Non-Euclidean Perceptual Space

It always amazed me that there has scarcely been a distinct characterisation of the geometry of human perception – how points and distances vary in space as perceived by us. There’s often talk about the vanishing point – the convergence of parallel lines in our visual field. This straightforwardly contradicts Playfair’s axiom, equivalent to Euclid’s 5th axiom in his description of geometry. Parallel lines cannot intersect in Euclidean geometry, essentially. Yet roads and train tracks pointing away shrink, until they vanish; is the far-away world unimportant and shallow to an ego-point capable of only seeing its own local environment, its neighbourhood? This is a feature of hyperbolic geometry: for a given point, everything is smaller the further one moves away from it, but this is true for any point. The centre of the universe is everywhere; seems so gratifying. But also lonely… with a prodigious unknown stretching out in every direction. Something is out-there, but I intrinsically have no idea what, and if any attempts are made to move elsewhere, one loses sight of another spatial subset; dynamic reconfigurations. Not only is every point different, but at each point everything else seems different.


How daft does Kant then seem in his argument that Euclidean geometry is a synthetic a priori fundamental to human perception, if he could just look outside his window and see it isn’t the case. Sure, looking down on a piece of paper on ones desk might make it seem like it prevails, but look at a colleague’s sitting even a little away and it’s not. How moderate, regulative, and paranoid does the assumption of a conductive Euclidean geometry then seems.

This is true even if one goes in the opposite direction. Take an airplane looking down upon the earth. The landscape seems flat, and it looks like a triangle between any three houses makes π degrees. So far so good. But now look up, into the vastness of black space, the nothingness and the stars. Again, never does Euclidean geometry apply – no – it crumbles under the sonorous and remote weight of the void.

The only way any focal point embedded in space is able to perceive its environment as Euclidean, is when that focal point is the universe itself – some paranoid abomination. As soon as non-absolute perception — difference, subsets, systems, concealment — is introduced, a given domain’s perception suddenly finds itself surrounded by a receding world and unknown exotic terrains. What lies beyond the horizon? At the end of the train tracks? Variety, a proliferation of universes: each eye, each perception – something different.


Moreover, psychedelic drugs alter the experience of space and time. Continuous strange change, a fluid fractal reality, dreamlike distortions. Metrics and shapes transfigure into even weirder kinds and configurations. Who now doubts visual space is not equal to “physical space”?

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I have found one paper addressing the same question – Is perceptual space inherently non-Euclidean?. It’s an interesting read.


In so far as the word “knowledge” has any meaning, the world is knowable; but it is interpretable otherwise, it has no meaning behind it, but countless meanings.- “Perspectivism.”

It is our needs that interpret the world; our drives and their For and Against. Every drive is a kind of lust to rule; each one has its perspective that it would like to compel all the other drives to accept as a norm.


Perspectivism is only a complex form of specificity. My idea is that every specific body strives to become master over all space and to extend its force (-its will to power:) and to thrust back all that resists its extension. But it continually encounters similar efforts on the part of other bodies and ends by coming to an arrangement (“union”) with those of them that are sufficiently related to it: thus they then conspire together for power. And the process goes on-

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