Euclid’s Mirror

A short story
by Terry Paul Pearce

A sphere sits on a concrete plain, and a man stands and peers into it. To the man, the surface of the sphere appears silvered, opaque and clear—all at once. He scrutinises the sphere closely. He cannot work out whether the sphere is a mirror, a window, or a picture; whether what he sees is on the surface, without, or within. What he sees there is a figure, much like himself. When he raises a hand, the figure does likewise. He takes a step forward; like a dance partner, his double steps to meet him.

All around the figure, he sees criss-cross gridlines, as of the paved surface on which he stands, stretching away from the figure just as they do from him. Behind the figure, as in his distance, are buildings; the city. The only difference between what he sees and what is around him is the way that the sphere's contents cleave to the curvature of its form, whilst everything around him adheres to a straighter, more familiar perspective. The arcs curving around the sides of the globe correspond perfectly to lines of grid, level parallels which constrict the boundless and bare plaza on which he stands, but those on which he stands stretch off towards vanishing points on the horizon. 

As he stands there, a conceit germinates in him, and it is this: the sphere contains within it a universe equal to the exterior one it reflects. Admittedly, the space outside is infinite, whilst that within is bounded by the circumference of the sphere. But, as the man reflects, he has never seen infinity. The scientists tell him that the universe in which he lives continues forever, but he sees only so far.

If every point in his universe has its twin in the sphere, the counterpart of a point on the infinitely distant rim of his universe could be found at the centre of the sphere. The increments one would need to traverse to get to the centre could merely get infinitesimally smaller, and therefore be unlimited. He thinks of a fable he once heard about a hare who gains on the tortoise he is racing, halving the distance, then halving it again, and halving it again. At that rate, the hare would reach the tortoise only in an infinite number of stages. He sees a parallel between the impossibility of the journey to the centre of the sphere and impossibility of the journey to the circumference of the universe. The same information, plotted in different geometries. The infinite, he whispers, is the analogue of the infinitesimal.

Worlds realign themselves in his mind. A jigsaw fits into place where before there was only chaos and formlessness. Time passes as he stands, staring, motionless.

His reflections turn to the being inside the sphere. Beings, plural, for surely if his doppelganger stares at him, then in the city behind, his wife is cooking dinner, his children are tucked up in bed, and all the nameless faces who share his joylessly anonymous journey to work every day are going about their business. In the city in the sphere, in the distance that does not seem distant. He wonders what they see when the gaze upon the surface of the sphere. It eludes him, at first. He is trying to make his brain fit into a dimension it is not used to, taking it on a forced march along both sides of a Moebius strip.

But then he has it. He finds another dimension to add to the realignment taking place, as if he has gone from solving a two dimensional puzzle to solving it in three. They would, of course, see it exactly as he does. He would be the one bounded by a circumference, living according to a strange geometry. He would be the one whose infinity was a centre. A chameleon can see through all the points of the compass, but does not see itself as having a strange field of vision. It would find eyes only on the front of its head strange. The mirror is perfect, he realises. As outside, so within; as within, so outside.

He gazes at the sphere, nodding, murmuring under his breath. His conjecture is perfect; it is self-consistent, has an undeniable internal logic. Then, however, a final realisation dawns. Knowing what he now knows, he sees that he can never be sure whether he is the man who gazes, or whether he is the figure seen inside. Whether his world is curved, or straight. Whether everything he knows is reflected here, or whether he is the reflection, existing outside of the physics he thinks he knows; in effect, not existing.

Perhaps this knowledge would madden some men, or crush them under its weight. This man, however, sees it differently. Balanced on the edge of infinity and nothingness, he feels the lightness lift him. Maybe the world he knows is real, maybe not. Maybe he is nothing but a reflection, maybe he sees the reflection. Maybe he is bounded in a nutshell, or perhaps he is the king of infinite space. It’s all, he realises, a matter of perspective.

He takes a last look at the sphere, the man, the flatness, the city in the distance. Then he turns on his heel and whistles as he walks home to his wife, feeling a world of possibility stretch out before him like parallel lines which will never meet.

 

Copyright © 2011 by Terry Paul Pearce
Story image: Adapted from an image by Ivan Mikhaylov, Dreamstime.com

 

Terry writes:

"I’ve always been interested in different perspectives on reality, especially as explained in scientific circles. I agree with Stephen Hawking that any description of reality is only a model, but need not be any less useful for that. It was in this context that, last Christmas, I was musing on a prompt set in a flash fiction competition: a photo of a huge mirrored ball – an open air art installation in a city square. At the time, I’d been reading about Non-Euclidian Geometry, and the possibilities that unfold in the story suggested themselves as soon as I inserted a character into the scene, peering into the sphere."

Terry Paul Pearce lives and works in London. He writes mainly short fiction, although he hopes to work up speed to a novel one of these days. Among his influences are Jorge Luis Borges, Italo Calvino, and Loren Eiseley. Previous publishers of his work include Grey Sparrow Press, The Foundling Review, Girls with Insurance, and The Battered Suitcase. Musings and links to all of his published work can be found at www.terrypaulpearce.blogspot.com.

{ 2 comments… read them below or add one }

G. K. Adams June 16, 2011 at 5:28 am

Great take on an old question, “Whether everything [we know] is reflected here, or whether [we are] the reflection, existing outside of the physics [we think we know]”. I like the resolution – satisfying.

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Cezarija Abartis June 19, 2011 at 6:51 pm

I loved the playfulness of “The infinite, he whispers, is the analogue of the infinitesimal.” I loved the allusion to Hamlet. Provocative!

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