The Science of Discworld IV

the inside. It sounds impossible. But we can make significant progress by borrowing some tricks from a fictional square and an ant.
    In 1884 the Victorian headmaster, clergyman and Shakespearean scholar Edwin Abbott Abbott fn1 published a curious little book,
Flatland
. It remains in print to this day, in numerous editions. Its protagonist, A. Square, fn2 lives in a world shaped like the Euclidean plane. His universe is two-dimensional, flat, and of infinite extent. Abbott made a few gestures towards figuring out plausible physics and biology in a two-dimensional world, but his main objectives were to satirise the rigid male-dominated class structure of Victorian society, and to explain the hot topic of the fourth dimension. With its mixture of satirical fantasy and science,
Flatland
has to be considered a serious contender for
The Science of Discworld 0
.
    Abbott’s scientific aims were accomplished through the vehicle of a dimensional analogy: that a three-dimensional creature trying to comprehend the fourth dimension is much the same as a two-dimensional creature trying to comprehend the third. We say ‘the’ for convenience: there is no reason for a fourth dimension to be unique. However,
Flatland
was, in its day, almost unique. There was one other tale of a two-dimensional world, Charles Howard Hinton’s
An Episode on Flatland: Or How a Plain Folk Discovered the Third Dimension
. Although this was published in 1907, Hinton had written several articles about the fourth dimension and analogies with a two-dimensional world shortly before Abbott’s
Flatland
appeared.
    There is circumstantial evidence that the two must have met, but neither of them claimed priority or was bothered by the other’swork. The fourth dimension was very much ‘in the air’ at that time, emerging as a serious concept from physics and mathematics, and attracting a variety of people ranging from ghost-hunters and spiritualists to hyperspace theologians. Just as we three-dimensional beings can gaze upon a flat sheet of paper without intersecting it, so a fourth dimension is an attractive location for ghosts, the spirit world or God.
    In Abbott’s narrative, A. Square strenuously denies that a third dimension is possible, let alone real, until a visiting Sphere bumps him out of his planar world into three-dimensional space. Inference didn’t do the trick; direct personal experience did. Abbott was telling his readers not to be unduly influenced by what the universe appears like to unaided human senses. We should not imagine that every possible world must be just like our own – or, more precisely, what we naively believe our world to be. In terms of Benford’s dichotomy – human-centred thinking, or universe-centred thinking – Abbott takes a universe-centred view.
    The spaces considered in
Flatland
obey the traditional geometry of Euclid – a topic that Abbott encountered as a schoolboy, and didn’t greatly enjoy. To remove this restriction on the shape of space, we require a more general image, which seems to have originated with the great mathematician Carl Friedrich Gauss. He discovered an elegant mathematical formula for the curvature of a surface: how bent it is near any given point. He considered this formula to be one of his greatest discoveries, and called it his
theorema egregium
– ‘remarkable theorem’. What made it remarkable was a striking feature: the formula did not depend on how the surface was embedded in surrounding space. It was intrinsic
to the surface alone
.
    This may not sound terribly radical, but it implies that space can be curved without being curved
round
anything else. Imagine a sphere, hovering in space. In your mind’s eye, it is visibly curved. This view of curvature comes naturally to the human imagination, but it depends on there being a surrounding space, somewhere forthe sphere to be curved
in
. Gauss’s formula blew this assumption out of the water: it showed that you can discover that a sphere is curved, without ever leaving its surface. The surrounding space, rather than being necessary for the surface to have some direction in which to bend, is irrelevant.
    According to his biographer Sartorius van Waltershausen, Gauss had a habit of explaining this point in terms of an ant that was confined to the surface. As far as the ant was concerned, nothing else existed. Nevertheless, by wandering around the surface with a tape measure (Gauss didn’t actually use that instrument, but let’s