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Alien Scientist: Being Square In Flatland

Author: Stephen Marshall, Issue: December 2004, Topic: Alien Scientist, Science

Of all possible kinds of alien encounter, surely one of the most, well, alien, must be meeting a being from another dimension. It stretches the imagination even to think of what an alien with an extra dimension would look like - it would not just be like a creature with an extra head, or a set of extra-terrestrial antennae. Yet, we can start thinking of what an extra-dimensional encounter might be like, by first considering the world of two dimensions.

In the classic book of geometric fantasy, Flatland, the narrator (known simply as 'A. Square') describes the life and times of a planar kingdom, whose inhabitants are an assortment of planar and linear shapes. The Flatlanders live in two-dimensional houses (or 'flats') with two-dimensional rooms. The flat rain falls horizontally on flat roofs.

Although we can imagine everyone and everything in Flatland as being, well, two-dimensional, to the Flatlanders themselves everything appears one-dimensional. That is, when a Square meets another Square - or a Circle or Triangle for that matter - all that each one perceives is a line in the plane. As each flat shape is only ever seen side-on, the two-dimensional shape (as such) has to be inferred from the intensity of the image of the visible line. This is just like the way three-dimensional beings effectively see in terms of a 2-D image - a Sphere appears to us in profile, as a circle - and we have to infer the third dimension.

Indeed, our Square hero himself encounters an alien being from another dimension, in the form of a Sphere, visiting from the 3-D world of Spaceland. The Sphere appears in Flatland, as if from nowhere, first as a single point, then expands to a circle, then shrinks to a point again and disappears again just as mysteriously.

This outlandish apparition and transformation is puzzling enough to the Square. But other kinds of three-dimensional visitor would be even more bamboozling, by varying in shape as well as size as they passed through Flatland, depending on their angle of entry.

A Cube passing through Flatland face down (and directly perpendicular to the plane) would appear instantaneously as a whole square, and remain a square for a while, then disappear just as suddenly. If the Cube passed though edge first, however, it would appear first as a line, expanding laterally into a rectangle, and then instantaneously appear as a square, before retreating again as a rectangle and line again, and disappearing once more. Finally, if the Cube passed though Flatland diagonally, vertex first, it would appear first as a point, then expand as a triangle, then a hexagon, and then return as a triangle to a point and then disappear. All these quite different shapes would be the result of the same Cube, simply passing though at different angles.

If he had the imagination, the Square might recognise himself in all this. Although he is only conscious of himself at the level of being a Square in Flatland, it is possible that he is in fact some aspect of a higher dimensional entity, like a cube or square prism - or a giant bendy crabstick or liquorice strand passing through the plane - leaving only a two-dimensional trace that simply appears as a square going about its business in Flatland.

As it is, however, the Square is only conscious of himself at a two-dimensional level. He can no more conceive of himself as an aspect of a 3-D being than he can somehow physically 'rise up' out of the plane, and escape to Spaceland.

To the Square, then, a being like a Sphere appears as someone quite magical. The three dimensional visitor is not only otherworldly because it comes from somewhere else - out of the plane. Intriguingly, the Sphere can see 'inside' the Square - and steal a glimpse of the interior of all the other flat shapes - interiors that are always invisible to the Flatlanders themselves. The Flatlander is mesmerised by such kinds of revelation. It might easily come to believe that the higher dimensional visitor was a 'higher being', all-seeing, and superior all round.

Yet, within the realm of Flatland itself, the three-dimensional visitor only ever manifests itself directly as a two-dimensional shape, such as a humble square or circle. Down in Flatland, the higher dimensional being is not so different from the Square, after all.

So perhaps we don't need to have any special imagination to recognise a higher dimensional being passing through our own dimension. Such a being might appear not so much an alien, after all, but someone just like us...

<< Science News: December 2004 | Master Index | Shopping Online >>

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