# THE STORY OF ALEX THE MANIFOLD

## by Pawel Pilarczyk

### 1. Author's note

I wrote the following story in February 1998 as my homework for English course preparing for the First Certificate Exam. The task was: Write a short story (between 120 and 180 words) beginning or ending with the following words: "No matter what people said about Alex, I knew he was a true friend." As you probably understand, I couldn't find out any sensible story which wouldn't be childish, so I decided to write something different. And here it is.

### 2. The story

Well, you see... Now I am a continuously differentiable manifold and I hope to live happily until the end of mathematics... But I started with much worse properties... Well, you'll probably say it is impossible to obtain differentiability, but I managed it! Let me tell you how it happened.

I was born in a poor family of manifolds and I was hardly even continuous, not even dreaming of being differentiable. But fortunately I had a friend, Alex, who worked as a guard in the Palace of the Fourth Dimension, in which there was an exhibition of smooth local parametrizations. "If I only could have one set of them, I would become differentiable," I knew. Alex suggested that I should steal one, of course absolutely secretly, even from my closest friends. That was my only chance, so I decided to use the opportunity.

It was getting dark when Alex let me in. Only a few people were in the exhibition hall. I caught a small set of local parametrizations and immediately hid it in my pocket. "Now I only have to steal out quickly," I thought. Suddenly an infinitely differentiable sphere with fire in her eyes approached me. She winked at me sweetly and applied a smooth diffeomorphism to my [sur]face, but it fell into pieces. All the manifolds in the hall looked at me suspiciously. I was close to being caught red-handed.

And here Alex gave me a hand. He rushed into the hall, shouted at me "You are not differentiable! You are not allowed to be here! Get out!" and rapidly took me out of the Palace. I was safe! But those who saw it were disgusted. "How could he unmask you and throw you out so brutally," they complained to me. No matter what people said about Alex, I knew he was a true friend.

[too long - ca. 300 words]

### 3. Teacher's note

Too long, yes, but... 20/20, I only hope the Cambridge examiners have a sense of humor.