Arts and Crafts in Four Dimensions

She returned the smile, then looked across the room to her youngest brother, Charles Wallace, and to their father, who were deep in concentration, bent over the model they were building of a tesseract: the square squared, and squared again: a construction of the dimension of time. It was a beautiful and complicated creation of steel wires and ball bearings and Lucite, parts of it revolving, parts of it swinging like pendulums.*

– Madeleine L'Engle, A Swiftly Tilting Planet

I wrote a couple of months ago about four-dimensional geometry. Today I'd like to continue our progress through transdimensional gulfs and sinister alien geometries by discussing the 120-cell in some detail, and also describing the workflow I used to print the three-dimensional sections and net shown below.

As usual when trying to understand the fourth dimension, it's easiest to proceed by way of analogy with lower dimensions. Imagine a two-dimensional creature, like A. Square of Flatland, existing in a planar universe. Such a creature would have an essentially one-dimensional field of vision, much as our field of vision is essentially two-dimensional (like a painting or a television screen). How would we describe a dodecahedron, that is, a polyhedron formed from twelve regular pentagons, to such a creature?

(Click to read more; I've got a lot going on in this post.)

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!

Keftu (Still) Indomitable!

Okay, ladies and gentlemen. The King of Nightspore's Crown is back and better than ever. In honor of the awesome new series name, let's peruse a few passages from...the King James Bible!

And Cain went out from the presence of the Lord, and dwelt in the land of Nod, on the east of Eden. And Cain knew his wife; and she conceived, and bare Enoch: and he builded a city, and called the name of the city, after the name of his son, Enoch. And unto Enoch was born Irad: and Irad begat Mehujael: and Mehujael begat Methusael: and Methusael begat Lamech. And Lamech took unto him two wives: the name of the one was Adah, and the name of the other Zillah. And Adah bare Jabal: he was the father of such as dwell in tents, and of such as have cattle. And his brother's name was Jubal: he was the father of all such as handle the harp and organ.

When I was about nine, I happened to see a hardcover Bible for sale in the mall. (This was back when you could buy books in malls.) I didn't have access to a Bible at the time, and this one caught my eye for some reason. I begged my mom to buy it for me. She did so, despite an initial reluctance out of fear that I would trash it or something. (I still have it, and it's still in perfect condition, so there.) (My parents were always suckers for buying me books, but they eventually discovered that I would buy them with my own allowance if necessary, so that source dried up.)

It was the King James Version. I set to work reading it at once. And man oh man, is there some weird shit in the Bible. I use the colorful metaphor advisedly, because scriptures are chock full of earthy images and bizarre carnal encounters, though the prudery of the translation hid certain, ah, matters, from my impressionable mind. I do remember asking my mother what "begat" and "slayeth" meant.

And it came to pass, when men began to multiply on the face of the earth, and daughters were born unto them, that the sons of God saw the daughters of men that they were fair; and they took them wives of all which they chose. There were nephilim in the earth in those days; and also after that, when the sons of God came in unto the daughters of men, and they bare children to them, the same became mighty men which were of old, men of renown.

I read Madeleine L'Engle's Many Waters at around the same time, and even dressed up as Japheth the son of Noah for Halloween. Yes, I was a weird kid. But ever since, I've been fascinated with the first eleven chapters of Genesis. My Ant–, er, Enoch stories reflect that. I don't know what it is about Norse / Teutonic / Medieval settings and fantasy – Tolkien is to blame, I guess – but me? I'm all about the Greek and the Semitic (cf. here and here). I'm especially interested in the ways in which the religions of the surrounding cultures bled into the Bronze Age traditions that went into what we call the Bible.

And the whole earth was of one language, and of one speech. And it came to pass, as they journeyed from the east, that they found a plain in the land of Shinar; and they dwelt there. And they said one to another, Go to, let us make brick, and burn them thoroughly. And they had brick for stone, and slime had they for mortar. And they said, Go to, let us build us a city and a tower, whose top may reach unto heaven; and let us make us a name, lest we be scattered abroad upon the face of the whole earth.

Anyways, these days I'm not reading my Bible so much as J. B. Bury's A History of Greece (1900). I've also been working on Lord Foul's Bane, but I have to say, it's been kind of a slog. And I read Bleak House this year, people. I can't fault the author, really. Epic fantasy has just gotten kind of boring to me. Even edgy epic fantasy.

Wait, don't I write epic fantasy? I don't know. To me it's different somehow. More along the lines of The Book of the New Sun than The Lord of the Rings. Anyway, I used to eat up epic fantasy, but these days it's tough going. Maybe reading Cars and Trucks and Things That Go too many times has shortened my attention span.

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.

Mining the Bible

The Book of Tobit isn't the most well-known book of the Bible. For one thing, it's very short; for another, it's not in all Bibles. Catholics refer to it as "deuterocanonical," indicating that it is not in the current Hebrew Bible, but regard it as canon; to Protestants it is "apocryphal," hence non-canonical.

The events Tobit describes are placed in the 8th century BC, but it was probably written in the 3rd or 2nd century, or at any rate sometime after the return from the exile. It was written in Aramaic, but most modern editions are based on one of several ancient Greek versions. The Aramaic and Hebrew versions were thought lost until fragments were found in Cave IV at Qumran. St. Jerome claimed to have based his version for the Latin Vulgate on an Aramaic copy.

The book is, according to most scholars, a kind of religious fairy tale:

Tobit is a righteous Israelite living in Nineveh after being deported by Shalmaneser, the king of Assyria who conquered the northern kingdom. He adheres to the Mosaic law, practices charity, and is especially solicitous regarding the burial of the dead. This gets him in trouble with Sennacherib, Shalmaneser's successor, forcing him to flee for his life. (There are some historical errors here, but I'm presenting the story as-is.) After Sennacherib's assassination, Tobit's nephew Ahiqar (a Near Eastern folk hero) pulls some strings to allow for his return. He takes up his dead-burying activities again, prompting the derision of his neighbors. After burying a man strangled in the marketplace, he sleeps in the open, and birds poop in his eyes. This causes cataracts and, eventually, blindness. He prays for death.

Meanwhile, a young Israelite woman named Sarah, who lives in Ecbatana in Media (in modern-day Iran), is experiencing troubles of her own. She's gotten married seven times, but every time, just as she was preparing to go to bed with her new husband, the demon Asmodeus (from the Persian aeshma daeva, demon of wrath) appeared and killed the groom to prevent their consummating their union. Her maid accuses her of having strangled them all. She prays for death.

Both prayers are heard by God, who sends the angel Raphael to make things right.

Tobit sends his son Tobias to Media to retrieve a large sum of money he deposited there many years ago. Tobias enlists the service of a young man (Raphael in mortal disguise) who claims to know the roads. While en route, a fish tries to eat Tobiah's foot as he's bathing in the Tigris River. He catches the fish, and Raphael advises him to keep the liver, the heart, and the gall. The first two repel demons when roasted, and the third is a cure for cataracts.

They reach Sarah's house. Tobiah marries Sarah and they go to the bridal chamber. He places the liver and heart on embers prepared for incense. Asmodeus is driven by the smoke into Upper Egypt, where he is bound by Raphael. Tobiah and Sarah rise to say a prayer together. It's a prayer of thanksgiving that goes from the cosmic to the intimate, dwelling on the heavens and the earth, Adam and Eve, sexual complementarity, mutual support, sincere love, and the hope of growing old together.

Sarah says, "Amen," and they get in bed together. Sarah's father, who had ordered a grave to be dug in the night, has it filled in when he discovers the happy outcome. The money is retrieved and the couple goes to Nineveh. Tobiah applies the gall to his father's eyes and is able to peel the cataracts off his eyeballs. The angel then reveals his true identity:

I am Raphael, one of the seven holy angels who present the prayers of the saints and enter into the presence of the glory of the Holy One.

He also gives some insight into the nature of his supposed corporeality:

All these days I merely appeared to you and did not eat or drink, but you were seeing a vision.

Years later, from his deathbed, Tobit admonishes his son to return to Ecbatana, for the prophet has preached the destruction of Nineveh. So Tobiah and Sarah and their family return to Media, where they hear about the fall of Nineveh and praise God.

All in all, it's a strange, beautiful tale full of bizarre happenings and vivid details that passes freely and unapologetically from the scatological, the visceral, and the sexual to the angelic and the divine. It's entertaining – it could almost be the basis for a story in the Arabian Nights – at the same time as being edifying and thought-provoking. More than anything, viewed purely as a story, it's a wonderful mine for writers.

John O'Neill of Black Gate fame recently mentioned his surprise that fantasy authors don't do more with Biblical material:

When I was editing fiction for Black Gate, I was always a little surprised at how many writers were eager to tap the dead religions of Ancient Greece, Rome and Scandinavia, and how few seemed interested in the rich storytelling of the Bible. Maybe it was an overabundance of respect — or, more likely, a lack of real familiarity with the source material.

I suspect that, culturally speaking, it's just too close, for believers as well as non-believers. There's a sense of ownership. We know the Bible, or so we think. It's old hat.

The truth is, the Bible has a lot of weird, sexy, and violent parts that don't come up in Sunday school. The Old Testament is a Bronze-Age epic of love and revenge and worship and war, rich in historical detail and local color, with poetic images that are some of the most beautiful in any language. It's a bridge from the cosmic to the historical, establishing a supernatural link and parallel between the Temple liturgy and the creation of the universe.

Even the New Testament has more strangeness to mine than most people suspect. Think, for instance, of the exorcism of the Gerasene demoniac, in which Jesus casts the demon called Legion ("for we are many") out of a tomb-dwelling demoniac, but, at its request, allows it to enter a herd of swine, which promptly drown themselves in the sea. That story alone prompts countless questions about the nature of the world of spirits; it makes you feel like you're catching just a glimpse into a strange and frightening plane with its own secret laws.

One fantasy author who did do a lot with Biblical material is Madeleine L'Engle. I read her Time Quartet many times when I was a pre-teen. My favorite was Many Waters, in which the "normal" Murray twins Dennys and Sandy are accidentally time-warped into the days leading up to the Deluge, where they meet the patriarchs, fall in love with a beautiful young woman, and come into contact with nephilim, seraphim, virtual unicorns, and tiny wooly mammoths. When I was in the fourth grade I began a project of reading the King James Version of the Bible, and Many Waters went hand-in-hand with the strange things I was encountering there. I even dressed as Japheth for Book Day in the fifth grade.

I happen to do a lot with Biblical material myself. The world of Antellus represents a blend of Greek and Semitic mythology. Like L'Engle, I'm especially drawn to the first part of Genesis, from the creation accounts (there's two of them, you know) down to the Tower of Babel. Robert Alter's Genesis: Translation and Commentary (W. W. Norton) is an excellent non-religious translation. I'm also fascinated by the mythologies of other Semitic peoples, especially the Arabian/Islamic jinn, on which I base my nephelim, as well as some of the other ancient Jewish traditions. But I have a special affection for Tobit, which is one reason I chose my pen name as I did.

So, go read your Bible. If you don't have one, get one. I'm sure you can find someone more than willing to give you a nice one for free, even if you tell them that you just want to mine it for material. It just might not have the deuterocanonical books in it...