Sphere of Influence

Just over a hundred years ago, the quaintly named Reverend Edwin Abbott Abbott wrote a slim volume that has captured a diverse readership ever since. Flatland (1884) is the story of a two-dimensional world and one of its citizens, who has a revelatory experience and comes to understand something of his spatial restriction. The book is a kind of fantasy yarn which has subsequently bedevilled cocktail party geometry and spawned a whole genre in the process. Yet for a contemporary reader, and originally for the author as well, the gimmickry and trigonometric stuntwork are not really the heart of the book. Flatland is fascinating because it leaves the reader with a real sense of the limitations of human cognition and the need for intellectual humility.

Abbott was a prolific writer who produced over 40 books during his 60-year writing career, including a guide to Shakespearean grammar and numerous theological texts. He became headmaster at the City of London School at 26 and remained there for 25 years, and it was towards the end of this period that he wrote Flatland. But significantly, and despite its zealous maths teacher following, he wrote no other science books at all. Flatland was the product of a theologian rather than a schoolmaster.

The book is divided into two sections. In the first, the author - Mr A. Square - describes life in Flatland; in the second, he undergoes a revelatory experience on the eve of the third millennium. In Flatland, the people are different shapes and their social status determined by internal angle: isosceles triangles, equipped with sharp ends, are soldiers; equilateral triangles are tradesmen; squares are professionals; and the high priests are circles. Special humiliation is reserved for women who, being mere straight lines, are both potentially invisible as well as lethally sharp. In Flatland therefore, all women must sing a 'peace-song' and walk with an exaggerated swing of the 'hips' in order to alert others of their location. But this structural sexism is just the start of Flatland's cultural woes. As a highly conservative theocracy, there is institutionalised eugenics, strict codes of colouration (yes, the fashion police become a reality) and every Victorian's nightmare - the possibility of a slightly irregular hexagonal child in your aristocratic attic, requiring expensive and risky corrective surgery.

On the basis of the first half of the book, Flatland is often positioned as a lampoon and linked to satires such as Swift's much earlier Gulliver's Travels (1726). But despite superficial parallels, the only fundamental connection is that they are both at pains to portray the limitation and blindness that existed in their societies. Otherwise Flatland is not a straight satire, and it is hard to imagine Abbott's contemporaries gaining any specific political insights from it. His treatment of Victorian resistance to the rights of women, for example, is not entirely convincing. Abbott, who was president of the Teacher Training Society, campaigned to improve women's access to universities, but in Flatland he provides a weak Catch-22 argument: anyone against educating women must surely agree with educating men, yet a man's education begins in the nursery under the control of a woman who in turn should be educated for the role. Hardly the most incisive and withering assault on blinkered Victorian values to be found in the English language. Likewise, there is little response to the ethical implications contemporary scientific thought beyond mathematics - Darwin had published the Origin of Species in 1859, 25 years before Flatland, and survival of the fittest was an established though controversial concept. (Abbott might well have been in a position to describe how, after many generations, the circles had been overthrown by terrifying spiky warlords. Disappointing as it seems, given the enigmatic perfection that circles represent, the smart money would have been on asterisks.)

The second half of the book deals with Mr Square's revelations and his subsequent incarceration. He has a queer dream of a place called Lineland and then, as he sits alone during the final hours of 1999, he is visited by a sphere from Spaceland. Mr Square perceives the visitor as a circle of changing diameter which alarmingly appears and disappears into thin air. The sphere apparently makes a habit of appearing each millennium to preach the Gospel of Three Dimensions. It gives the square a mind-altering trip off the plane and out over Flatland, during which, despite some initial hostility, the square is converted. Finally, since discussion of a third dimension is forbidden by law in Flatland, he is imprisoned after describing his experiences. The book clearly begs the question as to whether we are similarly constrained and muses on the existence of fourth and higher dimensions.

Such ideas were not unprecedented at the time. As a Cambridge classicist, Abbott would have been familiar with Plato's Simile of the Cave from Republic, in which Plato describes prisoners whose entire knowledge of the outside world comprises the two dimensional shadows it throws upon their cell wall. Furthermore, 19th-century interest in non-Euclidean geometry had produced a number of allusions to life in two dimensions: over 50 years before Flatland was published; Gauss, for example, had encouraged readers to consider the measurements made by intelligent flatworms moving along the surface of a membrane in space. Victorian scientists had also mulled over the existence of a fourth dimension - the psychologist Fechner wrote Space has Four Dimensions in 1846, though he had time in mind, and the writer and bigamist Charles Hinton had also written an article on the fourth dimension. That Hinton was an acquaintance of Abbott's strongly suggests that Abbott became familiar with such scientific musings.

This slant has been instrumental in Flatland's popularity and in the process has unleashed a whole category of books which might be called 'visceral physics'. There are direct descendants like A. K. Dewdney's The Planiverse (1984), which covers many technical details such as how to play volleyball, open doors and use zippers in 2D. Then there are more distant relations in the school of Richard Feynman, a popular physicist who won a Nobel prize with a theory of the atom inspired by the wobbling motion of a plate being thrown across the Princeton canteen. Having almost certainly been lifted from the plane at some stage, Feynman returned to Earth with the TV series Fun to Imagine in which a bemused audience is invited to picture themselves shrunk to atomic size, immersed in boiling water and mesmerised by all those jiggling molecules.

For these types of Flatland reader, together with millions of Star Trek fans, the high moments of the book are the descriptions of living in 2D and the implications of extrapolating 3D geometry into 4D by analogy. For example, Abbott discusses the family of forms which we might call the cubes. Take a point. Make another point and connect them to make a line. Make another line and connect the vertices to make a square. Make another square and connect them to make a cube. Make another cube and connect them to make a tesseract - a 4D cube. And so on. Modern computer animation techniques can now model such shapes, and, using ordinary rules of projection, show us the 3D and 2D shadows that they cast as they move and rotate.

Although Abbott did not really write Flatland for a techie audience, thinking about the fourth dimension is a compelling subject and he is frequently quoted in the same breath as the subsequent scientific developments which have utilised it. Many are the physicists who boast of their ability to envisage that elusive otherness. But one suspects that, despite these claims, the reality of perceiving the fourth dimension remains as fleeting as the childhood endeavour of trying to see round the corner of a mirror. The fourth dimension remains an imposing symbol of our ability to deduce beyond that which we can conceive - a symbol of the frustrating gap between shadowy consciousness, whose soft focus so often makes our ideas seem beautiful, and their ugly, unconvincing offspring, which we discharge only after a traumatic, bottle-necked passage to the outside world.

But simply conceiving a fourth dimension does not prove its existence. Can a fourth dimension ever really be said to exist when philosophers cause us to doubt even our direct perceptions? In one sense, the question is facile since every point in space has a multitude of dimensions (hydrogen density, magnetic field strength, and so forth). The question needs to be rephrased: is there another 'space-like' dimension? How would we set about verifying the reality of something which we are not designed to perceive and which therefore can only be inferred from indirect evidence? The examples most commonly cited as indications of Abbott's prescience are String theory and its reincarnation, M theory. These postulate that atomic particles are not points but lines stretching out in other dimensions we cannot see but in which they interact. However, there is a real sense that these descriptions are only analogies and until such theories have been demonstrated as virtually irrefutable, the fourth dimension must remain only a possibility.

Arguably it is this armchair science angle which has produced the largest audience for Flatland and driven the book towards cult status. It is also this aspect which has led to frequent comparisons between Abbott and Lewis Carroll. But as with Swift, the real connection lies in their portrayal of the limited perspectives of Victorian society. Beyond that, Carroll was a mathematician and logician writing stories for children, whilst Abbott was a theologian using Flatland as an allegory to enlighten his peers. Dimensional speculations are an indulgence rather than material to Abbott's book - dimensionality is used as a convenient device and his main objective seems to have been more with morality than geometry. During his life he had shown a keen interest in the popular Victorian pastimes of magic and mediums. In particular, it was the relationship between illusion and miracles that caught his attention. He felt strongly that miracles should not form an integral part of religious belief because they were both misleading as evidence of holiness and highly distracting to the main messages of the Bible.

As such, he was alarmed that Victorian scientific criticism of the Bible was unnecessarily undermining its credibility and wrote Flatland to highlight the division that should be made between miracles and morality. It is a belt-and-braces attack. On one hand Abbott argues that physical superiority does not imply moral superiority and that consequently the two must be divorced in our evaluations. On the other hand, he argues that all societies have limited perspectives and that the reckless rejection of anything incongruent with those societies' beliefs is at best unwarranted and more likely a blunder. The miracles might have been true records, and even if they were not, they were separate from the ethical and religious importance of Christianity - we cannot know the Transcendental directly anyway. How Abbott felt about the religious significance of miracles comes across clearly when A. Square is discussing his experiences off the plane with the visiting sphere. The square declares 'Behold, I am become as a God. For the wise men in our country say that to see all things, or as they express it, OMNIVIDENCE, is the attribute of God alone'. The sphere replies: 'This omnividence, as you call it [...] does it make you more just, more merciful, less selfish, more loving? Not in the least. Then how does it make you more divine?' For Abbott, miracles are right up there with the Golden Calf.

His feelings about the limited nature of society pervades many of his books, but in Flatland the most limited creature is probably the point that lives in Pointland. It holds that most popular of opinions - that it is the universe. It purrs to itself: 'Infinite beatitude of existence! It is; and there is none else beside It'. As we suspected, happy people are indeed deluded. But whilst the Point is portrayed as plainly ridiculous, Abbott also explores the frequently tragic consequences of narrow-mindedness. Ten years after Flatland, he wrote two related stories, Philochristus and Onesimus. In the first, the storyteller is a Pharisee who meets Christ, and in the second the protagonist is the Greek slave who meets St Paul and is then mentioned in his Epistle to Philemon. Like Mr Square, both characters are transformed by their encounters, and following their revelations are rejected and abused by their former brethren. For the truly inspired, Abbott foresees only frustration and persecution at the hands of their bigoted and complacent peers. It is this cautionary tale which should make the book required reading for our generation.

There has been a long-fought rear-guard action to maintain the primacy of mankind in the universe. Naturally, humans want to see themselves in the middle of things since from their perspective that is indeed where they are. Even in our own agnostic era, this intuition remains pervasive. Yet this kind of assumption is at bottom a conceit which seems to have become the unwitting instrument of mankind's worst excesses against itself. Simple tests can remind us of just how specifically we were designed. As you sit and read, your mind is conscious of a complete and detailed picture of the world around you. Yet if you hold this page a few inches away from the centre of your vision the writing is a blur - demonstrating the fallacy of your perceptual completeness. Every aspect of the mind is similarly partial, and with a moment's reflection we can sense this. Our mind is not so completely guided by our consciousness as we think and nor is that consciousness operating as continuously and smoothly as we believe. Our memories are inaccurate, incomplete and significantly rewritten. Our reasoning is focussed on diagnosing patterns and causality, incapable of handling more than a few variables simultaneously, and frequently draws automatically from a bag of pre-programmed heuristics. We have minds pared to the absolute minimum necessary for the survival and procreation of our forebears and not a neuron more.

Today, in our abstract and mechanised world, examples of these mental constraints abound. Everywhere there is an inability to synthesise information, conflicting accounts of events and failures to co-ordinate society. People are willingly ignorant of these cognitive limitations and the appalling distortions they are hatching into a world whose complexity and sophistication has rendered the original blueprint obsolete. People are struggling with wildly outmoded legacy wetware.

So, if ever a parable of our deficiencies were needed it must surely be today. But how are we to tread that thin line between mental humility and outright gullibility? If he were a headmaster today, it would be interesting to see how Abbott might react to the Fundamentalist demand that Genesis be taught as an alternative to evolutionary theory in his school. One suspects he would have disapproved. Abbott did not advocate irrationality, but open mindedness. He wanted us to judge ideas, but more importantly to base that judgement on a recognition of our own undoubted yet uncharted limitations. As the sphere says of the point: 'Yet mark his perfect self-contentment, and hence learn this lesson, that to be self-contented is to be vile and ignorant, and that to aspire is better than to be blindly and impotently happy.'