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Fantastic Geometry: The Perfection of the Impossible


Sometimes reason dreams with perfect but self-contradictory polyhedrons.

One of Borges’ best stories (almost) ends like this:

“There are three lines too many in your labyrinth,” he said at last. “I know of a Greek labyrinth that is but one straight line. So many philosophers have been lost upon that line that a mere detective might be pardoned if he became lost as well."

This ending is from “Death and the Compass”, a text from 1942 , which is part of Fictions. It is a detective story, rich with symbols pertaining to the Jewish tradition; the Porteño culture of the first half of the twentieth century and, especially, it is rich in allusions to geometry and some of its concepts. Borges, fascinated by speculative thought, found an enormous intellectual pleasure in the axioms and theorems of Euclidian geometry, in its contradictions and paradoxes that could bring their apparent theoretic perfection to an end.

In “Death and the Compass”, symmetry becomes a reason for nightmares: for its main character, detective Erik Lönnrot, a house with double stairways and repeated identical rooms results intolerable, ominous and, in addition, infinite.

A similar effect of the use of geometry as a literary resource is found in “The Immortal”, where the traveller arrives in a city of absurd architecture with dead-end corridors, unreachable windows and inverted stairs, a description that —coincidentally or not— seems to adjust to some of M.C Escher’s best known drawings (Relativity, 1953; Waterfall, 1961; Ascent and Descent, 1950), where, paradoxically, movement does not lead anywhere.

Perhaps this is how fantastic geometry could be defined: from paradox, situated in that ambiguous land of understanding where the procedures of reason turn on themselves. Escher’s drawings are perfectly geometrical, but they soon distance themselves from rules to generate a world where their contradictions are possible. The same thing happens with Oscar Reutersvärd’s figures or with Renaissance artists Wenzel Jamnitzer, Johannes Lenker and Lorenz Stoer‘s polyhedrons.

In a certain way, these creations could be taken as a mere form of entertainment or as an idle skill. Nontheless, even this consideration tells us about a somewhat more defiant courage. Why should we reduce to entertainment or mere awe exercises which prove the fallibility of a system we are taught is impeccable?

If, as Goya wanted, the sleep of reason produces monsters, one of those could take the shapes of fantastic geometry. But regardless if that dream becomes a nightmare or not, like in Borges’ stories, it is enough for its irruption to show us those territories unfettered form from the control of logic and yet still normed by its most elementary rules.

Lacan used to say that the unconscious is structured like language. Perhaps something similar could be thought of fantastic geometry: placed in the bosom of lines and planes, even there it is capable of breaking with said rigidity to make the perfection of chaos bloom.

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