The Unfathomable Mysteries of ‘Flatland’

Erasmo Calzadilla

I just finished enjoying a fascinating animated film for adults, “Flatland” (2007), based on the work “Square” (written in 1884 by Edwin A. Abbott). I’m taking advantage of this to continue a long and interesting discussion with some friends from here.

Flatland is a two-dimensional world inhabited, as is logical, by flat beings. One of the nations of Flatland suffers a bloody political crisis.

The leading character in the film, A Square, meets a voluminous three-dimensional sphere from Spaceland who tries to explain to him the depth of bodies. However it’s not until he gives A Square a little tour of three-dimensional space to see his own world from above that the protagonist understands.

To his surprise, the three-dimensional universe is not a divine kingdom, like he first believed, but another more complex level of reality, with as many or more problems than even in Flatland.

It then occurs to him that the third dimension couldn’t be the final, that perhaps there exist a fourth, a fifth and even infinite dimensions. Thinking beings of universes with “n” dimensions could understand the worlds with less dimensions (n -1), but not with more (n+1). It’s not stated explicitly but it remains as a kind of implied epistemological hierarchy.

Lying here, like in everything, is a more or less concealed political thesis, one taking place with implications in the human environment, and based — as one would expect — on questionable foundations(*).

I propose, but I will not demonstrate, that any being of “n” dimensions is incapable of not only understanding more “complex” beings but also those “simpler” than themselves (for the very reason that they would cease being simple). This is demonstrated in the film when the voluminous being tries to explain tri-dimensionality to the two-dimensional one.

For that, he appeals to the cards. These are very thin, but by putting one card on top of another one ends up forming a voluminous body. Bi-dimensionality for the three-dimensional being is no more than little-developed tri-dimensionality. It reduces the world of the other to a singular and simplified variant of its own. But the issue is that superimposed infinite planes will never generate a voluminous body.

Recalling Thomas Kuhn, the well-known philosopher of science, I would say that an epistemological incommensurability persists between worlds with a diverse number of dimensions; or generalizing, between different epistemological paradigms. “Complex” worlds are not a mere enhancement of “simple” ones.

The Messiah (the mediator) between two different universes has to share the qualities of both worlds (something practically unthinkable since they’re contradictory); they cannot be a mere literacy-spreading missionary.

Defending the irreducibility of the Flatlanders, we are on the path of those medieval thinkers who insisted on the persistence of the individual soul after death, not integrated to the soul of souls: God; of Don Juan and Castaneda struggling to not being swallowed by the eagle; of the colonies becoming emancipated from the colonial world view; of dissidents denying national interests – in short, of all those who refuse to be easily included.

Sure, I also know that right now I’m putting myself in a global position.

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(*) The foundations of my proposition are no less doubtful.