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
[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.