The Walrus and the Mandelbrot Peter R. Cromwell

"The time has come," the Walrus said, "to talk of many things.

"Irrational dimension is a thing that puzzles me.

And yet, it's still self-similar-- you've seen it all before.

"We first remove the middle third and then repeat to get

four segments (each is one-ninth long), but we're not finished yet.

Continue this, and in the end you'll make the Cantor set.

"Next, take this line and make a hump, three parts replaced by four,

Do this again on each new line, repeat for evermore.

Then three of these together make . . . a snowflake that won't thaw!

"This snowflake shape will fit inside this circle over here,

and yet its edge has boundless length. Have I made myself clear?"

"This fractal stuff," the Walrus said, "is just a trifle queer.

4 0 THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 2 (~ 1995 Springer-Verlag New York

"A fractal shape contains itself although reduced in size.

Enlarge, zoom in, it reappears in slightly different guise.

These copies can be found in inexhaustible supplies.

"But now," the Mandelbrot sat down, "I think we should adjorn.

Although I've shown you much today there's still a lot to learn

of strange attractors, gasket sets, and how to make a fern."

The pictures they had drawn lay on the table in a heap.

The Walrus ventured timidly "Are these for me to keep?"

The Mandelbrot did not reply for he had gone to sleep.