The Curve of the Snowflake

By William Grey Walter
W.W.Norton & Company, Inc., New York, 1956. Original title Further Outlook.

The Koch snowflake plays an important role in this science fiction novel. It first makes an appearance when one of the characters (Simon) asserts there was a rational and simple explanation of every paradox, and another character (Punch) offers to give Simon a rational but paradoxical proposition in two dimensions.

Simon, always a fool for simplicity, accepted. Punch took an envelope out of his pocket and scribbled on the back of it. He said, 'This has a simple arithmetical proof but no rational explanation of the paradox.' He gave it to Simon. Simon read it, looked at Punch with raised eyebrows, hunched his shoulders, shook his head sadly, and got up and left the room without a word....

I took the envelope. On the back of it I read, in Punch's scrawl: "The boundary of a finite area can be a line of infinite length."

I handed it back to [Paula] with a smile.

"Even I have heard that one," I said. "I'm only surprised Simon hadn't....The snowflake curve, they call it. One of a family of pathological curves.

I drew her the classical equilateral triangle with which the process begins.

"I trisect each side of the triangle; I draw an equilateral triangle on the middle third of each side, pointing outward, and rub out the lines common to old and new triangles. I trisect each side of the figure again and repeat the operation. You can go on doing this as long as eye can see or microscope show--go on arithmetically adding to the sides without end. But the area which it contains remains finite here on this paper. That's the paradox. There's a simple arithmetical proof which, briefly, shows that the sides increase in number with each operation--for ever and ever."

Later in the novel, Simon encounters a strange object that others had described as a flying saucer. He describes his account of the incident this way:

But now, peering at it through the heather from the brookside, I began to suspect that it was something more complex than a lake, tent, diamond, or balloon. Its symmetry was pyramidal; the misty effect was apparently due to a peculiar multiplicity of surface angles.

A giant snowflake, I said to myself, almost expecting it to melt at the thought. I recalled how Punch Andrews had once caught me out in my ignorance of the snowflake curve. Topology is my weak point in mathematics--the war came before I could get round to it. Jim Bursley had explained the snowflake curve to me and we had discussed the projection of such a curve into three dimensions. Would the properties of such a body account for the seeming insubstantiality of this one, its thistledown appearance, its myriad surface points? Was this in fact a vessel built on snowflake lines? And why had I never thought of that design?...

There was no escaping the conclusion that the effect was due to an apparently infinite extension of the angular elaboration of the vessel's envelope. The perfection of its landing facility was a natural function of its structure!

Rapidly going over what I could recall of Jim Bursley's information about pathological curves confirmed the conjecture. The snowball curve, derived from an equilateral triangle, is a perimeter of infinite length enclosing a finite area. The angles or points of the perimeter are uncountable. An equilateral triangle projected integrally in the third dimension is a triangular pyramid of equal surfaces. The three-dimensional snowflake derived from this pyramid--hence its diamantine appearance--is a finite volume enclosed in a surface of infinite area. The convolutions of such a surface, to be gathered around its defined content extrude a number of discrete angles or points beyond all possibility of computation. The pressure on each point is infinitesimal, unmeasurably small; the total external pressure exerted on any part of the surface is an aggregate of infinitesimal values, itself infinitesimal.

When I reached this conclusion, I, who know all there is to know about aircraft construction the world over, found myself forced to take the logical step from which I had turned aside to worry about such little things as international jealousies; I had to acknowledge a suspicion that this was no terrestrial vessel, no human creation.