Exercise on a Sphere


The Christmas party is done; still in a paper hat,
I empty an ashtray and push back a chair;
Finish my glass? No, the champagne is flat
Like the air in this room, the stale, the exhausted air.

Yet, like the wine, only an hour ago,
It bubbled with laughter, bright with the criss and cross
Of meeting eyes, of words tossed to and fro.
Alone with myself there are no words left to toss.

Alone with what self? For now my eye is caught
By a silvered ball hung in the Christmas tree.
Riemann and Lobachevski share the thought
Of the watcher there who is, and is not, me.

My party is over, but his has just begun;
His lips twist back in a non-Euclidean grin.
The bulbous nose is a parody of my own;
His forehead dwindles; he wags a minikin chin.

The ghosts of Bolyai and Cayley, Gauss and Klein
Have distorted his table, the frames of picture and door;
Nothing is flat in his world; there is no straight line;
Parallels meet; there are no fixed shapes any more.

From the wizened arm that hangs from his shoulder there sprout
Monstrous fingers to meet mine touching his sphere;
And an insect voice more daunting than a shout,
Squeaks: “Welcome, brother, you were expected here.

“You think this world of ours a grotesque, perhaps;
A crazy fiction to laugh at and forget;
Projective geometry making improbable maps
Of all your eyes trust to be true—and yet

“Nothing is arbitrary here: you see, for a fact,
How, as you move, my movements correspond.
All details match, by mathematically exact
Equations, your orthometric world beyond.

“But what if your world is the mirror, the mirror I hate,
A world of terror without limit or bound,
Which makes my brain turn, even to contemplate:
Depths, heights, infinites no thought can sound;

“Euclidean forms, which to my eyes appear
Like the fixed archetypes of Plato’s dream?
To me, in the closed space of a finite sphere
Yours is a nightmare from which I wake only to scream.

“So I tell you: the world you live in, it seems to me—
And you will not like the alternative I provide—
Is a sphere whose radius is infinity,
With only imaginary space outside.

“Your world is the false map, imitating mine,
Yourself the grotesque projection of what I know
On the fiction of an infinite straight line.
Can you disprove or prove that it is not so?

“You cannot, of course; yours is the small world, friend,
Its instruments too crude, its means too rough
To test my hypothesis that space must bend:
You cannot measure triangles big enough.

“Step over this threshold; come into our cosier sphere,
This non-zero curvatured space we take to be real;
Look back now, like us, at the world you left from here,
The world that you live in, by which you judge and feel.

“Look well at those features, consider the ways you act;
They are all distortions, your art, your morals, your law;
Those values you took for incontrovertible fact
Here fail to support the conclusions you wished to draw,

“For you live in the new age of Provisional Man;
You may choose from alternative systems, which suits you best,
But can you disprove the other we offer or can
You bring the truth of your own to a final test?

“Of course, you cannot: yours is the small world still;
History changes your systems like a cloak,
The social contract, the individual will
May wrestle like snakes but change their shapes like smoke.

“And even if one day one system should prove itself true,
You will not live long enough, brother; none of us will.
If man needs assurance to live by, what can he do?
To believe what he knows may be false, is belief that may kill.

“His need for certainty is a disease past cure.
Where are those fixed world-pictures of the past
Which, as they changed, gave place to others as sure?
Christmas is over, Christmas is over at last!”

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