Flatland

 I can't remember where I heard about Flatland.  I suspect in more than one place.  

Edwin Abbott Abbott was famous in his own time as the principal of a prestigious London school and a writer of school textbooks.  It's fascinating that almost a century after his death, the only reason he is remembered is for a a little book he published in 1884 which the editors of the British Dictionary of National Biography didn't even feel was worth noting in their entry on him.

Flatland: A Romance in Many Dimensions is, at one level, an extended set of mathematical jokes.  They begin on the title page, where we learn that the tale is narrated by A. Square, a play on Abbott's own name (in mathematical notation his initials could be rendered EA²).  I would not be surprised if many of the geometrical illustrations in the book began their lives as jokes to liven up dull geometry lessons for his pupils.

The first part of the story describes the land of Flatland through the eyes of A. Square who is, of course, just what his name says.  Flatland is a land which has only two dimensions, length and breadth.  Its inhabitants are geometrical figures, with a clear social hierarchy based on the number of sides a figure has.  The working class are triangles, with subtle gradings based on the degree to which they approximate an equilateral.  Above them, the social hierarchy stretches through the levels of polygons until finally we reach a ruling elite of circles.  Women, unfortunately, are portrayed as straight lines.  It's not clear if Abbott is being misogynistic or parodying misogyny but sadly I suspect the former.

A. Square provides his readers with a cheery description of some of the social mores of Flatland - the ways people distinguish between individuals, the protocols for avoiding being impaled on a sharp angle (particularly for low class triangles and women, whose ends can be deadly) the subtle skill of determining the angle of a junction of lines by sight and the less subtle art of feeling a shape's sides to determine its rank.  There are geometrical diagrams to help, with lines, points and measures.  

It's all jolly fun, but the second part of the book presents a rapid change of tone.  First of all, in a dream A. Square pays a visit to Lineland, a land in which there is only one dimension, and has a chat with its king.  He tries unsuccessfully to persuade the king that in fact there are two dimensions, but of course the Lines are unable to perceive them and so think he is just being impertinent.  

Then, as he is sitting in his home awaiting the turn of the Flatland Millennium, A. Square is visited in his turn by a Sphere, a missionary from the realm of Spaceland.  This august being is taking advantage of a once-in-a-millennium opportunity to enlighten the people of Flatland about the true nature of the universe.  He attempts to prove the existence of  third dimension mathematically, then by dramatically passing through the plane of Flatland.  Nothing works, and finally he is forced to lift the Square bodily out of Flatland and show him.  

What happens next is the most fascinating part of the book.  One the one hand, the inhabitants of Flatland, and particularly the ruling circles, believe that A. Square is not merely foolish and deluded but positively dangerous, and they lock him up.  On the other, he draws the next logical inference from the revelation - if there are three dimensions and not, as he had always thought, merely two, then why should there not also be a fourth, and a fifth, and so on?  He discusses this with the Sphere and demonstrates the possibilities to him mathematically, but is rebuffed in his turn - the Sphere considers the idea absurd.  

Apart from the fact that it's good fun, I think there are a number of reasons why Flatland is still in print after 140 years.  The first is that on the level of mathematics, it is quite prescient.  In 1884 the idea that there could be more than three dimensions was a controversial outlier.  Yet only 20 years later Einstein published his first papers on the Theory of Relativity, which treats time as a fourth dimension.  Then in the 1970s John Schwarz made the initial ventures into String Theory, the mathematics of which require ten dimensions.  What was a mere mathematical game for Abbott is now serious physics and cosmology.

But there is another level too.  Abbott is asking us to examine ourselves and the way we think.  We assume the world is limited to what we experience, but what if this is not so?  What if beyond our own frame of reference there is a whole universe, infinitely richer and more complex than our imagination allows?  Saying this might get you labelled a crank or a criminal by the powers that be, but it is often the cranks and dissenters who turn out to be right.

Then there is the coda.  In another dream A. Square is given a glimpse of Pointland, the realm in which there are no dimensions and only a single Point.  The Point is filled with its own sufficiency and uniqueness, speaking of itself in the third person.

Infinite beatitude of existence! It is; and there is none else beside It.... It fills all Space, and what It fills, It is.  What It thinks, that It utters; and what It utters, that It hears; and It itself is Thinker, Utterer, Hearer, Thought, Word, Audition; it is the One, and yet the All in All.

In vain does A. Square try to jolt the Point from its self-absorbed soliloquy.

Silence, silence, contemptible Creature.  You call yourself the All in All, but you are the Nothing; your so-called Universe is a mere speck in a Line, and a Line is a mere shadow as compared with...

Here the Sphere stops the Square in mid-harangue and invites him to listen to the response.

Ah, the joy, ah the joy of Thought.  What can It not achieve by thinking!  Its own Thought coming to Itself, suggestive of Its disparagement, thereby to enhance Its happiness!.... Ah, the divine creative power of the All in One! Ah, the joy, the joy of Being!

This is, perhaps, superficially a joke about how a point must feel, and at the next level a warning about the perils of self-absorption.  But the theological language is unmistakable.  This is the self-referential, self-sufficient, unapproachable God which needs nothing else, is unaffected by anything else, which (in extreme versions) is itself everything.  Abbott's single Point is a pinprick which lets the air out of this conception of God - such a God would in fact be a virtual nothing, a contemptible creature so limited that even the wall-eyed king of Lineland is more alive, more changeable, more powerful.  

Abbott would like us, by contrast, to imagine a universe not smaller and more limited than ourselves, but larger even than we can imagine, one that exists in dimensions we can't even conceive.  Even the Sphere, which initially seems Godlike to Square, is ultimately shown to be a limited, finite being.  If, as in Anselm's thought experiment, God is that than which nothing greater can be imagined, God must exist in more dimensions than the sphere knows, more dimensions than even the mathematics of String Theory suggests.  Only thus could we have a truly great God, existing in our three dimensions but also in dimensions which we cannot even conceive, never mind visit.