Flatland

Selected extracts from a review of “Flatland: The Movie”, by Ian Stewart in the Notices of the American Mathematical Society.
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Despite its apparent social conformity, Victorian England also gave the world some of its greatest innovations. Darwin’s Origin of Species, first published in 1859, made a persuasive case against supernatural creation by pointing out the mechanism of natural selection. Charles Lyell’s 1830 Principles of Geology, following pioneering work of James Hutton, made it clear that our planet is far older than had previously been assumed. Advances began to be made in the humane treatment of prisoners and the poor. Women began to free themselves from centuries of subservience. In fact, many basic rights and ideas that intelligent, informed people now take for granted emerged from the Victorians’ struggle to reconcile their rigid social structure with a changing understanding of the natural world.

Mathematics, too, changed. When Victoria came to the throne in 1837, mathematics was almost entirely concrete, its concepts mostly modelled on the natural world. Analysis had embraced complex numbers and progressed as far as elliptic functions, but its logical foundations remained obscure, and it was really just calculus. Geometry remained rooted in human perceptions of physical space, although it was now projective as well as Euclidean. Algebra was largely about the solution of polynomial equations with numerical coefficients. There were revolutions waiting in the wings—Bernhard Bolzano was pioneering logical rigour in calculus; Janós Bolyai and Nicolai Ivanovich Lobachevsky had invented non-Euclidean geometries, though these were still neither welcomed nor understood; Evariste Galois had begun to free algebra from the shackles of numerical interpretation. But these revolutions were at best incipient.

… By 1901, when Victoria died, mathematics had become almost unrecognisable. Abstractions were replacing the concrete. Non-Euclidean geometry was part of the mainstream, joined by differential geometry and even the beginnings of topology. Karl Weierstrass and his successors had reformulated analysis, and infinite processes were no longer a puzzle. Georg Cantor had dazzled the world, and baffled it, with set theory and transfinite numbers. Algebra now included finite fields, rings of algebraic integers, and groups. William Rowan Hamilton had struggled his way to the invention of quaternions, Sophus Lie had invented his eponymous groups, Henri Poincaré had encountered the obstacle that we now call his conjecture, and David Hilbert had produced his hit-list of twenty-three unsolved problems. Almost everything that was taken for granted at the start of Victoria’s reign was up for grabs when it ended. Allegedly “self-evident” propositions proved to be false, allegedly universal imperatives were revealed as parochial conventions, and alleged impossibilities were not only possible, but unavoidable. The late Victorian era, in short, was a period of remarkable progress and free thinking. Science and the Church came to a gentleman’s agreement not to tread on each others’ toes, and many a country clergyman became the world expert on six types of beetle or the reproductive habits of slugs. Scientific advances were discussed along with the price of sugar and the increasingly parlous state of the British Empire at garden parties and polite social gatherings.

…The most serious heresy in Flatland’s totalitarian theocracy is belief in the Third Dimension. Every new millennium a Sphere from this nonexistent realm visits Flatland and causes trouble, and the priests suppress public knowledge of such incidents to preserve their own power. Poor Mr. Square (Abbott never tells us what “A” stands for, because his protagonist is merely a square) gets caught up in these events and is set up for the book’s main mathematical theme, a dimensional analogy. Our own difficulties in contemplating the fourth dimension are analogous to those of A. Square contemplating the third. Abbott uses this analogy to explain fourdimensional space to his readers, while appearing to be explaining three-dimensional space to A. Square. It seems likely that he got the basic ideas from Charles Howard Hinton, an incorrigible rogue and accomplished geometer who wrote widely about four dimensions. In 1880 Hinton published an article “What is the Fourth Dimension” in the Dublin University Magazine, and it was reprinted in the Cheltenham Ladies College Gazette a year later. In 1884 it was reprinted yet again as a pamphlet, with the subtitle “Ghosts explained”.

…The Sphere convinces A. Square that the third dimension exists, but he succeeds only when he takes the drastic step of pushing A. Square out of his plane into the wider realm of Space. Even the most ardent disbeliever in the fourth dimension might well change their minds in similar circumstances. But the ending is dark and tragic. A. Square, on his return to Flatland, proclaims the Truth of the Third Dimension and ends up in jail, having failed to convince anybody else, and at times doubts his own sanity: “It is part of the martyrdom which I endure for the cause of the Truth that there are seasons of mental weakness, when Cubes and spheres flit away into the background of scarce-possible existences; when the Land of Three dimensions seems almost as visionary as the Land of One or none; nay, when even this hard wall that bars me from my freedom, these very tablets on which I am writing, and all the substantial realities of Flatland itself, appear no better than the offspring of a diseased imagination, or the baseless fabric of a dream.”
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