Views of a Tesseract

…the breadth, and length, and depth, and height…

– Ephesians 3:18

And the city lieth foursquare, and the length is as large as the breadth: and he measured the city with the reed, twelve thousand furlongs. The length and the breadth and the height of it are equal. And he measured the wall thereof, an hundred and forty and four cubits, according to the measure of a man, that is, of the angel. And the building of the wall of it was of jasper: and the city was pure gold, like unto clear glass.

– Revelation 21:16-18

Un homme qui y consacrerait son existence arriverait peut-être à se peindre la quatrième dimension. [A man who devoted his life to it could perhaps succeed in picturing to himself the fourth dimension.]

– Henri Poincaré

This spring I have scaled the awful, sanity-threatening Unknown Kadaths of the fourth dimension in a desperate, god-provoking quest to visualize the six regular polytopes.

What is a polytope, you ask? The word polytope is the general term in the sequence whose first terms are the line segment (dimension one), the polygon (dimension two), and the polyhedron (dimension three). A regular polytope is a polytope which is "completely symmetric."

Theatetus, a contemporary of Plato, proved that there are exactly five regular polyhedra: the tetrahedron, the octahedron, the icosahedron, the cube, and the dodecahedron. They are called the Platonic solids because Plato identified each of the first four with a material element (fire, air, water, earth), and the fifth with "the delineation of the universe" [Timaeus]. Their construction is the crowning achievement of Euclid's Elements, written in about 300 BC. But the world had to wait more than two thousand years for the "discovery" of their analogues in the fourth dimension.

Fourth-dimensional geometry, thought it might seem mysterious to the uninitiated, is defined axiomatically, just like Euclid's three-dimensional geometry, and has an intuitive basis. It was first described by Ludwig Schläfli, a Swiss mathematician, in the 1850s, but his work remained relatively inaccessible and unknown. Then, between 1880 and 1900, the geometry of higher dimensions was rediscovered in nine different publications written independently of each other. The time, it seems, was ripe. It was the dawn of a new era.

Not that era. [source]

This phenomenon of numerous researchers all suddenly reaching the same conclusion at the same time, though surprising when it happens, isn't all that uncommon in the history of math, science, and technology. What's striking is the way four-dimensional geometry fired the popular imagination, which seems in some cases to have outstripped academia.

Last year I blogged about Flatland: A Romance of Many Dimensions, a strange geometrical fantasy written by the English schoolmaster Edwin A. Abbott (1838-1926) and published in 1884. In it, Abbott gives what must be the first popular description of the tesseract, or four-dimensional hypercube, by way of analogy.

In One Dimension, did not a moving Point produce a Line with TWO terminal points?

In Two Dimensions, did not a moving Line produce a Square with FOUR terminal points?

In Three Dimensions, did not a moving Square produce – did not this eye of mine behold it – that blessed Being, a Cube, with EIGHT terminal points?

And in Four Dimensions shall not a moving Cube – alas, for Analogy, and alas for the Progress of Truth, if it be not so – shall not, I say, the motion of a divine Cube result in a still more divine Organization with SIXTEEN terminal points?

Behold the infallible confirmation of the Series, 2, 4, 8, 16: is not this a Geometrical Progression? Is not this – if I might quote my Lord's own words – "strictly according to Analogy"?

Again, was I not taught by my Lord that as in a Line there are TWO bounding Points, and in a Square there are FOUR bounding Lines, so in a Cube there must be SIX bounding Squares? Behold once more the confirming Series, 2, 4, 6: is not this an Arithmetical Progression? And consequently does it not of necessity follow that the more divine offspring of the divine Cube in the Land of Four Dimensions, must have 8 bounding Cubes: and is not this also, as my Lord has taught me to believe, "strictly according to Analogy"?

How much exactly did Abbott know of contemporary research? One imagines he must have encountered something, but I can't seem to find anything definite. A matter for further research, I suppose.

Inspired by Abbott, a high school teacher and amateur mathematician by the name of Charles Howard Hinton (1853-1907) wrote a number of "scientific romances" exploring higher dimensions. It was Hinton who coined the term tesseract, and his book A New Era of Thought, published in 1888, provides a detailed account of the hypercube's structure. It also offers a mystical interpretation of the fourth dimension, following to some extent in Abbott's footsteps, but with considerably greater gravity and self-importance.

We have been subject to a limitation of the most absurd character. Let us open our eyes and see the facts.

Now, it requires some training to open the eyes. For many years I worked at the subject without the slightest success. All was mere formalism. But by adopting the simplest means, and by a more thorough knowledge of space, the whole flashed clear.

Space shapes can only be symbolical of four-dimensional shapes; and if we do not deal with space shapes directly, but only treat them by symbols on the plane – as in analytical geometry – we are trying to get a perception of higher space through symbols of symbols, and the task is hopeless. But a direct study of space leads us to the knowledge of higher space. And with the knowledge of higher space there come into our ken boundless possibilities. All those things may be real, whereof saints and philosophers have dreamed.

Hinton was read by Jorge Luis Borges, and his book is mentioned "Tlön, Uqbar, Orbis Tertius."

Through his father, described by some as a religious crank, Hinton came to know the family of the late George Boole, the father of algebraic logic, whose untimely death had left his wife, Mary Everest* Boole, with their five daughters to raise. Mrs. Boole's interests ranged from mathematics to mysticism to politics; she wrote a number of pedagogical works, organized controversial discussion groups, and hobnobbed with the denizens of the fringes. Among these were the polygamy advocate James Hinton and his son Howard.

Howard married the eldest daughter, Mary Ellen Boole, in 1880, and they had four children together. A few later, he married a second woman under an assumed name, had two children with her, was convicted of bigamy, spent a few days in jail, lost his job, and moved to the United States with his (first) wife to become a university professor. He died unexpectedly in 1907, and Mary Ellen committed suicide the next year.

H. S. M. Coxeter's Regular Polytopes, published (in its second edition) in 1963, remains the main authority on its subject. I've entertained myself by constructing the various solids he describes in it.

More importantly for us, each chapter concludes with historical notes. There Coxeter discusses Alicia Boole Stott (1860-1940), another of George Boole's daughters, with whom he was personally acquainted in her later years. Curiously, though he mentions both Hinton and his book (in deprecatory terms), he says nothing about the family connection or about the fact that Stott assisted in finishing and publishing A New Era in Human Thought when Hinton left the country.

When Alice was about thirteen the five girls were reunited with their mother (whose books reveal her as one of the pioneers of modern pedagogy) in a poor, dark, dirty, and uncomfortable lodging in London. There was no possibility of education in the ordinary sense, but Mrs. Boole's friendship with James Hinton attracted to the house a continual stream of social crusaders and cranks. It was during those years that Hinton's son Howard brought a lot of small wooden cubes, and set the youngest three girls the task of memorizing the arbitrary list of Latin words by which he named them, and piling them into shapes. To Ethel, and possibly Lucy too, this was a meaningless bore; but it inspired Alice (at the age of about eighteen) to an extraordinarily intimate grasp of four-dimensional geometry. Howard Hinton wrote several books on higher space, including a considerable amount of mystical interpretation. His disciple did not care to follow him along these other lines of thought, but soon surpassed him in geometrical knowledge.

In 1890, she married an actuary and "led a life of drudgery" [Coxeter] as a wife and mother with a small income. But she continued to explore the fourth dimension as a kind of hobby, building cardboard models of three-dimensional "slices" of four-dimensional figures. Somehow her husband came across the work of the Dutch mathematician Pieter Hendrik Schoute, whose published diagrams mirrored her models. She contacted Schoute and the two began a long and fruitful collaboration. As Coxeter puts it,

Mrs. Stott's power of geometrical visualization supplemented Schoute's more orthodox methods, so they were an ideal team.

It was she who coined the term polytope.

Among other "enthusiasts" (as opposed to academicians) who contributed to four-dimensional geometry, Coxeter mentions Paul S. Donchian, an Armenian American.

His great-grandfather was a jeweller at the court of the Sultan of Turkey, and many of his other ancestors were oriental jewellers and handicraftsmen. He was born in Hartford, Connecticut, in 1895. His mathematical training ended with high school geometry and algebra, but he was always interested in scientific subjects. He inherited the rug business established by his father, and operated it for forty years. At about the age of thirty he suddenly began to experience a number of startling and challenging dreams of the previsionary type soon to be described by Dunne in 'An Experiment with Time'. In an attempt to solve the problems thus presented, he determined to make a thorough analysis of the geometry of hyper-space.

Donchian built delicate three-dimensional models of four-dimensional polytopes which were displayed at expositions in Chicago and Pittsburgh, several pictures of which appear in Coxeter's book.

I built a wire-solder model of the hypercube many years ago, using what I suppose are the same principles, though I didn't know it at the time. It remains in good shape, but it's in my parents' possession, and I don't have a picture of it handy.

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These days I'm working on a set of 3D printer files reproducing Stott's model of the 120-cell, a polytope composed of 120 dodecahedral cells. From her 1900 paper "On Certain Series of Sections  of the Regular Four-dimensional Hypersolids," I've created the virtual constructions from which I'll derive the vertex coordinates.

The following image represents a series of slices slices cut by hyperplanes parallel to a dodecahedral cell, starting with the cell itself (at the center of the image) and ending with the "equatorial" slice midway up the polytope (at the outside of the image). In my file the layers are numbered from VIII to XIV, in accord with the partial nets illustrated in her paper shown above.

And here is part of the "net" from which the 120-cell can be "folded." The "equatorial" layer of dodecahedra (not shown) fits in the interstices, with one for each edge of the dodecahedral cell forming the "base." A second set identical to the one shown then "caps" the 120-cell above the equator.

However, I find that I'm not the first to attempt reconstructing Stott's fascinating models. Well, I'll do the 600-cell as well, and that will be impressive. Here is my projection of the 600-cell to the plane.

I hope to recreate it in string art, the use of which in teaching children was pioneered by Mary Everest Boole.

Here are some of my printed polyhedra, which I built myself in Blender: we have a compound of five tetrahedra, a compound of five cubes, a compound of five tetrahedra (edges only), a great dodecahedron, and four rhombic dodecahedra, but no hypersolids yet. (Chessboard chosen advisedly: see below.)

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The geometry of the fourth dimension has made appearances in a number of imaginative works. Aside from Flatland, the earliest instance is probably The Time Machine by H. G. Wells, published in 1898. Unfortunately, he makes the rather common mistake of conflating temporal extension with a fourth spacial dimension.

"Well, I do not mind telling you I have been at work upon this geometry of Four Dimensions for some time. Some of my results are curious. For instance, here is a portrait of a man at eight years old, another at fifteen, another at seventeen, another at twenty-three, and so on. All these are evidently sections, as it were, Three-Dimensional representations of his Four-Dimensioned being, which is a fixed and unalterable thing.

"Scientific people," proceeded the Time Traveller, after the pause required for the proper assimilation of this, "know very well that Time is only a kind of Space. Here is a popular scientific diagram, a weather record. This line I trace with my finger shows the movement of the barometer. Yesterday it was so high, yesterday night it fell, then this morning it rose again, and so gently upward to here. Surely the mercury did not trace this line in any of the dimensions of Space generally recognized? But certainly it traced such a line, and that line, therefore, we must conclude was along the Time-Dimension."

H. P. Lovecraft gives a much better account of the fourth dimension in "The Dreams in the Witch House," published in 1933 and described by its Weird Tales tagline as "a story of mathematics, witchcraft and Walpurgis Night, in which the horror creeps and grows." Whatever you think of Lovecraft as a writer, one thing you can say is this: he knows when to be explicit and when to be vague and ominous. It serves him well here.

Toward the end of March he began to pick up in his mathematics, though the other studies bothered him increasingly. He was getting an intuitive knack for solving Riemannian equations, and astonished Professor Upham by his comprehension of fourth-dimensional and other problems which had floored all the rest of the class. One afternoon there was a discussion of possible freakish curvatures in space, and of theoretical points of approach or even contact between our part of the cosmos and various other regions as distant as the farthest stars or the transgalactic gulfs themselves – or even as fabulously remote as the tentatively conceivable cosmic units beyond the whole Einsteinian space-time continuum. Gilman's handling of this theme filled everyone with admiration, even though some of his hypothetical illustrations caused an increase in the always plentiful gossip about his nervous and solitary eccentricity. What made the students shake their heads was his sober theory that a man might – given mathematical knowledge admittedly beyond all likelihood of human acquirement – step deliberately from the earth to any other celestial body which might lie at one of an infinity of specific points in the cosmic pattern.

Reminds me of my own college days! Ha ha, actually, it doesn't. I spent an entire year of my life working a problem of 10- and 26-dimensional geometry, got stuck on a minus sign for most of its duration, and finally had to give up and start a new problem. My dissertation advisor may very well have wondered about my nervous and solitary eccentricity, and my fellow students may have shaken their heads at my theories, but not for the reasons Gilman found himself the source of such disturbance…

[source]

The net of a tesseract figures in Robert A. Heinlein's 1941 story "And He Built a Crooked House," in which an architect builds a house in the shape of the three-dimensional "net" of a tesseract (from which the polytope can be "folded" much as a cube is folded from a two-dimensional cruciform net); an earthquake causes it to collapse into an actual tesseract from which other worlds can be reached. The story was anthologized in Fantasia Mathematica in 1958.

Madeleine L'Engle's A Wrinkle in Time, which contains the most well-known tesseract (and verbs the word as tesser), was published five years later, in 1963. I wonder if L'Engle got her idea (which is rather garbled) from the Heinlein story?

Meg sighed. "Just explain it to me."

"Okay," Charles said. "What is the first dimension?"

"Well, a line."

"Okay.  And the second dimension?"

"Well, you'd square the line. A flat square would be in the second dimension."

"And the third?"

"Well, you'd square the second dimension. Then the square wouldn't be flat any more. It would have a bottom, and sides, and a top."

"And the fourth?"

"Well, I guess if you want to put it into mathematical terms, you'd square the square. But you can't take a pencil and draw it the way you can the first three. I know it's got something to do with Einstein and time. I guess maybe you could call the fourth dimension Time."

"That's right," Charles said. "Good girl. Okay, then, for the fifth dimension you'd square the fourth, wouldn't you?"

"I guess so."

"Well the fifth dimension's a tesseract. You add that to the other four dimensions and you can travel through space without having to go the long way around. In other words, to put it into Euclid, or old-fashioned plane geometry, a straight line is not the shortest distance between two points."

Terrible! Just imagine an inhabitant of Flatland speaking like that: "The third dimension is Time. The fourth dimension's a cube. You add that to the three dimensions and you can travel through the plane without having to go the long way around. In other words, to put it into linear terms, a straight line is not the shortest distance between two points." Ugh! A novel is not a math textbook, it is true, but, for me, it's harder to overlook such nonsense than scientific speculation. There's nothing like 1 + 1 = 3 to break the suspension of disbelief. (Not that it's a bad book mind you.)

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[source]

Higher-dimensional geometry appears in art as well. Cubism is an oft-cited example, but geometry figures more directly in Salvador Dali's Crucifixion (Corpus Hypercubus), which depicts Christ crucified on the net of a tesseract. Just as the net permits us to approach what lies beyond our comprehension, in God, so does the Incarnation provides a "picture" of God comprehensible to humankind. That's how the picture usually seems to be interpreted.

Well, somehow this has turned into one of those posts of mine in which I draw connections between whatever unrelated topics I happen to be interested in. Here it's higher-dimensional geometry, science fiction and fantasy, the early twentieth century, and art. I do still want to describe the catalog of regular polytopes, but that will have to wait for a subsequent post.

* The mountain was named after her uncle. Quite a dynasty!

Fantasy Cathedrals and McMansions

My post about "serial epic fantasy" a few weeks ago has gotten more (non-robot) hits than most of my recent posts, so I thought I would revisit the topic, which is, after all, a perennial one on this blog. In fact, it goes back to my second post ever, when I threw down my gauntlet before the literary world.

At the time I wrote that, I had just despaired of ever finding an agent or publisher for my gargantuan first novel, which was 150K words after trimming it down by half and seriously boring. It was about autism, autogyros, quantum mechanics, and various characters that eventually found a home in my Enoch stories. At the back of my mind, I knew that it was too inwardly focused to be readable. I ultimately relegated my autistic recluse to the void in favor of a sanguine pugilist who leaps before he looks and basically does everything I would never do.

So that post was (and is) a kind of manifesto. Manifestos are fun, especially when they're full of hyperbole and self-importance. So let's do another! I'll begin with an unapologetic catalog of my twenty favorite fantasy novels, listed in chronological order:

• Phantastes by George MacDonald (1858)
• She by H. Rider Haggard (1887)
• The Time Machine by H. G. Wells (1895)
• The House on the Borderland by William Hope Hodgson (1908)
• The Night Land by William Hope Hodgson (1912)
• A Princess of Mars by Edgar Rice Burroughs (1912)
• Thuvia, Maid of Mars by Edgar Rice Burroughs (1916)
• A Voyage to Arcturus by David Lindsay (1920)
• The Worm Ouroboros by E. R. Eddison (1922)
• The King of Elfland's Daughter by Lord Dunsany (1924) 
• The Dream-Quest of Unknown Kadath by H. P. Lovecraft (1927)
• At the Mountains of Madness by H. P. Lovecraft (1931)
• Out of the Silent Planet by C. S. Lewis (1938)
• Perelandra by C. S. Lewis (1943)
• Titus Groan by Mervyn Peake (1946)
• Gormenghast by Mervyn Peake (1950)
• The Lord of the Rings by J. R. R. Tolkien (1955)
• The Last Unicorn by Peter S. Beagle (1968)
• Little, Big by John Crowley (1981)
• The Book of the New Sun by Gene Wolfe (1983)
• Mythago Wood by Robert Holdstock (1984)

Well, okay, that was twenty-one. To get on that list, you have to have made a permanent impression my imagination, and I have to have reread you as a guilty pleasure at least once. A list of my favorite short fiction would include pieces by George MacDonald, Robert W. Chambers, Arthur Machen, Lord Dunsany, H. P. Lovecraft, Robert E. Howard, Clark Ashton Smith, and C. L. Moore. I tend to gravitate toward the weird and ornate, like "The Repairer of Reputations" or "The Fortress Unvanquishable, Save for Sacnoth" or "The Tower of the Elephant" or "The Demon in the Flower."

So I think it's fair to say that I'm biased toward the what came before the Tolkien watershed. When I read what other people have to say about pre-Tolkien novels, I often get the feeling that they're viewed as ungainly precursors, as the dinosaur skeletons at the museum entrance. They're the work of primitives who didn't quite know what they were about, Giottos and Fra Angelicos to our modern Raphaels and Caravaggios. They're tolerated or even admired, but as a kind of academic curiosity that one delves into occasionally as an act of literary penance.

My preferences are quite the opposite. I do read modern fantasy. I discover hidden gems that way. They're out there, to be sure. But my first and only real love is the pre-Tolkien canon. How many times have I read A Voyage to Arcturus or Phantastes or The Worm Ouroboros? Too many to count. If I read modern fantasy, it's because I'm looking for something to satisfy the hunger whetted by these works.

But, more often than not, genre fiction cheats this desire. It's got no bite, no danger, no weirdness. It's tame. To put it in galline terms, it's caponized and clipped. It lacks the grotesque stylistic bosses, the glowing digressions, the awkward framing devices and dumb shows of the classics.

I could point to myriad ways in which the works of the pre-Tolkien canon influenced one another. But each draws far more from philosophy and science and mythology. They were part of a real literary movement rather than a genre. A literary movement is a living thing that grows according to its own inner logic and is necessarily bound in time; a genre is a dead thing, a pigeonhole in a commercial classification system depending on the presence or absence of various material elements. The difference between fantastic literature and a lot of genre fiction is the difference between a Gothic cathedral and a McMansion.

And in case I seem entirely dismissive of more recent efforts, let me emphasize that I'm not talking about everything that's out there right now. I've read things that I've liked very much. But just look at the sheer quantity of it all. The kinds of works I like can't be mass produced or made to order. They're like lightning strikes. And one thing I note about the majority of the authors I listed above is that their primary career was not fiction writing. If it had been, they would have starved.

Let me also emphasize that I'm not into nostalgia. I have no desire to try to "get back" to anything. If a true fantasy novel is to be written in our time, it must take root right here in the twenty-first century, drawing its nourishment from the world around us even if in reaction against it. There are no "good old days" to hearken back to. The world in which fantasy first flourished was a dirty and brutal place, and the master fantasists were people of their time. Even the one author most responsible for creating the modern fantasy novel, the aggressively backward-looking William Morris, was a social activist whose efforts in manufacture reform were echoed in the ultra-modern Bauhaus half a century later.

As Tolkien points out, the Escape for which fantasy is often blamed has "Disgust, Anger, Condemnation, and Revolt" as its companions. True fantasy is not safe. It's revolutionary. But genre fiction can't be revolutionary. It mixes and matches the established tropes, making novel arrangements rather than creating new worlds. It might, if quite certain of its audience, cautiously advance a few progressive or conservative talking points, but it will never ever do anything that makes it unclear which tribe it's supposed to appeal to, because that is the one unforgivable sin.

It's easy enough to talk about all this on my blog, and I do often enough. The quality of my novels is debatable, but at any rate they're out there. Sometimes you just have to get up on a rooftop and shout about what you're doing and why you're doing it.

Plus, the robots were getting kind of rambunctious in my absence. Go away, robots. Dance somewhere else.

Related: A Festival of Wrap-Around Cover Art

Flatland: A Romance of Many Dimensions

I CALL our world Flatland, not because we call it so, but to make its nature clearer to you, my happy readers, who are privileged to live in Space.

Having recently re-read Edwin A. Abbott's Flatland: A Romance of Many Dimensions (1884), a late Victorian novella which I assign as reading in the geometry course I teach every spring, I am struck with the fact that here is a bona fide fantasy that rarely makes the canons of fantasy fiction. British fantasists like H. Rider Haggard, William Morris, and George MacDonald are always named, though rarely (one suspects) read. Earlier satires like Gulliver's Travels and Utopia receive honorable mention. But Flatland, which is enjoyable as both a fantasy and a satire, is sadly excluded, and its author, a clergyman, unknown to fantasy-lovers.

It is easy to see why self-appointed historiographers like L. Sprague de Camp (Literary Swordsmen and Sorcerers) and Lin Carter (Imaginary Worlds) left it by the wayside, if they were aware of it at all, and why the latter excluded it even from very eclectic collections like Golden Cities Far and Dragons, Elves, and Heroes. A novel that takes place in a two-dimensional universe wouldn't have been in their line, fixated on material elements as they were, despite the large role world-building plays in the work. There really is nothing quite like Flatland.

The narrator, A. Square, begins by helping the reader imagine what life in Flatland is like.

Imagine a vast sheet of paper on which straight Lines, Triangles, Squares, Pentagons, Hexagons, and other figures, instead of remaining fixed in their places, move freely about, on or in the surface, but without the power of rising above or sinking below it, very much like shadows – only hard and with luminous edges – and you will then have a pretty correct notion of my country and countrymen.

A busy thoroughfare in Flatland

Alas, a few years ago, I should have said "my universe": but now my mind has been opened to higher views of things.

With perceptions limited entirely to the plane, the world appears to its denizens as a line, much as our three-dimensional world is perceived by us through a two-dimensional field of vision (as in a television screen, which is flat).

As there is neither sun with us, nor any light of such a kind as to make shadows, we have none of the helps to the sight that you have in Spaceland. If our friend comes closer to us we see his line becomes larger; if he leaves us it becomes smaller: but still he looks like a straight line; be he a Triangle, Square, Pentagon, Hexagon, Circle, what you will – a straight Line he looks and nothing else.

This exposition of Flatland society and history continues through half the book, touching on the stratification of social classes according to number of sides, the operation of schools and prisons, domestic arrangements, relations between the sexes, the rise of Chromatistes and the art of painting, the Universal Colour Bill, the machinations of the Chief Circle Pantocyclus, the violent suppression of the chromatic sedition, and so forth.

Before the Sanitary and Social Board.

The satire of late Victorian society is heavy but not (to me, at least) altogether transparent. For instance, women in Flatland are both despised and feared – despised, for they are regarded as irrational and foolishly sentimental, and feared, for their bodies are extremely sharp line segments, and they are capable of unthinkingly slaughtering their own families if provoked. A. Square belabors the point in several passages, but it seems plain from the forward that this is to be taken ironically. What precisely Abbott was driving at escapes me, unless it was to ridicule Victorian mores by showing a society in which the strait confinement of women really was a cogent necessity, though even this is questioned within the narrative itself.

A well-bred Hexagon yielding to a Lady.

The second half of the book presents a sequence of visions and visitations. In the first, A. Square descends upon Lineland in a dream, coming to revile its King for his narrow-minded inability to conceive of more than one dimension

"Besotted Being! You think yourself the perfection of existence, while you are in reality the most imperfect and imbecile. You profess to see, whereas you can see nothing but a Point! You plume yourself on inferring the existence of a Straight Line; but I can see Straight Lines, and infer the existence of Angles, Triangles, Squares, Pentagons, Hexagons, and even Circles. Why waste more words? Suffice it that I am the completion of your incomplete self. You are a Line, but I am a Line of Lines, called in my country a Square: and even I, infinitely superior though I am to you, am of little account among the great nobles of Flatland, whence I have come to visit you, in the hope of enlightening your ignorance."

Hearing these words the King advanced towards me with a menacing cry as if to pierce me through the diagonal; and in that same moment there arose from myriads of his subjects a multitudinous war-cry, increasing in vehemence till at last methought it rivaled the roar of an army of a hundred thousand Isosceles, and the artillery of a thousand Pentagons. Spell-bound and motionless, I could neither speak nor move to avert the impending destruction; and still the noise grew louder, and the King came closer, when I awoke to find the breakfast-bell recalling me to the realities of Flatland.

Our narrator is then visited in his turn by a Sphere from Spaceland, who appears to him as a circle (or priest) who can change sizes at will, and before whom A. Square is no better off than the denizens of Lineland were before him. When arguments fail, the visitant resorts to deeds:

"The higher I mount, and the further I go from your Plane, the more I can see, though of course I see it on a smaller scale. For example, I am ascending; now I can see your neighbour the Hexagon and his family in their several apartments; now I see the inside of the Theatre, ten doors off, from which the audience is only just departing; and on the other side a Circle in his study, sitting at his books. Now I shall come back to you. And, as a crowning proof, what do you say to my giving you a touch, just the least touch, in your stomach? It will not seriously injure you, and the slight pain you may suffer cannot be compared with the mental benefit you will receive."

 

Before I could utter a word of remonstrance, I felt a shooting pain in my inside, and a demoniacal laugh seemed to issue from within me. A moment afterwards the sharp agony had ceased, leaving nothing but a dull ache behind, and the Stranger began to reappear, saying, as he gradually increased in size, "There, I have not hurt you much, have I? If you are not convinced now, I don't know what will convince you. What say you?"

Though at first bewildered by his subsequent elevation above Flatland, which permits him to see "through" walls, A. Square is eventually led to posit the existence of more than three spacial dimensions.

[T]ake me to that blessed Region where I in Thought shall see the insides of all solid things. There, before my ravished eye, a Cube, moving in some altogether new direction, but strictly according to Analogy, so as to make every particle of his interior pass through a new kind of Space, with a wake of its own – shall create a still more perfect perfection than himself, with sixteen terminal Extrasolid angles, and Eight solid Cubes for his Perimeter. And once there, shall we stay our upward course? In that blessed region of Four Dimensions, shall we linger on the threshold of the Fifth, and not enter therein? Ah, no! Let us rather resolve that our ambition shall soar with our corporal ascent. Then, yielding to our intellectual onset, the gates of the Sixth Dimension shall fly open; after that a Seventh, and then an Eighth –

The Sphere, overcome with ire at this impertinence, returns A. Square to Flatland. There the narrator inevitably shares the fate of all enthusiastic visionaries out of step with their ruling classes when he tries to spread the "Gospel of Three Dimensions."

The "still more perfect perfection" of the cube referred to above is the regular polytope now known as a hypercube or tesseract. The latter term was coined by Charles Howard Hinton in 1888. Incidentally, this figure (or, rather, its five-dimensional analogue) plays a large role in A Wrinkle in Time by Madeleine L'Engle, in which the dimensional analogy is pursued considerably less competently; that novel also glances upon a two-dimensional world, and would seem to be partly inspired by Flatland. However, in treating time as a fourth "spacial" dimension, it adopts the erroneous conception of space-time expounded upon by H. G. Wells in The Time Machine (1895).

Two recalcitrant revolutionaries executed after trial.

Abbott's lucid approach to the fourth dimension by way of analogy is almost astonishing, considering that it comes from a clergyman (presumably) untutored in such matters, and well before the theories of Ludwig Schläfli were well known. Schläfli, a Swiss mathematician, originated the idea of regular polytopes in the 1850s, but his work did not receive recognition until much later. In Regular Polytopes (1947), H. S. M. Coxeter notes that regular polytopes were independently rediscovered by nine different mathematicians between the years 1881 and 1900, even as Flatland was being written. The time, evidently, was ripe.

Though relatively obscure at its publication, Flatland went on to receive widespread acclaim among mathematicians and physicists in the twentieth century. In our own time, theoretical physics has for a number of years been feeling its way toward the possibility that more than three spacial dimensions play a role in the structure of the universe. My doctoral work focused on higher-dimensional geometry and applications to particle physics, so this is something I know a bit about.

But the book as a whole is quite enjoyable purely as a work of speculative fiction. At every point it compels the reader to ponder what life would be like in a two-dimensional world.

There being no sun nor other heavenly bodies, it is impossible for us to determine the North in the usual way; but we have a method of our own. By a Law of Nature with us, there is a constant attraction to the South; and, although in temperate climates this is very slight – so that even a Woman in reasonable health can journey several furlongs northward without much difficulty – yet the hampering effect of the southward attraction is quite sufficient to serve as a compass in most parts of our earth. Moreover, the rain (which falls at stated intervals) coming always from the North, is an additional assistance; and in the towns we have the guidance of the houses, which of course have their side-walls running for the most part North and South, so that the roofs may keep off the rain from the North. In the country, where there are no houses, the trunks of the trees serve as some sort of guide.

Stephen Hawking suggests in The Universe in a Nutshell that such a creature would be unable to digest food, since a "tube" through the body would separate the unfortunate polygon into two halves. But, perhaps, like flatworms, the Flatlanders expel waste material through the mouth, which, Abbott tells us, also serves as the eye, indicating a physiology markedly different from our three-dimensional preconceptions.


Two small country houses with an antiquated square outbuilding.

Other questions arise. Writing is mentioned, for instance. Flatland writing must needs be one-dimensional, however; of what does this writing consist? Something like printed Morse code, perhaps? What are books like? Does Flatland geometry predominantly consist of the study of magnitudes on a line, much as ours consists of shapes in a plane? What would they think of the Cantor set, I wonder?

 

Whatever could the hills and mines mentioned by the narrator be like? And what of trees, which are referred to as growing from south to north? And so on.

Alas, the book is much too short to begin answering such questions. Not that most other readers would be as interested in them as I am. I happen to know the ins and outs of classical Euclidean geometry and its modern extensions and generalizations pretty well, and I can imagine any number of brave new worlds for A. Square to explore. Others have tried their hands at sequels before now; perhaps I shall join their number some day.

Lately, though, I've been piecing together digital collages of Flatland life in spare moments here and there, ostensibly to put an illustrated version of Flatland on my faculty website for my students to peruse. (These are the color pictures on this post; the drawings are Abbott's.) I have to say, I rather like the results, not that that means much.

When I took art as a teenager, I quickly found myself at the front of the class; however, every year, my teacher would begin by tasking us with forming a composition out of geometrical shapes, and I inevitably received F's on these assignments without ever knowing why. It was quite maddening.

In recent years, I've spent a good bit of time pondering the relation between artistic and mathematical abstraction, as a perusal of my artsy posts will show. (See here and here, for instance.) These collages are, like my fractals, a step toward abstraction in art from the far side. In making them I'm reminded of my old composition assignments. What I'm trying to say is, I hope I wouldn't still get F's on them.

Getting There

I posted recently about my chosen vein of speculative fiction, namely, planetary romance, or the sword-and-planet novel. One peculiarity of early practitioners of this high and lonely art is the variety of means they employed in getting their protagonists to the Other Side.

Among the earliest we have Jules Verne (From the Earth to the Moon and its sequel) and H. G. Wells (The First Men in the Moon), both of whom use mechanical means. Verne, who was obsessed with gadgets and "realism," has his travelers fired out of a cannon (and, truth be told, they never actually land on the moon), while Wells, who was more interested in ideas and settings, uses a rather absurd physical artifice, giving mere lip-service to theoretical constraints, much as he does in The Time-Machine. Verne, I believe, took him to task for this, but wrongly, to my mind – I find that I can forgive the author any amount of ridiculousness as to means of transport, so long as his style is sufficient to bear it along, and he attends to realism once we get there. Chandler wrote that plausibility is mostly a matter of style, and I agree with him.

And anyway, look at the historical antecedents, such as they are. Ariosto has his hero drawn by hippogriffs to the moon, and Dante is conducted by celestial spirits. Perseus – the archetypal voyager to the Other Side and slayer of fearsome alien foes – flies on winged sandals lent him by the gods, while St. Brendan executes his wonder-voyage through the North Atlantic in a flimsy leather coracle. The whole point is to get there, never mind how.

David Lindsay follows Wells rather than Verne in the absurdity of his conveyance in A Voyage to Arcturus, but the account is mysterious and quite compelling. Once again, I find it easy to suspend my disbelief in the hands of a competent storyteller. And, once we're there, we're caught up in the protagonist's physical and spiritual reactions to the alien landscape of Tormance, and Earth is all but forgotten.

Other writers didn't trouble themselves to employ even the most superficial of mechanical means. In The Worm Ouroboros, E. R. Eddison has Lessingham borne to Mercury in a hippogriff-drawn chariot conducted by a wise martlet (after which he was promptly forgotten). On the other side of the Atlantic we have Edgar Rice Burroughs, whose John Carter has only to look upon the Red Planet with a certain measure of longing to find himself there, though stark naked and seemingly in a different body. Otis Adelbert Kline, a Burroughs imitator from the thirties, uses telepathic body-exchange to get his down-and-out character to Mars. His artifice is less successful, in that we're apparently expected to forget about the fate of the Martian prince trapped in the dispossessed, jilted hero's body back on earth.

Most of the examples we've cited so far have the device of a framing story that involves the principals but is quickly forgotten, which (to me) adds a certain charm not to be found in other veins of literature. Here, then, are the keynotes of classic planetary romance:

1. The main structure of the tale is that of an everyman observer in an alien milieu.
2. The observer must be someone with whom we share rapport, hence must originate from Earth or an Earth-based society in something close to our own day and age.
3. The observer must be immersed in the alien milieu, hence must effectively be cut off from mundane society – marooned, as it were. The milieu must be a closed environment, like a desert island.
4. Because it's the immersion that matters, the mode of conveyance is of little import. The simpler and quicker, the better.

Some time after the "classic" period, C. S. Lewis wrote two planetary romances, consciously using both mechanical and spiritual means. In Out of the Silent Planet, Ransom is kidnapped by a rogue professor and his gentleman accomplice and taken to Mars in a spaceship. The ship is just as absurd as anything in Wells or Lindsay, but much more artfully introduced and described. The interplanetary voyage itself is starkly beautiful; I can't think of any other description of space that compares with it. In Perelandra Lewis doesn't bother with technology at all, and has his hero conveyed to Venus in a stone sarcophagus carried by angels. By his own account, this was inspired by a reading of A Voyage to Arcturus. Again, the point is to get there, never mind how.

Later writers were apparently uncomfortable with such flights of fancy, I suppose because the growing canon of science fiction and the advances in rocketry were daily making this scientifictional license less acceptable and less necessary. The protagonists in works like Leigh Brackett's The Sword of Rhiannon, Ursula Le Guin's Rocannon's World, and Frank Herbert's Dune get there by means that are purely mechanical and (within the genre) plausible. So, sadly, we have no more hippogriffs, astral projection, angelic sarcophagi, telepathic mind-exchanges, or backyard rocketships. They've lost something essential, though, because a keynote of the subgenre is that travel to the Other Side has to be exceptional. Otherwise we just have space opera or some other form of science fiction.

Some "planetary romances," instead of taking place elsewhere, take place elsewhen, in the very far future. Examples include H. G. Wells' The Time-Machine, William Hope Hodgson's The Night Land, A. E. van Vogt's The Book of Ptath (a.k.a. Two Hundred Million A.D.), Brian Aldiss' The Long Afternoon of Earth, and Gene Wolfe's Book of the New Sun. The keynotes are much the same. There is usually some kind of framing story and a fanciful mode of conveyance, or at least (as in Wolfe's case) some playful description of how the account was obtained.

For me personally, what really defines the subgenre is the sense of remoteness, of otherness. I like an observer I can identify with, or at least some kind of reference point with what I'm familiar with, however tenuous, but equally important is my total immersion in the other sphere. These two factors go hand in hand. A novel taking place in some other planet not linked to our own in any way would be out-and-out fantasy (like The Dark Crystal, say); a novel taking place in a planet linked to our own but not utterly severed in time or space is science fiction or alternate history.

Planetary romance: it's its own thing.

A Common Thread

I like to link ideas. What do the following passages have in common? Do you see it?

"Chrysaor was joined in love to Callirrhoe, the daughter of glorious Ocean, and begot three-headed Geryones. […] And in a hollow cave she bare another monster, irresistible, in no wise like either to mortal men or to the undying gods, even the goddess fierce Echidna who is half a nymph with glancing eyes and fair cheeks, and half again a huge snake, great and awful, with speckled skin, eating raw flesh beneath the secret parts of the holy earth. And there she has a cave deep down under a hollow rock far from the deathless gods and mortal men. There, then, did the gods appoint her a glorious house to dwell in: and she keeps guard in Arima beneath the earth, grim Echidna, a nymph who dies not nor grows old all her days. Men say that Typhaon the terrible, outrageous and lawless, was joined in love to her, the maid with glancing eyes. So she conceived and brought forth fierce offspring; first she bare Orthus the hound of Geryones, and then again she bare a second, a monster not to be overcome and that may not be described, Cerberus who eats raw flesh, the brazen-voiced hound of Hades, fifty-headed, relentless and strong. And again she bore a third, the evil-minded Hydra of Lerna, whom the goddess, white-armed Hera nourished, being angry beyond measure with the mighty Heracles. […] She was the mother of Chimaera who breathed raging fire, a creature fearful, great, swift-footed and strong, who had three heads, one of a grim-eyed lion; in her hinderpart, a dragon; and in her middle, a goat, breathing forth a fearful blast of blazing fire. […] Echidna was subject in love to Orthus and brought forth the deadly Sphinx which destroyed the Cadmeans, and the Nemean lion, which Hera, the good wife of Zeus, brought up and made to haunt the hills of Nemea, a plague to men."

– Hesiod's Theogony

"Come, now, tell me, where hast thou proved thyself a seer? Why, when the Watcher was here who wove dark song, didst thou say nothing that could free this folk? Yet the riddle, at least, was not for the first comer to read; there was need of a seer's skill; and none such thou was found to have, either by help of birds, or as known from any god: no, I came, I, Oedipus the ignorant, and made her mute, when I had seized the answer by my wit, untaught of birds."

– Sophocles, Oedipus the King

"Then someone suggested that their plaything should be exhibited in the nearest building, and so I was led past the sphinx of white marble, which had seemed to watch me all the while with a smile at my astonishment, towards a vast grey edifice of fretted stone."

"Above me towered the sphinx, upon the bronze pedestal, white, shining, leprous, in the light of the rising moon. It seemed to smile in mockery of my dismay."

"I stopped very gently and sat upon the Time Machine, looking round. The sky was no longer blue. North-eastward it was inky black, and out of the blackness shone brightly and steadily the pale white stars. Overhead it was a deep Indian red and starless, and south-eastward it grew brighter to a glowing scarlet where, cut by the horizon, lay the huge hull of the sun, red and motionless. The rocks about me were of a harsh reddish colour, and all the trace of life that I could see at first was the intensely green vegetation that covered every projecting point on their south-eastern face. It was the same rich green that one sees on forest moss or on the lichen in caves: plants which like these grow in a perpetual twilight."

— H. G. Wells, The Time Machine

"I stood in one of the embrasures of the Last Redoubt—that great Pyramid of grey metal which held the last millions of this world from the Powers of the Slayers."

"To the North-West I looked, and in the wide field of my glass, saw plain the bright glare of the fire from the Red Pit, shine upwards against the underside of the vast chin of the North-West Watcher—The Watching Thing of the North-West… 'That which hath Watched from the Beginning, and until the opening of the Gateway of Eternity' came into my thoughts, as I looked through the glass…"

"[O]lden sciences […], disturbing the unmeasurable Outward Powers, […] allowed to pass the Barrier of Life some of those Monsters and Ab-human creatures, which are so wondrously cushioned from us at this normal present. And thus there had materialized, and in other cases developed, grotesque and horrible Creatures, which now beset the humans of this world. And where there was no power to take on material form, there had been allowed to certain dreadful Forces to have power to affect the life of the human spirit. And this growing very dreadful, and the world full of lawlessness and degeneracy, there had banded together the sound millions, and built the Last Redoubt; there in the twilight of the world."

"And then, so it would seem, as that Eternal Night lengthened itself upon the world, the power of terror grew and strengthened. And fresh and greater monsters developed and bred out of all space and Outward Dimensions, attracted, even as it might be Infernal sharks, by that lonely and mighty hill of humanity, facing its end—so near to the Eternal, and yet so far deferred in the minds and to the senses of those humans. And thus hath it been ever."

— William Hope Hodgson, The Night Land

"At length they came to the door upon the outer court, and they halted. Even from where they stood they felt the malice of the Watchers beating on them, black silent shapes on either side of the gate through which the glare of Mordor dimly showed. […]
     "Sam drew out the elven-glass of Galadriel again. As if to do honour to his hardihood, and to grace with splendour his faithful brown hobbit-hand that had done such deeds, the phial blazed forth suddenly, so that all the shadowy court was lit with a dazzling radiance like lightning; but it remained steady and did not pass.
     "'Gilthoniel, A Elbereth!' Sam cried. For, why he did not know, his thought sprang back suddenly to the Elves in the Shire, and the song that drove away the Black Rider in the trees.
     "'Aiya elenion ancalima!' cried Frodo once again behind him.
     "The will of the Watchers was broken with a suddenness like the snapping of a cord, and Frodo and Sam stumbled forward. Then they ran. Through the gate and past the great seated figures with their glittering eyes."

— J. R. R. Tolkien, The Lord of the Rings

The Sphinx of Remote Posterity

I think I encountered H. G. Wells’ The Time Machine when I was in middle school. I had seen the first part of a film version on PBS. It ended with the Traveller being silently approached by the queerly inhuman humans of the unthinkable future. The image haunted me, and I went immediately to our town library to check it out. As I recall, I read it in one sitting. I was not yet a teenager then, but my enjoyment of the novel is unabated to this day, which to me is the sign of a truly good book.

It is a melancholy tale touched with weird loveliness, a haunting myth of the ephemerality of mankind’s place in the universe. What can compare with the Time-Traveller’s loneliness amid the simple-minded Eloi in their meaningless ruins and gardens presided over by the marble sphinx? The plot and resolution are poignant enough to make us take interest, but never so obtrusive as to distract from the contemplation of time’s abyss.

And then there is the awful moment when the Traveller escapes at last and rushes far, far into the future, into an age when the earth has ceased in its turning and the sun has become a motionless red disc in an eternally black sky; when nearly all life has become extinct, and the world is inhabited only by giant crabs; when the sea is a dead dark pool without wind or waves. Here we see the genesis of the Dying Earth of The Night Land and Jack Vance’s tales; here we see a precursor of Lewis’ dead city of Charn.

Of what fabric is the mantle that hangs like enchantment over entropic tales of earth’s last days? It is a question I often ask, for my own novel has an anticlimactic-apocalyptic aspect. Why do we take pleasure in such stories? Perhaps the feeling is akin to the sense of awe that ovecomes one when walking through the grassy spaces of Tintern Abbey or Stonehenge. Ruins please us where the intact structures would not. A decadent pleasure, suited to jaded generations. We want our cathedrals old and gray, stained with the patina of centuries, a preference the builders would not have understood. For the Gothic cathedral was a riot of color and white stone in its day.*



At any rate, there is something about The Time Machine that affects me in much the same way that good fantasy novels do. Tolkien, in his ”On Fairy Stories,” even concedes that it comes close to being fantasy, though he stops short at classifying it as part of the genre. Of course it is officially classified as science fiction because it involves “science” and machines and not “magic.” But I contend that what classifies a novel as fantasy should not be the presence or absence of this or that material element. It should be its end as a work of art, what sort of beauty it aims at. Many fantasy novels that concern “magic” nowadays are really just banal technology stories, whereas books like The Time Machine are within the sphere of what I call fantasy.

*Perhaps this reduction to schadenfruede unduly harsh. Perhaps it would be more correct to speak of catharsis. Our anxieties about our civilization are vicariously purged in the contemplation of its eventual demise.