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 discovered that women are really strings of fractals."

Apologies to Edwin A Abbott