Archive for February, 2009

Some initial thoughts on Flatland. So, to begin at the beginning with the title page…

Many dimensions

Many dimensions

Strange that this has occasioned so little critical comment. Iain Stewart’s excellent annotated edition of the text locates the Shakepearean quotes (Hamlet, Act I Scene v, the appearance of the ghost, and Titus Andronicus, Act III Scene i), both of which are fairly obviously puns and perhaps only tangentially connected to their context. I’m intrigued by the illustration – is it a map? – of a nebulous mass, perhaps fog, perhaps clouds.

It might well be a map. Flatland, ‘a Romance of Many Dimensions’, was published in October 1884. As such it arrived not terribly long into the ‘romantic revival’ of the 1880s, inaugurated, according to most accounts, the previous year, with Robert Louis Stevenson’s Treasure Island. Before that, in the launch issue of Longman’s Magazine in November 1882, Stevenson had given a theoretical outline of his fictional practice with ‘A Gossip on Romance’, advocating a robust, masculine, adventuring, fiction delivering a ‘kaleidoscopic dance of images’ and recalling books read in the ‘bright troubled period of boyhood’. Stevenson’s advocacy of romance has subsequently been read in opposition to Henry James’s championing of the interiorized, feminine and despicably foreign (!) realist novel.

This brief sketch is sufficient for now to give an idea of one aspect of the context into which Flatland arrived: while the descriptive term romance had been used in the title of many earlier nineteenth century novels, and even proto-SF novels – Edward Maitland’s An Historical Romance of the Future (1873) being an (the only?) example of the latter – when Abbott subtitled his book a ‘romance’, he connected it to a very current trend in fiction publishing. There was good reason for so doing: Treasure Island had been a bestseller. The inclusion on the title page of Flatland of a map would have underlined the connection to Treasure Island in particular.

Treasure Island

Treasure Island

So is it fog, or is it clouds? The text contains fog – common, apparently, in the temperate regions of Flatland – but the closing illustration repeats the nebulous illustration with more Shakespeare, this time from Prospero’s speech in The Tempest, Act IV, Scene i:

You do look, my son, in a moved sort,

As if you were dismay’d: be cheerful, sir.

Our revels now are ended. These our actors,

As I foretold you, were all spirits, and

Are melted into air, into thin air:

And, like the baseless fabric of this vision,

The cloud-capp’d towers, the gorgeous palaces,

The solemn temples, the great globe itself,

Yea, all which it inherit, shall dissolve,

And, like this insubstantial pageant faded,

Leave not a rack behind. We are such stuff

As dreams are made on; and our little life

Is rounded with a sleep.

The baseless fabric of vision

The baseless fabric of vision

Thin air, then, and clouds. And I’d suggest that the classical scholar Abbott may also have had in mind an earlier passage of satire, from Aristophanes’ The Birds. In the following exchange the tyrannical Pithetaerus passes judgement on the geometer Meton in his attempt to enter Cloudcuckooland:

(Enter METON, With surveying instruments.)

METON: I have come to you…

PITHETAERUS (interrupting): Yet another pest! What have you come to do? What’s your plan? What’s the purpose of your journey? Why these splendid buskins?

METON: I want to survey the plains of the air for you and to parcel them into lots.

PITHETAERUS: In the name of the gods, who are you?

METON: Who am I? Meton, known throughout Greece and at Colonus.

PITHETAERUS: What are these things?

METON: Tools for measuring the air. In truth, the spaces in the air have precisely the form of a furnace. With this bent ruler I draw a line from top to bottom; from one of its points I describe a circle with the compass. Do you understand?

PITHETAERUS: Not in the least.

METON: With the straight ruler I set to work to inscribe a square within this circle; in its centre will be the market-place, into which all the straight streets will lead, converging to this centre like a star, which, although only orbicular, sends forth its rays in a straight line from all sides.

PITHETAERUS: A regular Thales!

Tools for measuring the air, indeed! This prompts a number of lines of thought. A bone of contention in discussions over higher space concerned its imaginary as opposed to its empirical nature. As an algebraic and then a geometric theory – in other words, as a mathematical construct – higher space remained comfortably ideal. With interventions from physics and Zollner’s catastrophic/catalytic misreading of four-dimensional space, the waters became muddied – or perhaps better to write that the airs became fogged. Was physical space actually four-dimensional?

Higher space existed in the interstices between the ideal and the empirical, as did the emergent sciences of mind, in which perception of space was a primary site of conflict. Abbott’s cloud, then, is thought, imagination, the higher space of mind in which the higher space of geometry existed, an analogue noted by William Spottiswoode in his 1878 address to the BAAS: ‘Or once more, when space already filled with material substances is mentally peopled with immaterial beings, may not the imagination be regarded as having added a new element to the capacity of space, a fourth dimension of which there is no evidence in experimental fact?’

As for Pithetaerus’s remark on Thales, this suggest routing discussions through Michel Serres, whose two essays on the origins of geometry consider the case of Thales and the discovery of mathematical analogy. But I believe that would justify another post entirely and this one already needs more work!

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[text of a paper given at the WiP conference at Birkbeck on 7 February 2009 – slides to be inserted as images at a later date]


In early 1878 Johnann Karl Friedrich Zöllner, professor of astrophysics at the University of Leipzig, published in The Quarterly Journal of Science, edited by the British chemist and spiritualist William Crookes, an account of an experiment he had undertaken with the spirit medium Henry Slade.

‘On Space of Four Dimensions’ argued that the human conception of space as three-dimensional was based upon experience and that as soon as experience appeared to contradict this conception we would be forced to revise our theories of space. Pausing briefly to appeal to the work of the mathematician Bernhard Riemann, whose paper ‘Concerning the Hypotheses Upon Which Geometry is Founded’ had become a posthumous cornerstone of the recently emerged non-Euclidean geometry, Zöllner went on to describe manipulations of a cord of string.

Confined to a plane, or two dimensions, a knot could be conceived as a simple twisting of the cord. Without access to the third dimension, this twisting or crossing could not be undone except by cutting the cord or passing it back through itself; with access to a third dimension, a simple rotation of the cord would suffice to undo the loop. This loop was the two dimensional equivalent of a knot and so extending space to three dimensions, and placing a three-dimensional knot in the cord, Zöllner argued that access to a fourth dimension of space would, by analogy, allow the knot to be undone without cutting the cord or passing it back through itself. Here, Zöllner was précising an 1874 paper by the young projective geometer Felix Klein, to whom I shall return later.

Zöllner continued his argument by borrowing highly selectively from the work of the mathematician Carl Friedrich Gauss and his own philosophical touchstone, Immanuel Kant. He quoted from Kant to demonstrate that because a space of four dimensions could be conceived, it would ‘probably’ exist, and likewise immaterial beings of the spiritual world.

His groundwork established, Zöllner proceeded to the core of his argument:

If a single cord has its ends tied together and sealed, an intelligent being, having the power voluntarily to produce on this cord four-dimensional bendings and movements, must be able, without loosening the seal, to tie one or more knots in this endless cord. Now, this experiment has been successfully made within the space of a few minutes in Leipzig, on the 17th December 1877, at 11 o’clock AM, in the presence of Mr. Henry Slade, the American.

Zöllner’s friend, and fellow witness, Gustave Theodor Fechner, was also convinced of the experiment’s success, describing it in his diaries as ‘above suspicion’. The spiritualist community in Germany and England was elated, and the triumph was trumpeted throughout its journals, and reported uncritically in the Daily Telegraph. Within a week British spiritualists, including T.L Nichols and the medium William Eglinton, claimed to have reproduced Zöllner’s ‘splendid manifestation’. At the heart of the longed-for scientific proof of the existence of both spirits and the fourth dimension, lay the not-so-humble knot.

The repercussions of Zöllner’s experiment echoed for some time. It became a highly contested episode in the history of Psychic Research and is still cited in spiritualist literature as evidence of scientific plausibility of the spiritual hypothesis. Scientific border policing and counter-currents of legitimisation and deligitimisation typical of hybrid scientisms surrounded accounts of Zöllner’s experiment. Evidence suggestive of its illegitimacy lay buried within Zöllner’s own accounts of his work, which stretch to over 4000 pages in the original German. I am grateful, therefore, to Eleanor Sidgwick, wife of Henry, one of the founders of the Society for Psychical Research, for undertaking the work of sifting through these papers in 1886. Mrs Sidgwick reported to the Society:

That Professor Zöllner did not always perceive and avoid important sources of possible error may, I think, be inferred from his writings. For instance, in describing the séance on December 17th, 1877, wherein he obtained four knots in a string of which the ends were tied and sealed together, he omits to mention that the experiment had been tried and failed before. We learn that this was so, accidentally, as it were, from his mentioning it in another place and in another connection, where he tells us that it was a long time before the spirits understood what kind of knot was required of them, and that before they did so he obtained knots, but not such as he wanted – knots, I infer, which could be made by ordinary beings without undoing the string.

Putting to one side for now the scientific legitimacy of Zöllner’s work, I want to focus on his interest in the knot. I want to untie its threads and trace them back, asking how the knot became proof of something so fantastic as fourth dimensional intelligence. In so doing, one theoretical touchstone has presented itself in an irresistible fashion. In Pandora’s Hope, the French sociologist Bruno Latour explores scientific concepts through metaphors of knots in networks. Latour argues against the idea of ‘pure’ science, seeing bound together in the knots of scientific concepts the heterogeneous processes on which they depend, processes he lists as ‘instruments, colleagues, allies, public’ and the knot itself. Indeed an equation is the perfect knot, so many different elements so tightly bound into one expression.

Latour’s way of thinking is particularly useful in approaching what might immediately be recognised as a hybrid, scientistic concept: he has used a historicist approach to the history of science to re-examine lost or abandoned theories. In this instance, as will become apparent, we’re examining disqualified theories on both sides of the argument. Furthermore, Latour’s insistence on the permeability of the object-subject relationship, on the significance of the non-human in science, is clearly appropriate to the technological object of the knot itself, the dense structure doing so much work at the heart of this account.

If you’ll permit me to really stretch Latour’s metaphor, I propose to pick out three strands from this particular knot: magic, matter and higher space.

The first strand of the splice follows the history of spiritualist performance and its shadow play in stage conjuring. For both mediums and conjurors, the knot was a tool and the tying of knots an essential skill. The Spiritualist Davenport Brothers who exhibited to their audience purported spiritual manifestations occurring inside a cabinet, such as the playing of instruments, were bound by ropes before being shut into their box. Their bindings were proof to their audience that these manifestations could not be performed by the brothers themselves but must be created by the spirits they summoned.

When the Davenports performed in Cheltenham in 1864, a young man called John Nevil Maskelyne interrupted their performance to announce that he knew how it was done. Two months later Maskelyne and his partner Cooke gave a performance with their own spirit cabinet. Special attention was given to their bindings, knotted by a sailor, with additional ropes tied and sealed with wax by other members of the audience. Maskelyne and Cooke nevertheless reproduced the manifestations the Davenports had earlier claimed to be due to spirits, launching a stage conjuring career that continued well into the 20th century.

Maskelyne was not alone in mastering the knot tricks of spiritualist performers. In France, Jean Robert-Houdin, who had already successfully reproduced various aspects of the performances of the medium Daniel Home, announced upon witnessing the Davenport’s act in 1865 that ‘the article wherein lay all the deception was the rope.’ Houdin explained that along with various tricks of misdirection and contortion, the type of rope used, and specifically, the type of knot tied, allowed the brothers to free themselves from their bindings. Papers published in English translation posthumously in 1880 give illustration of the type of slip-knot, or cat’s paw, Houdin believed to have been used. The first book length guide to magical knots written by Harry Houdini in 1923 was seeded in this sketch.

In the context of spiritualism, then, the knot was part of the armoury of the conjuror, both professed and occluded. In this context it was an ambiguous object, capable of manipulation and being manipulated, the focal point of both confidence and doubt: highly unstable, in other words. Illusory.

Before I move on to matters more scientific, it is important to note that Maskelyne was a witness for the prosecution at the trial of Henry Slade in London in 1875, a highly publicised and sporadically farcical event at which the biologist Edwin Ray Lankester and his friend Herbert Donkin attempted to prove that they had caught the slate-writing medium in the act of concealing a pre-prepared slate during a seance. Lankester’s aim had been to discredit spiritualism, which had been leant considerable scientific legitimacy by the publicised interest of several leading scientists, not least William Crookes. So Maskelyne and Slade had encountered each other across a courtroom before Zöllner and Slade met.

The second strand of the splice. In a roughly synchronous timeframe, the knot was also coming under close scrutiny from the physicists William Thomson and Peter Guthrie Tait. In his 1867 paper given before the Royal Society of Edinburgh, ‘On Vortex Atoms’, Thomson proposed that atoms were vortex rings, that matter was a mode of motion in space. He demonstrated diagrams and models of various knotted or knitted vortex atoms, and while I don’t have an image of the models that accompanied that talk, this sketch from his notebook gives some idea of what he envisaged in terms of knotted or knittedness. Thomson wrote:

two ring atoms linked together or one knotted in any manner with its ends meeting, constitute a system which, however it may be altered in shape, can never deviate from its own peculiarity of multiple continuity, it being impossible for the matter in any line of vortex motion to go through the line of any other matter in such motion or any other part of its own line.

Thomson had been in part inspired by a demonstration of smoke rings by his colleague Peter Guthrie Tait, and Tait in turn took his lead from Thomson. If atoms were knots, Tait reasoned, then further research into knots was required. Tait’s first realization, now known to anyone who’s ever attempted to untangle fairy lights or headphone cords, was that some objects appearing to be knots, are in fact merely tangles, and can be unraveled by pulling. These he termed trivial knots, or unknots. Tait tabulated true knots according to the number of their crossings and presented his research to the Royal Society of Edinburgh in 1876, instigating the branch of mathematics now known as knot theory.

Mathematical knots, then, in distinction to magical knots, were closed. The putative cords in which they were tied had no ends. They were unambiguous; indeed, they distinguished between true knottedness and unknottedness. They were unbreakable in their own space. And they were – possibly – the very bedrock of matter.

The third strand of the splice. By the time Tait’s work was published later that year, he had received notice from Felix Christian Klein, whose name you might recall in connection with Zöllner. Klein, a young projective geometer working with geometries allowing for dimensions greater than three, had made, in Tait’s words, ‘the very singular discovery that in space of four dimensions there cannot be knots.’

In Klein’s own description he had demonstrated that: ‘the presence of a knot can be considered an essential (i.e., invariant under deformations) property of a closed curve only if one is restricted to move in three-dimensional space; in four-dimensional space a closed curve can be unknotted by deformation.’ In other words knots, or closed space curves, could be untied by deformation, or movements that did not involve cutting the knot or passing the cord through itself, in space of four dimensions.

In the mid-1870s Klein had also informed Zöllner of his research. According to Klein: ‘Zoellner took up my remark with an enthusiasm that was unintelligible to me. He thought he had a means of experimentally proving the “existence of the fourth dimension”’.

Zöllner, in fact, had previously engaged with both the fourth dimension and Thomson and Tait. In his 1872 book Uber die Natur der Cometen, Zöllner had launched a highly personal, unscientific attack against various British physicists including Thomson and Tait, and their German friend and translator Helmholtz. His vituperative comments occasioned a front page editorial in Nature describing the ‘numberless sources of amusement which the work affords’ and letters of apology from Helmholtz in which he imputed charges of insanity against Zöllner.

And so, back to the beginning of my presentation, six years later, when the same scientist published work dealing with spiritualism and knots, what was the response in the British scientific press? Understandably, it was too much for Tait, who seized the opportunity of an amateur’s dabbling in the field of knots to belatedly retaliate. Tait’s review of Zöllner’s work in Nature in 1878 is a hatchet job of the first order, best summed up by the extravagant typography of his response to the knot experiment:

He has held the two ends of a cord (sealed together) in his hands while trefoil knots, genuine, IRREDUCIBLE TREFOIL KNOTS, of which he gives us a picture, were developed upon it!

For Tait, who had glimpsed the vastness of the mathematics of knots, the simple trefoil knot, the very first knot on his table, was evidently not very impressive. And this was several years before he had the advantage of Mrs Sidgwick’s close readings of Zöllner’s experimental accounts. What Tait would have made of the trivial knots previously produced under Slade’s aegis, we can only guess.


So what does this tangled tale reveal?

My own research is focused on higher space, the fourth dimension in which Zöllner’s knots were supposedly created. I see Zöllner’s intervention as crucially transformative for higher space. Not only did his work put the fourth dimension on the front page of daily newspapers, for the first and possibly the final time, but it inflected all subsequent understanding of this already difficult concept. Unravelling the knots in his cord reveals some of the resources drawn together in this moment and subsequently internalised in the cultural fourth dimension. The conjuring context of the knot is particularly important, and often obscured by its shadow brother, spiritualism. The fourth dimension is magical, but a crucial element of this magic is the performed, dramatised, fictive magic of the stage conjuror, as opposed to the theorised, scientistic magic of the occult.

Zöllner’s knots are also demonstrative of the hybrid construction of higher space and higher spatial theories and objects, drawing together geometry, scientific rivalries, national politics, occult belief systems: this is typically Latourian. Latour’s critique of a hard and fast division between reality and language is informative when we consider the knot. Latour argues that: ‘We never detect the rupture between things and signs and we never face the imposition of arbitrary and discrete signs on shapeless and continuous matter.’

This resonates particularly acutely with regard to the seductive idea of the atom as a knot, an idea in which Thomson and Tait tied word and world so tightly.

Before closing, I believe my abstract promised visual demonstration of how the knots were produced in the cord. And I think if we look here [attempt at sub-Tommy Cooper magic trick]

There is no more appropriate summary to this set of stories than that provided by the most aloof and brilliant of Victorian physicists. James Clerk Maxwell had long chuckled from the sidelines at his friend Tait’s obsession with knots, channelling his mockery through a favourite form: humourous verse. The very last poem Clerk-Maxwell wrote before his death combined an informed overview of spatial theories enjoying currency at the time with a critique of Tait’s own pet theory of Continuity, an attempt to reach a compromise between religious faith and physical science, dependent upon the concept of the infinite as a stand-in for the Godhead, and outlined in two books co-written with Balfour Stewart, The Unseen Universe and its sequel, Paradoxical Philosophy.
Clerk Maxwell, no mean parodist, pastiched Prometheus Unbound thus:

My soul’s an amphicheiral knot
Upon a liquid vortex wrought
By Intellect in the Unseen residing,
While thou dost like a convict sit
With marlinspike untwisting it
Only to find my knottiness abiding,
Since all the tools for my untying
In four-dimensioned space are lying,
Where playful fancy intersperses
Whole avenues of universes,
Where Klein and Clifford fill the void
With one unbounded, finite homoloid,
Whereby the infinite is hopelessly destroyed.

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I’m now over a year into my research, and having covered what, for a humanities researcher, is hopefully the hardest stretch, the geometrical origins of the concept of higher-dimensioned space, it seems like an opportune time to start putting some of my own research online (although probably not the maths stuff until I’ve redrafted it many more times and had real mathematicians read it over). I’m beginning to re-engage with more comfortable territory – the perennially popular Flatland and the work of the more obscure, but no less cultish, Charles Howard Hinton, the remarkable figure whose writing first drew me into this field – so there may be interest from beyond my immediate family (and to be honest, that often flags, and who can rightly blame them for that?).

I suppose some initial comments on the working title would be useful. ‘The Fairyland of Geometry’ is a phrase owing to the American mathematician Simon Newcomb (see how easily the patterns of Victorian prose take hold?) Newcomb first used it in 1897 in an address to the American Mathematical Society, but evidently liked it so much he used it again for a popular article on higher space in Harper’s Magazine five years later. I like it, too, because it covers the necessary ground between pure maths and fantastic fiction, pausing for heavy breath in the mystical graveyard of the occult.

And I guess, since Newcomb uses the term hyperspace, I should explain why I use higher space. It’s partly a transatlantic thing, but also for reasons of consistency: higher space was Hinton’s preferred coinage; hyperspace was first used by G.B. Halsted in the American Journal of Mathematics in 1878 (although he was using the term pro-space not long before); Bertrand Russell’s An essay on the foundations of geometry (1897) preferred the term meta-space; and an occultist account of fourth dimensional clairvoyance from 1893 delighted in the title Throughth. Evidently higher space is prepositionally challenging: this is something I’ll certainly be returning to at greater length, so suffice to say I lean towards higher space not least for its relative clarity.

And finally the dates: not as arbitrary as they may at first seem. This is Hinton’s lifespan, and not only is it impossible to imagine a study like this without Hinton, it’s also a handy period container for most of what I want to do: to trace the transformations of the concepts of higher space from their origins in analytic geometry to their various manifestations in esoteric/occult texts, science fiction and popular culture, working up to and into the more thoroughly researched area of Modernism’s fourth dimensions and pulling short a good few years before Einstein muddies the water with relativity, damn his crazy eyes. And hair.

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