangle soldiers and the occasional raging female straight line segment might impale you on purpose, but I figured to steer clear of them.

The first creature I encountered appeared to be the Point. A Square considered him to be a miserable creature - living in the abyss of no dimensions, "he is himself his One and All, being really Nothing." I recalled that talking to the Point was pointless, and started to move on, but was surprised to be addressed, "Who said you could leave? I wasn't finished talking to you!"

"Are you talking to me?"

"I just said I was talking to you, if you'd listen to what I say instead of what you think I said, we'd both be a lot better off."

"But you're a point, how do you even know I exist?"

"Don't prejudge me, it only makes you look more stupid, Stupid," it flounced.

While it babbled on, I tried to investigate how a point could flounce. As I circled the creature, I mostly saw a point, but if I circled back I briefly saw a short segment. And there seemed to be another segment beyond. After much circling, and enduring much verbal abuse, I deduced that this was no Point. What I saw was merely the end of a female line segment - and a bent one at that. She had kept one end pointed towards me, perhaps to disguise or defend herself or perhaps to attack.
Or perhaps for all three. Being deformed, or irregular, was considered very bad in Flatland.

So I said, "I have no quarrel with you good woman, I must be on my way."

What happened next was astonishing. She, the segment, joined one end with the other to become roughly circular, and said, "My circular wisdom precludes your needs!"

Being a circle was a real cushy position in Flatland. Everyone just assumed you were smart and you could write your own ticket. Imitating a circle was bound to be a crime. And no wonder her segment was bent. A Square never mentioned this.

"I clearly see that you are no circle, just a line segment." It seemed wise not to mention that she was bent.

Straightening out as much as she could, the segment lashed, "Don't tell me what I am! I certainly know what I am! The question is do you know what you are?"

With that, I decided that there was naught to be learned by conversing any further. I hoped to find A Square, or wake up.

I made my way North, and happily she did not follow.

I came upon a large pentagonal structure, and made my way to the men's entrance. Two isosceles stood guard, so I approached a stolid square who was waiting in the ... square.

"Excuse me, but I am a visitor. Can you tell me the purpose of this structure?"

"Why yes, and welcome to a fellow square," he replied. (I knew it) I thought, but attempted to capitalize on his show of solidarity, "Why thank you and let me compliment you on your regular sides."

"Oh likewise," he gushed, "I thought the same thing when you approached." There was actually no way either of us could have known such a thing without becoming much more intimate, but good manners often lead to good results. "This is a Polygonal Hall of Debate," he continued. "Our polygons congregate to apply their obtuse ideas against each other. Many have found themselves much smoother and well-rounded as a result."

No one talked about it much, but the circles were really polygons with hundreds of sides - so many that they seemed circular, as did their arguments. All polygons wished they had been born circles, and hoped to father circles. The competition to have one's offspring recognized as a circle was vicious.

"I suppose there's no room for us in there?" I offered, with a smile.

"No, we have our four angles and they're all right with me," he laughed heartily, and I joined in. It had the sound of a very old joke. He had some business with an icosagon, but afterwards insisted that I visit his favorite pub.

After a few straight shots, I told him about my encounter with the bent female. He said, wistfully, "Ever it has been since we learned that we are ultimately constructed of fractals."

Chapter II

"You know about ... fractals?" I asked.

"We are not so backwards as it may seem - though some long for the time before fractals were revealed." He looked directly at me, "You spoke of regularity. I took that as a formality because you were a stranger. Have your people managed to avoid the upheavals?"

"Oh, we've had our upheavals. But few of our people know a fractal by name."

"Knowledge of fractals led to great changes here. What changed in the heavy South?'

"Umm." I had disquieting vision of isosceles soldiers pressing closer while my square fellow accused me of heresy. "Just ... science. A lot of folk can't accept what science tells them."

"Oh, Cyants is the one who told us about fractals. We wondered where he had gone."

(Science is a person here?) I just nodded.

"I see, I see. Of course you resist the implications of fractals. So did I. Cyants had an immediate appeal to the women and irregulars, but many of us regular folk resisted it."

"Resisted fractals?"

"Of course. If everything is ultimately fractals, what is the point of the perfection of our circles and the complexity of our polygons?" He answered his own question, "Any irregular now can feel that his fractals are just as complex, just as deserving as any circle."

"But it seems orderly here."

"It is on the surface. In truth many of us thought what they, we, did to irregulars was not fair. And the circles always took care of us regulars, but it was ... limiting. My wife appreciates being more than just a line segment, and I find her more ... companionable as well. But there is also something to be said for the old ways - some of them."

After a long silence, I decided not to overstay my welcome. "This is all very new to me. I thank you for your company, but I must be on my way." His angles seemed relieved.

"Of course. I wish you a safe journey. Are you still traveling North?"

"Yes"

"Don't tarry with Anonagon - he will just slow you down."

"A nonagon?" (Nine sides? Why was that bad?) "I've met some nonagons and we got along fine ..."

"No," he said, "Anonagon. He comes out of nowhere, brandishes his point and then disappears. He is harmless enough, but engaging him on his point will slow you down."

"He only has one point?"

"Yes, he claims to have many, of course, but that has never been verified."

"Very curious. Is this another fractal effect?"

"No, more of an affectation. You must hurry! Farewell!"

And with that we parted.

The terrain was flat (of course) and the going was easy enough, but a whirring sound persisted around me, like the fan on my old humidifier. Either I was waking up, or Flatland was in for some weather.

I caught some movement, and saw some tiny lines and angles making their way across my path. Did Flatland have a Lilliput? Was I a giant now?

They reminded me of insects. My recent friend hadn't mentioned insects, only Anonagon, and only one. And a personage he called Science. Though as I looked at the insects, in my head I heard him say it again. He had pronounced it "Sci-Ants. or maybe Psy-Ants."

Someone spoke sharply from behind, "Don't eat them!"

"I wasn't going to eat them, I was just looking closely."

"Don't deny it. You seem the type that would eat anything, even roaming numerals."

"Is that supposed to be some sort of point?" I asked, wary of Anonagon. He had more than one point, though. He was a triangle, not very acute, which was less threatening, but not regular, which could mean anything. "Are you a worker?"

"You'd like to pretend that I'm a worker so you can order me around. You squares like to assume things."

"So what are you?" I wondered if he was going to change shape.

"I am Angle Angle Side. We congruent triangles are too busy postulating to respond to your petty insults."

"You are congruent? With whom?"

"Isn't it self-evident?"

"OK, I guess with another triangle with the same AAS ..."

"AAS!"

"What?"

"I am Side Angle Side."

"You said Angle Angle Side before."

"I am Side Angle Side." Indeed he seemed to have reoriented his angles and sides.

"With whom are you congruent now?"

"With the same as before."

"Is there another SAS nearby?" One was bad enough.

"SAS!"

"You ... aren't SAS?" I was catching on.

"I am Angle Side Side! You keep trying to pigeonhole me. Well it won't work." Now he was changing a lot. Two of his angles seemed obtuse, impossible for a triangle. I saw no proof that he was still a closed triangle. Was this how Flatlanders died?

Then it hit me. Angle Side Side wasn't a valid postulate. "I think you should try Angle Side Angle before you hurt yourself."

"Don't tell me how to postulate!"

I wasn't going to waste any more time arguing with an obtuse ASS, so I said, "Suit yourself - I have to be on my way."

As I hurried away, it occurred to me that fractals were liberating, but some Flatlanders handled it a lot better than others.

Did this Psy-Ants know what he had started here?


Even more profuse apologies to Edwin A Abbott