## Variations on Euclid

The roof is the shortest distance between two walls.

*—le Parkour *maxim

Education is unthinkable without it’s geometry. The very gates to Plato’s Academy make the prohibition clear, “Let none ignorant of geometry enter here.” We could take it not merely as a shibboleth, the secret pronunciation that allows or disallows entrance to the clubhouse, but as a declaration of a more fundamental truth: education itself is only opened through its geometry. Otherwise one just has an olive grove. Gate and prohibition do not simply police the educational border, but constitute the space itself.

Education, then, pretending to the timelessness of Platonic Solids, is all about entrances and exits. If educational space, or rather, space as we know it, is Euclidean, there is a crack in it’s very core. Education is about slipping into, and climbing out of, Euclidean space. And yet, the outside itself is virtually unthinkable. With Euclid comes the Cartesian grid, spread to infinity. Indeed, “non-euclidean” geometry emerges almost as a kind of uncanny thought experiment, often treated as a kind of unholy wormhole. It too, has trouble escaping, posed as the eerie possibility from within: “what if parallel lines *were* to meet?”

The other of Euclid is thus phantasmagorical. Which is to say, our visions of alternative educations haunt our existing educational spaces like ghosts. We rearrange the chairs, like an invocation.

And yet, there is perhaps room to believe that there are educational alternatives that are neither euclidean nor not. A way of moving differently. Hyperbolic space, for example, in which space itself seems to fold back on itself, is fascinating not just for being excessive, but for the ways in which it evokes movement and relation not just in space, but *of* space.

Is it possible to imagine geometries not just contesting the *same* space, fighting over interpretation, but of engaging with each other?

Would this not have to entail an exploratory space, in which things are up for grabs? Education has long understood the need for the crack at the heart of geometry, but it has also quickly filled it in, dictating it’s terms. In Plato’s allegory of the cave, one escapes up a path, through a hole in the ground. But what is virtually unthinkable is an exploration of the cave itself.

Which is where we should pay careful attention to the antics of *le Parkour*. In many ways this movement of “free running” or gymnastic exploration of the urban environment is an extension of the Situationist’s engagement with urban space, with their practice of the *derive:* drifting in order to understand and transform the “psycho-geography” of the city. But what is interesting here, is how they do not cleave out an alternative space, fighting against Euclid, so much as they play up against it. Traverse it differently. It is a question of inflection. If a line is the shortest distance between two points, this does not say anything yet. But when one says that a roof is the shortest distance between two walls, everything is set in motion.

This is important. Because the language of escape, of leaving the city, of upward mobility, of traversing the educational path to the promised land, or of a simple return to nature, is not what it seems, instead often fixing the very terms of movement, cutting off exploration and variation.

But what if the roof is not just the thing over our head, but under our feet?