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Notes on the Fourth Dimension part1


Hyperspace, ghosts, and colourful cubes – Jon Crabb on the work of Charles Howard Hinton and the cultural history of higher dimensions.

The coloured cubes — known as “Tesseracts” — as depicted in the frontispiece to Hinton’s The Fourth Dimension (1904) – Source.

Men of broader intellect know that there is no sharp distinction betwixt the real and the unreal… – H.P. Lovecraft, The Tomb (1917)


La Belle Époque, a beautiful term for a Beautiful Age, as Light and Understanding replace Fear and Superstition, and Science and Art join hands in unholy matrimony and set out to discover the world anew. Trains become underground worms burrowing through the city, displacing medieval graves in the name of modernity; the Aéro-club de France send men into the heavens, amazing the public; Muybridge proves horses flytoo and wins a bet; Edison floods the world with light; biologists discover germs and defy Death; botanists grow tropical plants in Parisian glass-houses and affront Nature with hot-house orchids; the phonograph and the cinema fold Time and Space for the masses. And for some reason bicycles become rather popular. The world was getting smaller every day and the discoveries were getting bigger every week. How very diverting it all was…


In the land of Sona-Nyl there is neither time nor space, neither suffering nor death. – H.P. Lovecraft, The White Ship (1920)


During the period we now call the fin-de-siècle, worlds collided. Ideas were being killed off as much as being born. And in a sort of Hegelian logic of thesis/antithesis/synthesis, the most interesting ones arose as the offspring of wildly different parents. In particular, the last gasp of Victorian spiritualityinfused cutting-edge science with a certain sense of old-school mysticism. Theosophy was all the rage; Huysmans dragged Satan into modern Paris; and eccentric poets and scholars met in the British Museum Reading Room under the aegis of the Golden Dawn for a cup of tea and a spot of demonology. As a result of all this, certain commonly-accepted scientific terms we use today came out of quite weird and wonderful ideas being developed at the turn of the century. Such is the case with space, which fascinated mathematicians, philosophers, and artists with its unfathomable possibilities.

Outside of sheltered mathematical circles, the trend began rather innocuously in 1884, when Edwin A. Abbott published the satirical novella Flatland: A Romance of Many Dimensions under the pseudonym A. Square. In the fine tradition of English satire, he creates an alternative world as a sort of nonsense arena to lampoon the social structures of Victorian England. In this two-dimensional world, different classes are made up of different polygons, and the laws concerning sides and angles that maintain that hierarchy are pushed to absurd proportions. Initially, the work was only moderately popular, but it introduced thought experiments on how to visualise higher dimensions to the general public. It also paved the ground for a much more esoteric thinker who would have much more far-reaching effects with his own mystical brand of higher mathematics.

Cover of the first edition of Flatland (1884) – Source: City of London School Archive.

In April 1904, C. H. Hinton published The Fourth Dimension, a popular maths book based on concepts he had been developing since 1880 that sought to establish an additional spatial dimension to the three we know and love. This was not understood to be time as we’re so used to thinking of the fourth dimension nowadays; that idea came a bit later. Hinton was talking about an actual spatial dimension, a new geometry, physically existing, and even possible to see and experience; something that linked us all together and would result in a “New Era of Thought”. (Interestingly, that very same month in a hotel room in Cairo, Aleister Crowley talked to Egyptian Gods and proclaimed a “New Aeon” for mankind. For those of us who amuse ourselves by charting the subcultural backstreets of history, it seems as though a strange synchronicity briefly connected a mystic mathematician and a mathematical mystic — which is quite pleasing.)

Hinton begins his book by briefly relating the history of higher dimensions and non-Euclidean maths up to that point. Surprisingly, for a history of mathematicians, it’s actually quite entertaining. Here is one tale he tells of János Bolyai, a Hungarian mathematician who contributed important early work on non-Euclidean geometry before joining the army:

It is related of him that he was challenged by thirteen officers of his garrison, a thing not unlikely to happen considering how differently he thought from everyone else. He fought them all in succession – making it his only condition that he should be allowed to play on his violin for an interval between meeting each opponent. He disarmed or wounded all his antagonists. It can be easily imagined that a temperament such as his was not one congenial to his military superiors. He was retired in 1833.

Janos Bolyai: Appendix. Shelfmark: 545.091. Table of Figures – Source.

Mathematicians have definitely lost their flair. The notion of duelling with violinist mathematicians may seem absurd, but there was a growing unease about the apparently arbitrary nature of “reality” in light of new scientific discoveries. The discoverers appeared renegades. As the nineteenth century progressed, the world was robbed of more and more divine power and started looking worryingly like a ship adrift without its captain. Science at the frontiers threatened certain strongly-held assumptions about the universe. The puzzle of non-Euclidian geometry was even enough of a contemporary issue to appear in Dostoevsky’s Brothers Karamazov when Ivan discusses the ineffability of God:
But you must note this: if God exists and if He really did create the world, then, as we all know, He created it according to the geometry of Euclid and the human mind with the conception of only three dimensions in space. Yet there have been and still are geometricians and philosophers, and even some of the most distinguished, who doubt whether the whole universe, or to speak more widely, the whole of being, was only created in Euclid’s geometry; they even dare to dream that two parallel lines, which according to Euclid can never meet on earth, may meet somewhere in infinity… I have a Euclidian earthly mind, and how could I solve problems that are not of this world? — Dostoevsky, Brothers Karamzov (1880), Ch. 34 Part II.
Well Ivan, to quote Hinton, “it is indeed strange, the manner in which we must begin to think about the higher world”. His solution was a series of coloured cubes that when mentally assembled in sequence could be used to visualise a hypercube in the fourth dimension of hyperspace. He provides illustrations and gives instructions on how to make these cubes and uses the word “tesseract” to describe the four-dimensional object.
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