The Science of Discworld IV Judgement Da

SIXTEEN



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SPHERICALITY SURELY IS EVERYWHERE





The Reverend Stackpole’s appeal to the ubiquity of round objects strikes a chord. The storytelling ape has a strong preference for neat, simple geometric forms. Circles and spheres featured prominently in early theories of planetary motion, such as those of Ptolemy and his successors: see chapter 22. To some extent today’s science, with its neat, simple mathematical laws, derives from an ancient tradition in which particular shapes and numbers have mystical significance. Stackpole appeals to the sphericality of several objects that are not Discworld to argue that the Disc must also be spherical. He is using a ploy that is all too common among people trying to promote some belief system: introducing ‘evidence’ that undeniably exists, observing that it is consistent with his beliefs but skirting quietly round a big logical gap. Namely: is the belief system the only possible explanation for the evidence concerned, or is it consistent with alternatives?

When it came to the shape of the universe, the cosmologists of the early twentieth century were a bit like Stackpole. They assumed that the universe should be spherically symmetric – behave the same way in all directions – to keep the sums easy. When they put that assumption into the equations, and did the sums, the mathematics spat out a spherical universe. This shape quickly became the perceived wisdom. However, there was very little independent evidence to support their original assumption. The logic was – well, circular.

What shape is the universe, then?

It’s a big question. We somehow have to work out the shape of everything that exists, from one location on 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 Abbottfn1 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’s work. 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 for the 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 not be too purist) the ant could infer that its universe was curved. Not curved round anything: just curved.

We all learn at school that in Euclid’s geometry, the angles of any triangle add up to 180˚. This theorem is true for a flat plane, but false on curved surfaces. For example, on a sphere we can form a triangle by starting at the north pole, going south to the equator, going a quarter of the way round the equator, and returning to the north pole. The sides of this triangle are great circles on the sphere, which are the natural analogues of straight lines, being the shortest paths on the surface between given points. The angles of this triangle are all right angles: 90˚. So they add up to 270˚, not 180˚. Fair enough: a sphere is not a plane. But this example suggests that we might be able to work out that we are not on a plane by measuring triangles. And that’s what Gauss’s remarkable theorem says. The universe’s metric – the way distances behave, which can be determined by analysing the shapes and sizes of small triangles – can tell the ant exactly how curved its universe is. Just plug the measurements into his formula.

Gauss was immensely impressed by this discovery. His assistant Bernhard Riemann generalised the formula to spaces with any number of dimensions, opening up a new branch of mathematics called differential geometry. However, working out the curvature at every point of a space involves an awful lot of work, and mathematicians wondered if there might be a simpler way to get less detailed information. They tried to find a more flexible notion of ‘shape’ that would be easier to handle.

What they came up with is now called topology, and it led to a qualitative characterisation of shape that does not require numerical measurements. In this branch of mathematics, two shapes are considered to be the same if one of them can be continuously deformed into the other. For example, a doughnut (of the type that has a hole) is the same as a coffee-cup. Think about a cup made from some flexible substance that can easily be bent, compressed, or stretched. You can start by slowly flattening out the depression in the cup to make a disc, with the handle still attached to its edge. Then you can shrink the disc until it has the same thickness as the handle, forming a ring. Now fatten everything up a bit, and you end up with a doughnut shape. In fact, to a topologist, both shapes are a distorted form of a blob to which one handle has been attached.

The topological version of ‘shape’ asks whether the universe is a spherical lump, like an English doughnut, or a torus, like an American doughnut with a hole in it, or something more complicated.

It turns out that a topologically savvy ant can deduce a great deal about the shape of its world by pushing closed loops around and seeing what they do. If the space has a hole, the ant can wind a loop of string through it, and it is impossible to pull the loop away, always remaining on the surface, without breaking it. If the space has several holes, the ant can wind a separate loop through each, and use them to work out how many holes there are and how they are arranged. And if the space has no holes, the ant can push any closed loop around, without it ever leaving the surface, until it all piles up in the same place.

Ant-like thinking, which is restricted to the intrinsic features of a space, takes a bit of getting used to, but without it, modern cosmology makes no sense, because Einstein’s general theory of relativity reinterprets gravity as the curvature of spacetime, using Riemann’s generalisation of Gauss’s remarkable theorem.

Until now, we’ve used the word ‘curvature’ in a loose sense: how a shape bends. But now we must be more careful, because from an ant’s-eye view, curvature is a subtle concept, not quite what we might expect. In particular, an ant living on a cylinder would insist that its universe is not curved. A cylinder may look like a rolled-up sheet of paper to an external viewer, but the geometry of small triangles on a cylinder is exactly the same as it is on the Euclidean plane. Proof: unroll the paper. Lengths and angles, measured within the paper, do not change. So an ant living on a cylinder would consider it to be flat.

Mathematicians and cosmologists agree with the ant. However, a cylinder is definitely different from a plane in some respects. If the ant starts at a point on a cylinder, and heads off in just the right direction in what it considers to be a straight line, then after a time it gets back to where it started. The line wraps round the cylinder and returns to its origin. That’s not possible with a straight line on a plane. This is a topological difference, and Gaussian curvature cannot detect it.

We mention the cylinder because it’s familiar, but also because it has an important cousin called a flat torus – an oxymoron if ever there was one, since a torus is shaped like a doughnut with a hole, which is deliciously curved. But the name makes sense nonetheless. Metrically, the space is flat, no curvature; topologically, it’s a torus. To make a flat torus we conceptually glue the opposite edges of a square together, and squares are flat. This construction is analogous to the way computer games connect opposite edges of a screen together, so that when a monster or an alien spacecraft falls off one edge, it instantly reappears in the corresponding position on the opposite edge. Game programmers call this ‘wrapping round’, which is what it feels like, but exactly what you do not attempt to do literally unless you want to create a big mess of broken screen. Topologically, wrapping the vertical edges round converts the screen into a cylinder. Wrapping the horizontal edges round then joins the ends of the cylinder to make a torus. Now there’s no edge and the aliens can’t escape.

The flat torus is the simplest instance of a general method used by topologists to make complicated spaces from simpler ones. Take one or more simple shapes, then ‘glue’ them together by listing rules for which bit attaches where. It’s like flat-pack furniture: lots of pieces and a list of instructions like ‘insert shelf A into slot B’. But mathematically, the pieces and the list are all you need: you don’t actually have to assemble the furniture. Instead, you think about how it would behave if you did.

Until humanity invented space travel, we were in the same position as the ant with regard to the shape of the Earth. We still are in the same position as the ant with regard to the shape of the universe. Like the ant, we can nevertheless infer that shape by making suitable observations. However, observations alone are not enough; we also need to interpret them in the context of some coherent theory about the general nature of the world. If the ant doesn’t know it’s on a surface, Gauss’s formula isn’t much help.

At the moment, that context is general relativity, which explains gravity in terms of the curvature of spacetime. In a flat region of spacetime, particles travel in straight lines, just as they would in Newtonian physics if no forces were acting. If spacetime is warped, particles travel along curved paths, which in Newtonian physics would be a sign that a force is acting – such as gravity. Einstein threw away the forces and kept the curvature. In general relativity, a massive body, such as a star or a planet, bends spacetime; particles deviate from a straight line path because of the curvature, not because a force is acting on them. If you want to understand gravity, said Einstein, you have to understand the geometry of the universe.

In the early days of general relativity, cosmologists discovered a sensible shape for the universe, one that was consistent with relativity: a hypersphere. Topologically, this is like an ordinary sphere, by which they mean just the surface. A sphere is two-dimensional: two numbers are enough to specify any point on it. For example, latitude and longitude. But a hypersphere is three-dimensional. Mathematicians define a hypersphere using coordinate geometry. Unfortunately, it’s not a shape that lives naturally in ordinary space, so we can’t just make a model or draw a picture.

It’s not just a solid ball – a sphere plus the material of its interior. A sphere has no boundary, so neither should a hypersphere. Discworld, for instance, does have a boundary, where the world stops and the oceans fall off the edge. Our spherical world is different: it has no edge. Wherever you stand, you can look around you in all directions and see land or ocean. An ant, wandering through its spherical world, would not encounter a place where it runs out of universe. The same should be true of a hypersphere. But a solid ball does have a boundary: its surface. An ant that could travel at will through the interior of a ball – as we move through space unless something gets in the way – would run out of universe when it hit the surface at the other side.

For present purposes, all we really need to know about a hypersphere is that it’s the natural analogue of a sphere, but with one extra dimension. For a more specific image, we can think about how an ant might visualise a sphere, and beef everything up by one dimension – the same trick that A. Square uses in Flatland. A sphere is two hemispheres glued together at the equator. A hemisphere can be flattened out to form a flat disc: a circle plus its interior, and this is a continuous deformation. So a topologist can think of a sphere as two discs glued together along their edges, like a flying saucer. In three dimensions, the analogue of a disc is a solid ball. So we can make a hypersphere by conceptually gluing the surfaces of two solid balls together. This can’t be done in ordinary space using round balls, but mathematically we can specify a rule that associates each point on the surface of one ball with a corresponding point on the surface of the other ball. Then we pretend that corresponding points are the same, much as we ‘glued’ the edges of a square together to get a flat torus.

The hypersphere played a prominent role in the early work of Henri Poincaré, one of the creators of modern topology. He operated around the turn of the nineteenth century, and was one of the top two or three mathematicians of the day. He came perilously close to beating Einstein to special relativity.fn3 In the early 1900s, Poincaré set up many of the basic tools of topology. He knew that hyperspheres play a fundamental role in three-dimensional topology, just as spheres do in two-dimensional topology. In particular, a hypersphere has no ‘holes’ analogous to the hole in a doughnut, so in a sense it is the simplest three-dimensional topological space. Poincaré assumed, without proof, that the converse is also true: a three-dimensional topological space without holes must be a hypersphere.

In 1904, however, he discovered a more complicated shape, the dodecahedral space, which has no holes, but isn’t a hypersphere. The existence of this particular shape proved that his assumption was wrong. This unexpected setback led him to add one further condition, which he hoped would fully characterise the hypersphere. In two dimensions, a surface is a sphere if and only if every closed loop can be pushed around until it all piles up in the same place. Poincaré conjectured that the same property characterises a hypersphere in three dimensions. He was right, but it took mathematicians almost a century to prove it. In 2003 a young Russian, Grigori Perelman, succeeded in proving Poincaré’s conjecture. This entitled him to a million-dollar prize, which he famously declined.

Although a hyperspherical universe is the simplest and most obvious possibility, there’s not a great deal of observational evidence for it. A flat plane used to be the simplest and most obvious possibility for the Earth’s surface, and look where that got us. So cosmologists stopped tacitly assuming that the universe must be a hypersphere, and started to think about other possible shapes. One of the most widely publicised suggestions, for a short time, appealed to the news media because it indicated that the universe is shaped like a football. (For US readers: soccer ball.) Editors loved it, because although readers might not know much cosmology, they sure know what a football is.fn4

It’s not a sphere, you understand. A football – at that time, and not for long – had abandoned the old shape of eighteen rectangular panels sewn into a sort of cube, and adopted a snazzy new shape, twelve pentagons and twenty hexagons sewn or glued into a truncated icosahedron.fn5 This is a solid that goes back to ancient Greece, and with a name like that, it’s a good job you can refer to it as a football. Except that – well, actually it’s not a truncated icosahedron at all. It’s a three-dimensional hypersurface bearing a loose relationship to a truncated icosahedron. A football from another dimension.

To be specific, it is Poincaré’s dodecahedral space.

To make a dodecahedral space, you start with a dodecahedron. This is a solid with twelve faces, each a regular pentagon; like a football without the hexagons. Then you glue opposite faces together – something that is not possible with a real dodecahedron. Mathematically, there is a way to pretend that distinct faces are actually the same, without bending the thing to join them together, as we saw for the flat torus; topologists, however, insist on calling it ‘gluing’.

The dodecahedral space is an elaborate variation on a flat torus. Recall that we make a flat torus by taking a square and gluing opposite edges together. To get the dodecahedral space, which is not a surface but a three-dimensional object, we take a dodecahedron and glue opposite faces together. The result is a three-dimensional topological space. It has no boundary, just like a torus, and for the same reason: anything that is in danger of falling out through a face reappears inside at the opposite one, so there’s no way out. It has finite size. And, like a hypersphere, it has no holes, so if you are a slightly naive topologist you might be tempted to think it passes all the tests needed to be a hypersphere – but it isn’t a hypersphere, not even topologically.

Poincaré devised his dodecahedral space as a piece of pure mathematics, exposing a limitation of the topological methods available in his day – one that he set out to remedy. But in 2003 the dodecahedral space acquired brief notoriety and a potential application to cosmology when NASA’s Wilkinson Microwave Anisotropy Probe (WMAP) satellite was measuring fluctuations in the cosmic microwave background, a persistent hiss detected by radio telescopes that is interpreted as a relic of the Big Bang. The statistics of these tiny irregularities provide information about how matter clumped together in the early universe, acting as a seed from which stars and galaxies formed. WMAP can see far enough in space to, in effect, see back in time to about 380,000 years after the Big Bang.

At the time, most cosmologists thought that the universe was infinite. (Although this conflicts with the standard description of the Big Bang, there are ways to accommodate it, and ‘universes all the way out’ has the innate appeal already noted for ‘universes all the way back’ – which, ironically, is not what the Big Bang indicates.) However, the WMAP data suggested that the universe is finite. An infinite universe ought to support fluctuations of all sizes, but the data did not show any large waves. As a report in Nature said at the time, ‘you don’t see breakers in your bathtub’. The detailed data provided further clues about the likely shape of our breakerless bathtub universe. Working out the statistics of the fluctuations for a variety of potential shapes, the mathematician Jeffrey Weeks noticed that the dodecahedral space fitted the data very well, without any special pleading. Jean-Pierre Luminet’s group published an analysis showing that if this were correct, the universe would have to be about 30 billion light years across.fn6 This theory has since fallen out of favour thanks to further observations, but it was fun while it lasted.

We human ants can use another trick to infer the shape of space. If the universe is finite, some rays of light will eventually return to their point of origin. If you could look along one of these ‘closed geodesics’ (a geodesic is a shortest path) with a sufficiently high-powered telescope, and if light travelled infinitely fast, you would see the back of your own head. Taking the finite speed of light into account, patterns should occur in the cosmic microwave background, forming matching circles in the sky. The way these circles are arranged would provide information about the topology of space. Cosmologists and mathematicians have tried to find such circles, so far with no convincing successes. If the universe is finite but too big, we wouldn’t be able to see far enough to spot the circles anyway.

So the current answer to the question ‘what shape is the universe?’ is very simple. We don’t know. We don’t know whether it’s a hypersphere or something more elaborate. The universe is too big for us to observe it all, and our current understanding of cosmology, indeed of fundamental physics, wouldn’t be up to the task even if we could.

Some of the difficulties surrounding cosmology stem from a mix-and-match approach in which relativity is invoked at some stages and quantum mechanics at others, without recognising that they contradict each other. Theorists are reluctant to discard the tools they are accustomed to, even when those tools don’t seem to be working. But the shape of the universe is a problem that really requires a combination of these two great physical theories. And that brings us to the search for a unified field theory, or Theory of Everything, to which Einstein devoted many years of thought – without success. Somehow, relativity and quantum mechanics have to be modified to produce a consistent theory that agrees with each of them in the relevant domain.

Today’s front runner is string theory, which replaces point particles by tiny multidimensional shapes, as we discussed in The Science of Discworld III. Some versions of string theory require space to be nine-dimensional, so spacetime has to be ten-dimensional. The extra six dimensions of space are thought either to be curled up so tightly that we don’t notice them, or inaccessible to humans – in the same way that A. Square could not travel out of Flatland unaided, but needed a shove from the Sphere to push him into the third dimension. The formulations of string theory currently in vogue also introduce new ‘super-symmetry’ principles, predicting the existence of a host of new ‘sparticles’ to match the known particles. So an electron is paired with a selectron, and so on. So far, however, this prediction has not been confirmed. The LHC has looked for sparticles, but so far it has found precisely none of them.

One of the latest unification attempts, refreshingly different from most that have gone before, takes us neatly back to Flatland. The idea, common in mathematics and often fruitful, is to take inspiration from a cut-down toy problem. If unifying relativity and quantum mechanics is too hard using three-dimensional space, why not simplify the problems by looking at the non-physical but mathematically informative case of two-dimensional space? Plus one of time, naturally. It’s clear enough where to start. In order to unify two theories, you have to have two theories to unify. So what would gravity look like in Flatland, and what would quantum mechanics look like in Flatland? We hasten to add that Flatland, here, need not be A. Square’s Euclidean plane. Any two-dimensional space, any surface, would do. Indeed, other topologies are essential to get anything interesting.

It’s straightforward to write down sensible analogues of Einstein’s field equations when space is a surface. It’s very close to what Gauss did when he started the whole thing off, and his ant would have no trouble in devising the right equations, since they’re all about curvature. There are obvious analogies to follow; all you do is replace the number three by the number two at key points. In Roundworld, the Polish physicist Andrzej Staruszkiewicz wrote such equations down in 1963.

It turns out that gravity in two dimensions differs significantly from gravity in three. In three dimensions relativity predicts the existence of gravitational waves, which propagate at the speed of light. But there are no gravitational waves in two dimensions. In three dimensions, relativity predicts that any mass bends space into a rounded bump, so that anything that passes nearby will follow a curved path, as if it were attracted by Newtonian gravity. And an object that was at rest will fall into the gravitational well of the mass concerned. In two dimensions, however, gravity bends space into a cone. Moving bodies are deflected, but bodies at rest simply remain at rest. In three dimensions massive bodies collapse under their own gravitation to form black holes. In two dimensions, this is impossible.

These differences are things we can live with, but in three dimensions, gravitational waves are a useful way to link relativity to quantum theory. The absence of gravitational waves in two dimensions causes headaches, because it means there is nothing to quantise – nothing to use as a starting point for a quantum-mechanical formulation. Gravity should correspond to hypothetical particles called gravitons, and in quantum theory particles have ghostly companions – waves. No waves, no gravitons. But in 1989 Edward Witten, one of the architects of string theory, ran into other quantum problems involving fields that do not propagate waves. Two-dimensional gravity is like that, and it opened his eyes to a missing ingredient.

Topology.

Even when gravity can’t travel as a wave, it can have a huge effect on the shape of space. Witten’s experience with topological quantum field theories, where just this ingredient arises, suggested a way forward. The humble torus, in many ways the simplest non-trivial topological space, plays a key role. We’ve mentioned the flat torus, in which we glue together the opposite edges of a square. Squares are nice because they can be fitted with grids of smaller squares, which have a quantum feel to them because they are discrete – they come in tiny lumps. But you can make flat tori from other shapes too, namely parallelograms.

The shape of the parallelogram can be captured by a number called the modulus, which distinguishes long thin parallelograms from short fat ones. A different modulus gives a different torus. Although the tori obtained in this manner are flat, they have different metrics. They can’t be mapped into each other while keeping all distances the same. The effect of gravity in Torusland is not to create gravitons: it is to change the modulus, the shape of space.

Steven Carlipp has shown that in Torusland, there is an analogue of the Big Bang. But it doesn’t start with a point singularity. Instead, it begins as a circle: a torus with modulus zero. As time passes, the modulus increases, and the circle expands into a torus. Initially this looks like a bicycle tyre, and corresponds to a long thin parallelogram; it is heading towards a square, the standard model for a flat torus, which when curled up looks more like a bagel. So the long-term goal of the Flatland Big Bang turns out to be A. Square. Crucially, Carlipp quantised this entire process; that is, he formulated a quantum-mechanical analogue. That let theoretical physicists explore the relationship between quantum theory and gravity in a precise mathematical context.

Torusland sheds a great deal of light on the process of quantising a gravitational theory. One apparent casualty of this process, however, is time. The quantum wave function of Torusland does not involve time at all.

In The Science of Discworld III Chapter 6, we discussed Julian Barbour’s The End of Time, which proposes that time does not exist in a quantum world because there is only one universal wavefunction, not involving time. The book was widely interpreted as telling us that time is an illusion. ‘There can only be once-and-for-all probabilities,’ Barbour wrote. We argued that alongside the universal wavefunction, our universe has another basic quantum-theoretic feature, which describes how likely transitions between different states are. These transition probabilities show that some states are closer together than others, and that lets us arrange the events in a natural order, restoring a sensible notion of time.

Torusland supports this idea, because it has several sensible notions of time, even though its quantum wavefunction is timeless. Time can be measured using Torusland’s equivalent of GPS satellites, by using the lengths of curves between its version of the Big Bang and ‘now’, or by the current size of the universe. Torusland is not timeless at all. You just have to look at it in the right way. In fact, Torusland time leads to an intriguing thought: perhaps time is a consequence of gravity.

Another idea that Torusland casts doubt on is the holographic principle. This says that the quantum state of the entire observable universe can be ‘projected’ onto any black hole’s event horizon – the point of no return from which nothing can escape – so the universe’s three spatial dimensions can be reduced to just two. It’s like taking a photograph, with the startling property that the photograph faithfully represents all aspects of reality. In Roundworld, if someone shows you a photo of a field with a dozen sheep lying down, you can’t tell whether there are lambs hiding behind some of the sheep. But in this event-horizon photo of the universe, nothing can be hidden. The behaviour in two dimensions corresponds perfectly to that in three. The laws of physics change, but everything matches up.

This is a bit like the way a two-dimensional hologram creates a three-dimensional image, which is why this idea is called the holographic principle. It suggests that not only is the dimension of the universe an open question: it may not be well defined – the answers ‘two’ and ‘three’ may both be true at the same time. This idea has led to some advances in the way string theory represents gravity, and also to articles in the press stating ‘You are a hologram!’

Physicists began to suspect that a similar principle works in any number of dimensions. But it turns out that in Torusland, there is no holographic principle. A. Square may be flat, but he’s not a hologram. So maybe we’re not holograms either. Which would be nice.

Some even more radical ideas about the shape of our universe have just surfaced, threatening to overturn many deep-seated assumptions in cosmology. Instead of being a gigantic hypersphere, or a flat Euclidean space, the universe might be more like an etching by the Dutch artist Maurits Escher.

Welcome to the Escherverse.

A hypersphere is the iconic surface with constant positive curvature. There is also an iconic surface of constant negative curvature, called the hyperbolic plane. It can be visualised as a circular disc in the usual Euclidean plane, equipped with an unusual metric, in which the unit of measurement shrinks the closer you get to the boundary. Escher based some of his etchings on the hyperbolic plane. A famous one, which he called ‘Circle Limit IV’ but is usually referred to as ‘angels and devils’, tiles the disc with black devils and white angels. Near the middle these appear quite large; as they approach the boundary they shrink, so that in principle there would be infinitely many of them. In the metric of the hyperbolic plane, all devils are the same size, and so are all angels.

String theory tries to unify the three quantum-mechanical forces (weak, strong and electromagnetic) with the relativistic force of gravity, and gravity is all about curvature. So curvature plays a key role in string theory. However, attempts to marry string theory to relativistic cosmology tend to come to grief, because string theory works best in spaces with negative curvature, whereas positive curvature works better for the cosmos. Which is a nuisance.

At least, that’s what everyone thought.

But in 2012 Stephen Hawking, Thomas Hertog and James Hartle discovered that they could use a version of string theory to write down a quantum wavefunction for the universe – indeed, for all plausible variations on the universe – using a space with constant negative curvature. This is the Escherverse. It’s terrific mathematics, and it disproves some widely believed assumptions about the curvature of spacetime. Whether it will also work out as physics remains to be seen.

So what have we learned? That the shape of our universe is intimately related to the laws of nature, and its study sheds some light – and a lot more darkness – on possible ways to unify relativity and quantum theory. Mathematical models like Torusland and the Escherverse have opened up new possibilities by showing that some common assumptions are wrong. But despite all of these fascinating developments, we don’t know what shape our universe is. We don’t know whether it is finite or infinite. We don’t even know for sure what dimension it is, or even whether its dimension can be pinned down uniquely.

Like A. Square, trapped in Flatland, we are unable to step outside our world and view it unobstructed. But, also like him, we can learn a lot about the world despite that. On Discworld, creatures from the dungeon dimensions are only an incantation away; in Flatland a helpful Sphere may pop into view to help the story along. But Roundworld doesn’t run on narrativium, and an extra-universal visitor from hidden dimensions seems unlikely.

So we are stuck with our own resources: imagination, ingenuity, logic and respect for evidence. With these, we can hope to infer more about our universe. Is it finite or infinite? Is it four-dimensional or eleven-dimensional? Is it round, flat or hyperbolic?

For all we know right now, it might be banana-shaped.

fn1 Yes, two Abbotts. His father was Edwin Abbott. So was his son.

fn2 Abbott never said what the ‘A’ stood for. One theory is that A2 = AA = Abbott Abbott. In Ian’s modern sequel Flatterland it is ‘Albert’. Google ‘Albert Square’.

fn3 Some mathematicians think he did, but the physicists didn’t notice because he wasn’t a physicist.

fn4 So do the wizards: see Unseen Academicals.

fn5 By the 2006 World Cup it was made from 14 panels: six dumbbell-shaped ones and eight like the Isle of Man’s triple running-legs emblem. The underlying symmetry was again that of a cube. If you think that analysing symmetries of footballs is nerdy, look up the literature on symmetries of golf balls.

fn6 J-P Luminet, Jeffrey R. Weeks, Alain Riazuelo, Roland Lehoucq and Jean-Philippe Uzan, Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background, Nature 425 (2003) 593.