Flatland is a two-dimensional world populated by symmetrical Shapes. It is rigidly divided into classes, which are controlled by a brutal hierarchy. Isosceles Triangles are the proletariat – peasants, labourers and soldiers. Equilateral Triangles are middle class artisans. Rectangles and Pentagons become doctors and lawyers, and Hexagons tend to be executives and administrators. Polygons constitute the aristocracy, and Circles – which represent the highest state of perfection – are ordained as priests.

Written in 1884 by Edwin Abbott Abbott (note the double-barreled symmetry), this prophetic novella is a satire on Victorian culture. The narrator – a certain A. Square – guides us through the tortuous machinations of Flatland’s politics. Every level of society is obsessed with upward mobility, and Shapes can morph into more prestigious configurations through marriage, procreation, or sheer hard work. Flatland is a patriarchy that champions men and traduces women – reducing them to Straight Lines, but happily they can use their pointiness to stab men to death.

Our hero A. Square fantasizes about realms of existence that might be more fulfilling. He slips into a dream about a one-dimensional domain called Lineland where he bumps into the King. A. Square evokes the benefits of Flatland’s two-dimensionality, but His Majesty’s terms-of-reference are so limited he can’t even begin to imagine it.

During another snooze A. Square meets Sphere from Spaceland; a three-dimensional place which is defined by height, breadth and length. Sphere explains solid-ness but A. Square struggles to understand. However, while looking down on the unremitting symmetry of Flatland, he begins to comprehend inside-ness and outside-ness and is seized with an urge to tell his fellow citizens about these new horizons.

A. Square and Sphere descend to Flatland hoping to introduce Flatlanders to the delights of other dimensions. But the Conclave of the Council of Circles is dishing out punishments to anyone who dares to challenge the status quo. In despair, the dynamic duo return to Sphere’s homeland where A. Square is introduced to cubes, cones, pyramids, cylinders plus numerous polyhedra, and fancies he could become the Apostle of the Three Dimensions.

A. Square and Sphere zoom off again and soar through a vacuum towards Pointland, which is an Abyss of No Dimensions. The only occupant is The Point, but confined to zero-dimension he has no conception of substance, and cannot conceive of anything other than himself.

The story ends in Flatland with A. Square imprisoned for evangelizing about the joys of elsewheres. He relates his experiences in a memoir to help following generations think bigger, but the two-dimensional mindset he occupies does not have the lexical vocabulary to describe it, nor the visual dexterity to illustrate it. He tries to conjure up the wonders he saw but sinks into melancholy as the memories dwindle. The final entry includes a few fragments from Prospero (who incarnates a kind of Fifth Element in The Tempest) revealing that what others thought they witnessed…were actually chimeras he created. ‘The baseless fabric of this vision… melted into air, into thin air…Such stuff as dreams are made on.’

Back here on Earthland we experience three spatial dimensions governed by time. Einstein’s general theory of relativity proposes that space can expand, contract and bend. String theory suggests there are nine dimensions in space, and one in time. Our universe could be part of an infinite multiverse, but there are a finite number of ways particles can be arranged in space and time before they replicate. So…could there be endless versions of ourselves out there in the big blue yonder? Symmetrical twins duplicating our neuroses, anxieties and obsessions, or even living the lives we never got round to living?

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