Hinton’s hypercubes


Illustration from The Fourth Dimension (1906) by Charles Howard Hinton.

A slight return to the worlds of Borges. I happened to be re-reading some of the stories in The Book of Sand (1975), one of the later collections which includes the story Borges dedicated to HP Lovecraft, There are more things. Borges’ writings are nothing if not filled with references to other works of literature, philosophies, religions, and ideas in general; following up the less-familiar references would preoccupy a reader to the detriment of the writing so there’s a tendency when reading a Borges piece to take for granted many of those nuggets of esoteric information. I’ve read There are more things many times—it’s a favourite in part for having the additional thrill of Borges going Lovecraftian—but only realised with this reading that I can now fully appreciate the following:

Years later, he was to lend me Hinton’s treatises which attempt to demonstrate the reality of four-dimensional space by means of complicated exercises with multicoloured cubes. I shall never forget the prisms and pyramids that we erected on the floor of his study.

Prior to reading this I did know that Hinton was Charles Howard Hinton (1853–1907), the British mathematician and dimensional theorist. Hinton’s name tends to turn up in discussions of the work of his mystical contemporaries, notably the Theosophists who were more taken with his theories than those in the scientific fields. (A conviction for bigamy didn’t help his reputation.) But those multicoloured cubes were a mystery until the publication of (fittingly) the fourth number of Strange Attractor Journal. Among the usual selection of fascinating articles the book contains a piece by Mark Blacklock about Hinton’s ideas including those mysterious cubes. Drawings of the cubes first appeared in A New Era of Thought (1888) where Hinton proposes using them as aids to a series of mental exercises with which the reader may visualise the higher dimensions of space. Hinton invented the word “tesseract” to describe the four-dimensional structure projected from the faces of his three-dimensional cubes.


Crucifixion (Corpus Hypercubus, 1954) by Salvador Dalí.

Hinton may not have impressed his mathematical colleagues as much as he hoped but his ideas have an understandable attraction, as the Borges story demonstrates. The story concerns the refashioning of a Buenos Aires house for an unusual resident; thirty years earlier Robert Heinlein wrote “—And He Built a Crooked House—” in which an architect builds a house in the form of a four-dimensional hypercube: only the lowest cube attached to the ground is visible from the exterior. I read that story when I was a teenager, and was already acquainted with tesseracts thanks to school-friends who were maths whizzes; I was the arts whizz, and I think I was probably the first of us Dalí enthusiasts to discover the artist’s own take on the hypercube, Crucifixion (Corpus Hypercubus) from 1954. Borges and Heinlein in those stories were both writing their own forms of science fiction, and Dalí’s painting finds itself co-opted into another story with sf connections, The University of Death (1968) by JG Ballard, one of the chapters in The Atrocity Exhibition (1970):

He lit a gold-tipped cigarette, noticing that a photograph of Talbot had been cleverly montaged over a reproduction of Dalí’s ‘Hypercubic Christ’. Even the film festival had been devised as part of the scenario’s calculated psychodrama.

If we seem to have strayed a little then it’s worth noting that Borges was familiar with Ballard’s work: he included The Drowned Giant in the later editions of The Book of Fantasy, the anthology he edited with Adolfo Bioy Casares and Silvina Ocampo. The two writers also met on at least one occasion.


JLB and JGB, circa 1970. Photo by Sophie Baker.

Charles Hinton’s coloured cubes and tesseracts are described in detail in The Fourth Dimension (1906), a reworking of the ideas from A New Era of Thought, and also the source of the colour illustration that everyone reproduces. Mark Blacklock has his own multi-dimensional website where you can read about his construction of a set of three-dimensional Hinton cubes. As for the mental exercises, Blacklock’s piece in Strange Attractor contains an anecdotal warning that the auto-hypnotic system required to fully visualise Hinton’s dimensions can result in a degree of obsession dangerous to the balance of mind. Proceed with caution.

Previously on { feuilleton }
Borges and the cats
Invasion, a film by Hugo Santiago
Spiderweb, a film by Paul Miller
Books Borges never wrote
Borges and I
Borges documentary
Borges in Performance

Invasion, a film by Hugo Santiago


From a film adapted from Borges to a film co-written by the man himself. Invasion (1969) is not to be confused with the British science fiction film of the same name made three years earlier, this is an Argentinian production and a much stranger piece of work. Hugo Santiago is an Argentinian director who moved to France where he worked for a while as assistant to Robert Bresson before establishing himself with this debut feature. Borges’ co-writer was his friend and regular collaborator Adolfo Bioy Casares.

I’ve not watched this yet, the two copies on YouTube are in Spanish with no subtitles but this is a blog with an international readership so there’s no need to exclude it on that account. Even without subtitles the film is visually intriguing, possessed of a moody and enigmatic style which online reviewers compare favourably to Antonioni and some of the Nouvelle Vague directors.


In 1957, a small group of middle-aged men fight a clandestine battle against forces quietly invading and taking control of their city, Aquilea. Enigmatic in its story-telling, Hugo Santiago’s once-lost film obscures the motivations of either side, leaving only a series of moves and counter-moves that evokes past dictatorial oppression and those still to come.

This post at Make Mine Criterion—a site dedicated to potential Criterion DVD releases—goes into considerable detail examining the film they describe as “part fantasy, part science fiction, part political thriller”. Given this and the rest of the film’s evident qualities it’s surprising it isn’t better known. I only have one Borges biography, The Man in the Mirror of the Book (1996) by James Woodall, but there’s no mention in there of Santiago’s film or of The Others (1974), a later Santiago feature that was also scripted by Borges and Bioy Casares. For those wishing to pursue the film further there are subtitles available if you can be bothered searching for them then trying to synch them up with a YT rip. It might be better to look for a Spanish DVD and hope it has a good choice of subtitles.


Previously on { feuilleton }
Spiderweb, a film by Paul Miller
The Library of Babel by Érik Desmazières
Books Borges never wrote
Borges and I
Borges documentary
Borges in Performance

Marienbad hauntings


Last Year in Marienbad (1961). Via.

In our age of cultural plenitude it can be salutary to remember the time when many things were easy to discover but often impossible to experience; albums, books, and especially non-American films could all too frequently exist as rumours, referenced but always out of reach. Two films in particular dogged me for years in this remote manner: The Saragossa Manuscript (1965) by Wojciech Has, and Last Year at Marienbad (1961), the film that Alain Resnais made from a very novelistic screenplay by Alain Robbe-Grillet. Philip Strick alerted me to this pair of films with tantalising descriptions in a time-travel chapter of his book-length study, Science Fiction Movies (1976). Marienbad isn’t a time-travel film as such (a later Resnais film, Je t’aime, je t’aime [1968] does deal with the subject, however, and even features an actual time machine), but it is sufficiently open-ended to allow a science-fictional rationale into its enigmatic spaces. Strick’s book covered all the familiar SF territory as well as looking beyond the clichés of Hollywood and the SF genre, hence the inclusion not only of Marienbad and Saragossa, but also Je t’aime, je t’aime, Chris Marker’s La Jetée (1962), Bergman’s The Hour of the Wolf (1968), Pasolini’s Teorema (1968), and Buñuel’s Discreet Charm of the Bourgeoisie (1972). Most of these films, which were seldom shown on TV, I had to wait years to see.


Marienbad page from Strick’s Science Fiction Movies.

I was reading Strick’s book in 1979, and since I was bored with generic clichés, and also reading a lot of reprinted stories from New Worlds magazine, I became a little obsessed with these inaccessible films, Marienbad especially. It’s difficult to say what was so fascinating about a few words of description, and a single photograph, but the picture seemed an unlikely inclusion amid so many pages filled with robots and spaceships. It promised a film that approached the themes of science fiction at the same oblique angle as many of the stories in New Worlds. A couple of years later I found a copy of the Robbe-Grillet screenplay whose pages of dogged description read like the kind of forbidding and formal exercise that Brian Aldiss had attempted in Report on Probability A (1967), a novel that first appeared in New Worlds. Among other similarities, both works share a dismissive attitude to character, presenting a trio of ciphers indicated by no more than their gender, and some initial letters. This confluence of influences, Marienbad included, fed into the chunks of New Worlds-derived prose I was writing at the time, trying to fix inchoate impressions on the page. I always failed each time I returned to that photo from Marienbad, the real charge—as I didn’t see at the time—being a result of the gap between the promise of the image and the inaccessible film itself. Finally seeing Marienbad in the late 1980s was a curious thing, like meeting somebody face-to-face after years of remote correspondence; the same readjustments needed to be made to accept that this was the reality of the work of art, not Robbe-Grillet’s embryonic version, or my own baroque imaginings.


Screenplay book, 1962. Cover design by Roy Kuhlman.

If the above seems to strain for association by hauling a celebrated work of the Nouvelle Vague into a disreputable area then this essay by Thomas Beltzer is worth a read. Beltzer’s “Intertextual Meditation” compares Marienbad to The Invention of Morel (1940), a science-fiction novel by Adolfo Bioy Casares which Jorge Luis Borges described as “perfect” (and which I really ought to read). If I’ve not written much about Marienbad itself that’s because it really needs to be experienced rather than described or explained. It’s a film that’s easier to admire than actually enjoy—I need to be in the right mood to accept its formalities—and given the choice I’d often sooner watch Providence (1977). But where Providence and other Resnais films have inevitably dated, Last Year in Marienbad remains out of time, a 20th-century dream held captive in 18th-century architecture where the airless rococo chambers might easily share a labyrinth with the hotel waiting-room at the end of 2001: A Space Odyssey.

• Alain Resnais obituaries: The Guardian | The Telegraph
Last Year in Marienbad at film|captures
Marienbad (2012) by Julia Holter

Vampyroteuthis Infernalis by Vilém Flusser


Cover design by Michel Vrana.

This, then, is the book that arrived a fortnight ago when I just happened to be in the midst of a week of tentacle posts. Vampyroteuthis Infernalis: A Treatise, with a Report by the Institut Scientifique de Recherche Paranaturaliste was originally published in Germany in 1987. This new edition is the first translation into English (by Valentine A. Pakis) published by the University of Minnesota Press in their Posthumanities series. It’s 100 pages long with a supplement of squid illustrations by Louis Bec. It is, to say the least, an odd book:

Part scientific treatise, part spoof, part philosophical discourse, part fable, Vampyroteuthis Infernalis gives its author ample room to ruminate on human—and nonhuman—life. Considering the human condition along with the vampire squid/octopus condition seems appropriate because “we are both products of an absurd coincidence…we are poorly programmed beings full of defects,” Flusser writes. Among other things, “we are both banished from much of life’s domain: it into the abyss, we onto the surfaces of the continents. We have both lost our original home, the beach, and we both live in constrained conditions.”

I’m not familiar with Flusser’s other work since I read few academic texts but it seems safe to assume that Vampyroteuthis Infernalis is an exception among the author’s volumes of media and communication theory. The tone is light but not overly comic unless you regard as inherently amusing Flusser’s analysis of an obscure cephalopod—the Vampyroteuthis Infernalis (the name translates as “vampire squid from hell”)—as a useful tool for studying the human condition. The study so far as it goes is along the lines of some of the essays by Jorge Luis Borges rather than any lengthy disquisition, looking at the squid’s existence from a number of angles in order to draw comparisons with human life. You wouldn’t think it easy to talk about “squid culture” or “squid politics” but Flusser manages:

…we are able to imagine cultural structures (“Utopias”) in which even our biological constraints are done away with. The vampyroteuthis cannot fathom Utopias, for the structure of its society is not a cultural product (it is not a “factum”) but rather a biological given (a “datum”). When it engages in politics, it does so against its own “nature”—it commits a violent act against itself. In the end, however, is not all human political activity contra nature?

And so on. In Borges terms (for me he’s the obvious touchstone) the book is reminiscent of the Chronicles of Bustos Domecq (1967), a series of deadpan essays about absurd cultural developments credited to one “H. Bustos Domecq” but written by Borges and Adolfo Bioy Casares. Flusser lived in Brazil for a number of years so Borges may have been an inspiration. Like Borges, Flusser is learned enough to write convincingly about his subject before he starts evading the reader’s grasp. The opening of Vampyroteuthis Infernalis is a creditable and informative run through the Octopoda taxonomy; later we have references and terminology from Heidegger, and Wilhelm Reich makes a surprising appearance. Many of the parallels are ingenious, such as when Flusser compares our electronic media—the glowing screens of televisions and computer monitors—to the glowing chromatophors on the skin of the squid which the animal uses to communicate in the lightless depths of the sea. Flusser ends on another Borgesian note, describing his “fable” as offering “an image of the self reflected between two facing mirrors”. Perhaps that’s the best way to regard this book: a continual play of reflections all of which would vanish if one of the mirrors were removed.

Those who wish to lose themselves in the reflections can order the book in hardback or paperback direct from the University of Minnesota Press. Elsewhere there’s a fair amount of Vampyroteuthis Infernalis footage on YouTube which reveals the animal in question to be as wonderfully strange as its name would imply.

Previously on { feuilleton }
Le Poulpe Colossal
Fascinating tentacula

Borges and I


Another piece of revenant television to tick off the “When will I see this?” list. I mentioned David Wheatley’s film Borges and I back in January in a post about the director’s dramatisation of the life and work of René Magritte. Wheatley’s student film secured for him a job as a BBC director at a time when the Arena arts series was one of the best things being produced by the corporation. Borges and I was filmed in 1982 and broadcast a year later, an event I managed to miss to my considerable regret. Once again Ubuweb has turned up the goods with a copy from an American video tape. It’s not ideal—all the Spanish sequences would have been subtitled in the original broadcast—but I’m not going to complain. This 80-minute film is not only the best Anglophone documentary I’ve seen on Borges, it was produced in collaboration with the author who for much of the running time discusses his life and work in English. The tape copy also frustratingly lacks credits but the unseen American interviewer and narrator would appear to be translator and collaborator Norman Thomas di Giovanni, a writer who later found himself and his work marginalised by the Borges estate. Between the interviews and readings there are dramatised sequences from The Meeting, Funes, the Memorious, The South, The Circular Ruins, Death and the Compass, and The Sartorial Revolution, one of the collaborations with Adolfo Bioy Casares. Plus, of course, the expected complement of mirrors, tigers and a duel with knives. The budget must have been generous: scenes were shot in Argentina and Uruguay, and we also see Borges at his favourite lodging in Paris: L’Hôtel in the Rue des Beaux-Arts, a building which now bears plaques celebrating the visits of Borges and another famous literary resident, Oscar Wilde.

Previously on { feuilleton }
René Magritte by David Wheatley
L’Hôtel, Paris
Borges documentary
Borges in Performance