The very idea of it is utterly inconceivable.”
Today, at the end of the twentieth century, scientists are not so complacent as a century ago. We are particularly aware of the limits of human sensory perception. Our instruments detect X rays and radio waves at frequencies that the eye cannot see, measure relativistic time dilation at speeds far greater than human travel, confirm that subatomic particles behave as if they were in two places at once. Biologists now have the ability to alter the cellular instructions of animals, or to clone others from single cells in a test tube. We freely acknowledge that the world is far stranger than it seems. Thus Schwarz’s string theory of ten spatial dimensions, Stephen Hawking’s calculations of the evaporation of black holes, and Alan Guth’s theory of an exponentially expanding universe are all taken seriously.
But Abbott, if we read him deeply, has challenged us to question more than our tenets of geometry and physics. If the very dimensionality of space is open to question, then what beliefs remain sacred? What else should we question? For example: Is there really a sharp division between animate and inanimate matter? Could human consciousness be some kind of collective phenomenon, even though each of us has the strong sensation of individual thoughts and minds? Does the earth behave as a single living organism, with all of its physical and biological systems purposefully connected (as proposed in the “Gaia Hypothesis”)? Do nonphysical dimensions exist? Does modern technology diminish, rather than enhance, the quality of life? I confess that I do not know how to ask these kinds of questions, or even what areas of thought they involve. I cannot conceive of a world with these possibilities. And that is the point. The inhabitants of Flatland could not conceive of a third dimension. By definition, it is extremely difficult to imagine worlds outside of our experience. For that, we are as likely to receive guidance from our artists and philosophers, as from our mathematicians and scientists.
Alice’s Adventures in Wonderland by Lewis Carroll, first published in 1865
Through the Looking Glass, and What Alice Found There by Lewis Carroll, first published in 1872
Mr. Tompkins in Wonderland by George Gamow, first published in 1940 (Cambridge: Cambridge University Press, 1965)
Mr. Tompkins Explores the Atom by George Gamow, first published in 1945 (Cambridge: Cambridge University Press, 1965)
“Euclid, Newton, and Einstein” by W.G. in Nature, no. 2624, vol. 104, p. 627 (February 12, 1920)
Einstein’s Dreams by Alan Lightman (New York: Pantheon, 1993)
Gulliver’s Travels by Jonathan Swift, first published in 1726 (New York: Bantam, 1981)
FLATLAND
To
The Inhabitants of SPACE IN GENERAL
And H. C. IN PARTICULAR
This Work is Dedicated
By a Humble Native of Flatland
In the Hope that
Even as he was Initiated into the Mysteries
Of THREE Dimensions
Having been previously conversant
With ONLY TWO
So the Citizens of that Celestial Region
May aspire yet higher and higher
To the Secrets of FOUR FIVE OR EVEN SIX Dimensions
Thereby contributing
To the Enlargement of THE IMAGINATION
And the possible Development
Of that most rare and excellent Gift of MODESTY
Among the Superior Races
Of SOLID HUMANITY
If my poor Flatland friend retained the vigour of mind which he enjoyed when he began to compose these Memoirs, I should not now need to represent him in this preface, in which he desires, firstly, to return his thanks to his readers and critics in Spaceland, whose appreciation has, with unexpected celerity, required a second edition of his work; secondly, to apologize for certain errors and misprints (for which, however, he is not entirely responsible); and, thirdly, to explain one or two misconceptions. But he is not the Square he once was. Years of imprisonment, and the still heavier burden of general incredulity and mockery, have combined with the natural decay of old age to erase from his mind many of the thoughts and notions, and much also of the terminology, which he acquired during his short stay in Spaceland. He has, therefore, requested me to reply in his behalf to two special objections, one of an intellectual, the other of a moral nature.
The first objection is, that a Flatlander, seeing a Line, sees something that must be thick to the eye as well as long to the eye (otherwise it would not be visible, if it had not some thickness); and consequently he ought (it is argued) to acknowledge that his countrymen are not only long and broad, but also (though doubtless in a very slight degree) thick or high. This objection is plausible, and, to Spacelanders, almost irresistible, so that, I confess, when I first heard it, I knew not what to reply. But my poor old friend’s answer appears to me completely to meet it.
‘I admit,’ said he—when I mentioned to him this objection—‘I admit the truth of your critic’s facts, but I deny his conclusions. It is true that we have really in Flatland a Third unrecognized Dimension called “height”, just as it is also true that you have really in Spaceland a Fourth unrecognized Dimension, called by no name at present, but which I will call “extra-height”. But we can no more take cognizance of our “height” than you can of your “extra-height”. Even I—who have been in Spaceland, and have had the privilege of understanding for twenty-four hours the meaning of “height”—even I cannot now comprehend it, nor realize it by the sense of sight or by any process of reason; I can but apprehend it by faith.
‘The reason is obvious. Dimension implies direction, implies measurement, implies the more and the less. Now, all our lines are equally and infinitesimally thick (or high, whichever you like); consequently, there is nothing in them to lead our minds to the conception of that Dimension. No “delicate micrometer”—as has been suggested by one too hasty Spaceland critic—would in the least avail us; for we should not know what to measure, nor in what direction. When we see a Line, we see something that is long and bright; brightness, as well as length, is necessary to the existence of a Line; if the brightness vanishes, the Line is extinguished.
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