Carload of Conquistadors

Life beyond art has become a perfect s**tstorm of late, which has left me some time and energy for said art, but none, alas, for blogging.

Just to give a fairly innocuous example, I took the fam on a two-week camping trip in June, only to collide with a majestic leaping bull elk in a national park, totaling our beloved Ordoñezmobile and leaving us (me, my wife, our three kids) stranded, first on the side of a mountain (with all our gear, the subjects of intense scrutiny by park police), then in a campsite, for days, with no way to get more food (fortunately we'd just stocked up), no rental car being available due to a massive hailstorm having struck the region, with black bears literally snuffling around our tents at night and leaving muddy paw prints on top of our bear box in the morning, in which we had thankfully thought to enclose our three-year-old's carseat, which is undoubtedly the most smellable thing we own, after all of which we were granted, not a minivan or an SUV as we had been promised by our insurance company, but, ironically, an Impala (another leaping ungulate!) in which we were amazingly able to fit our gear and dependents (as the Egyptians fit their blocks of stone into the pyramids) although we had to bestow our faithful roofbag upon a chance-met traveler at a hotel.

Of course we continued our trip after securing our distant rental car through a devious use of shuttle buses and a laconic and somewhat unreliable taxi driver, because we Ordoñeces aren't quitters, although we were put to the test when we discovered a subsequent campsite, which we'd reserved months before, to have been inundated by a flash flood which had deposited a smooth, foot-deep sheet of white badlands silt, out of which the picnic table rose like something from The Drowned World, which, in another dose of irony, I happened to be reading at the time; then, to take another page from The Drowned World, we arrived back at our home in south Texas, which this summer is like the point of heat focused by a magnifying lens, only to find our air conditioner broken.

And that was my vacation.

I am nevertheless happy to report that the newest installment in the Tashyas saga will grace the e-pages of Heroic Fantasy Quarterly this fall, in my story "Raft of Conquistadors," which relates the first fateful landfall of Francisco Carvajal y Lopez in what's now the Lone Star State. The action takes place on Padre Island, the bleak and strangely remote barrier island that has kept south Texas, from Corpus Christi to Brownsville, sparsely inhabited for generations. The story will be illustrated by me.

Although it may be of interest to no one but myself, these stories excite me because they let me evoke all the varied landscapes and climates that "Tashyas" has to offer; I've been making somewhat frequent trips to the Corpus Christi area since I was a little kid, when my dad would take me on collecting expeditions for his marine biology class. Right now I'm working on a Carvajal story set in central Texas, set in the vicinity of...no, I won't say it.

This summer I've also been working on the cover painting for Ark of the Hexaemeron, which is still in the process of being written, but which will, when complete, be the first heroic fantasy epic that I know of to feature three-dimensional manifold topology (sorry). I haven't done much else in the way of illustration or art-for-art's-sake lately, but I have been doing some mathematical sculptures for a little museum I've created at my college, inspired by the beautiful models and displays math departments used to curate around the year 1900, e.g., here and here.

Here, for instance, is the "dodecahedron family," printed on my 3D printer and subsequently painted:

Going from left to right, top to bottom, we have: the icosahedron, the dual icosahedron / dodecahedron, the triacontahedron (the convex hull of the dual pair, colored as the compound of five cubes), the dodecahedron, the icosidodecahedron (the quasiregular solid associated with the dual pair), the right-handed compound of five tetrahedra (colored as the triacontahedron), the compound of five cubes, and the skeletal compound of five tetrahedra.

While painting these I listened to Ride the Pink Horse by Dorothy B. Hughes, an excellent noir set in a New Mexico town modeled partly on Santa Fe.

I have also been experimenting with paper-folding techniques. What especially interest me are the algebraic aspects; theoretically, origami is quite similar to the compass-and-straightedge constructions you did in high school. I've been doing origami for a long time, having been introduced to it as a kid by a Japanese friend of the family, though I don't pretend to be proficient. Just recently, though, I've discovered "kusudama" (a term that gets thrown around rather loosely) or modular origami.

Here, for instance, is the compound of five tetrahedra, assembled, without glue, from 10 sheets of square origami paper cut into thirty 1:3 rectangles, using the instructions found here.

Each strut is one 1:3 rectangle; three struts fit into one another at each point.

The five tetrahedra are not connected to one another in any way. The tricky part was getting them to intersect correctly while connecting the struts at the vertices.

There are some books on modular origami out there; the best I've found are by Ekaterina Lukasheva, who apparently has a math background. Here's one I've made according to her instructions:

It's assembled from 15 square sheets of origami paper cut into two 1:2 rectangles apiece. The 30 rectangles are all folded in exactly the same way and attached to one another without glue. The patterns are chosen according to the compound of five cubes, with six rectangles of each pattern, and one pattern for each of the five cubes. In the end, the piece is roughly the same as a triacontahedron.

Blogging will continue to be light, but I'll probably soon post some thoughts about the things I've been reading and watching.

The Four-Dimensional Lord of Dance

I wrote two posts last year dealing with the fourth dimension:

• Views of a Tesseract
•  Arts and Crafts in Four Dimensions

The focus was mathematical, but along the way I looked at how the fourth (spacial) dimension appears in the works of authors like H. P. Lovecraft, Robert Heinlein, H. G. Wells, Jorge Luis Borges, and, most memorably, Madeleine L'Engle, whose A Wrinkle in Time is about to appear as an uninspired-looking Disney movie in March (sigh). I also talked about mathematical visionaries and mystics like Paul S. Donchian and Charles Howard Hinton, both of whom made real contributions to the field, if only in the sense that they developed and humanized what the academics were saying in their inaccessible research articles, and both of whom might be labeled as cranks or crackpots.

Since then I've done a little research on Hinton, Donchian, et al., and have found a number of other links between the idea of a fourth spacial dimension and various forms of spirituality or mysticism. For instance, the German astronomer Friedrich Zöllner (1834 – 1882) apparently tried to use the fourth dimension explain Spiritualist phenomena. In his eagerness, he was imposed upon by the medium Henry Slade in experiments that have since been debunked. Fantasy and horror authors in their turn used the claims of Spiritualism in their stories; some, like Arthur Machen and Algernon Blackwood, actually subscribed to its views. Hinton, who wrote a number of "scientific romances" himself, was a post-Christian altruist who speculated that spiritual agencies might work by means of the fourth dimension and believed in something like eternal return.

Some Christians of the late Victorian era, disconcerted by the advance of materialism, attempted to colonize the fourth dimension themselves. For other Christians, such as the liberal theologian Edwin Abbott Abbott (author of Flatland: A Romance of Many Dimensions, a book much admired by Hinton), higher spacial dimensions were merely a metaphor for gradual way in which the human mind must approach divine truths. 

[source]

Salvador Dalí appears to have used the fourth dimension in a similar way, in his famous 1954 painting Crucifixion (Corpus Hypercubus), which portrays Christ crucified on the net of a tesseract / hypercube / 8-cell (Schläfli symbol {4,3,3}) hovering over a square grid (Schläfli symbol {4,4}), illustrating the incomprehensibility of God to man.

I have in my hands a Dover edition of Speculations on the Fourth Dimension: Selected Writings of Charles H. Hinton, which includes several of Hinton's scientific romances. It's edited and has an excellent introduction by Rudolph v.B. Rucker, also known as Rudy Rucker, author of the Ware Tetralogy and a modern tribute to Flatland and all-around sci-fi author of note. So no doubt I'll soon be posting about all of this yet again.

*

My post on four-dimensional arts and crafts includes an account of my building the sections and net of a 120-cell. More recently, I've printed and built the sections and net of a 24-cell, which is a regular four-dimensional polytope built from twenty-four octahedra.

The sections proceed as follows, with colors given as the craft paints I bought at Wal-Mart: (I) the octahedral cell at the "south pole" (Parchment); (II) the truncated octahedral section cut by a hyperplane through the midpoints of the edges "above" the south pole (Parchment and Real Brown); (III) the cuboctahedral equatorial section cut by a hyperplane through the set of vertices to which these edges connect (Look At Me Blue and Real Brown); (IV) the truncated octahedral section analogous to Section II but in the "northern hemisphere" (Look At Me Blue and Real Brown); and (V) the octahedral cell at the "north pole" (Coffee Latte).

The net has the "south pole" at the center and the "north pole" at the base. For reasons fully known only to my subconscious, but partly inspired by Dalí's painting above, I decided to model it after traditional depictions of the Hindu god Shiva as Nataraja or Lord of Dance, with three-fold rotational symmetry.

Shiva is the destroyer, and his dance is the cosmic dance of creation / destruction. That puts me in mind of the line from the Bhagavad Gita, uttered by Krishna, quoted by Robert Oppenheimer, and used by me in the title of a short story: "Now I am become Death, destroyer of worlds."

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!

The Bone King at HFQ

There's a Far Side cartoon with a crowd of art patrons admiring a painting in a gallery. In the back, one old lady is whispering to another, "My boy made the frame." In that spirit, I announce the appearance of my illustration to "The Enemy's True Name" by Mark Silcox at Heroic Fantasy Quarterly. That's right: I made the picture.

The Bone King: drawn in ink, digitally colored. Go check it out! And read the story, too!

As I've said, I've been toying with Photoshop workflow used by people in the comics industry. I've recently become interested in comics and in digital illustration in general. For me, it seems to work best when I begin with a hand-drawn picture with cross-hatched shading and build up a layer of flats, with highlights and shadows in separate layers. Not quite the same as painting, but it's exhilarating to actually be able to change things that need to be changed.

In other artsy news, I'm having an exhibition at a certain university out in Alpine, Texas, this August and September. More details to come. Stop by if you're in the area! I'm also still working on my Enoch and Carvajal stories – I've written several of the latter – but I've been too busy with school to blog regularly. Hopefully that will change once the semester is over.

Right now I'm wrapping up a graduate course on regular polytopes. My school recently purchased a large 3D printer which, for lack of a better place (and because of my shameless begging), is set up in my office, and I'm working on producing some of the interesting solids and compounds we explored this spring. I've also been composing a post on higher-dimensional geometry and genre fiction that I'll have to wrap up some time soon...

Heart of the Hollow Earth

I recently watched Apocalypse Now for the first time. It is undoubtedly the best and most beautiful film I've seen in some time. People generally regard it as the best Vietnam film. Though gritty and realistic in its details, and, to some extent, inspired by real events, it represents an almost mythical vision, floating from The Ride of the Valkyries to Dante's Inferno and man's primordial roots in the jungle.

What made me want to see it was a comparison someone made between it and my Tashyas story. Actually, as I've said, I was thinking about Herzog's Aguirre, the Wrath of God and Conrad's Heart of Darkness. As is well known, though, Francis Ford Coppola cited both of these as major influences on Apocalypse Now. As a matter of fact, it's fair to say that Apocalypse Now is nothing more than a film treatment of Heart of Darkness, translating the Company's involvement in equatorial Africa into that of the United States in Vietnam, each presented just as imbecilic, futile, and destructive as human endeavors tend to be.

I've started to regard Heart of Darkness as a kind of modern myth. It has two basic motifs: the dark journey inward, and the great man who goes native and betrays humanity. The moral is that civilization is a veneer over something very dark indeed. Like the plot of Red Harvest (which I regard as another modern myth), it seems to have become part of the dream-logic of our culture. Aguirre and Apocalypse Now have been mentioned; I'm reminded also of Ridley Scott's Alien (whose ship and shuttle are named from Conrad stories) and James Cameron's popcorn-selling sequel. And, as a very recent and not-quite-legitimate descendant, we have Kong: Skull Island, which I went to see at the $4.00 matinee last week.

It's pretty plain that the makers of Skull Island were wanting to evoke and/or perfectly willing to plunder the visuals and general atmosphere of Apocalypse Now. It opens at the close of the Vietnam war, and the first scenes are chock-full of in-your-face historical details that let you know exactly what era you're looking at, while the jungle scenes are overlaid with the predictable rock songs so that you don't forget that this is the Vietnam era despite the overwhelming chronological ambiguity. (Me, to my eight-year-old son: "They played a Creedence song in that movie I just saw. Can you guess which one?" Him, without a moment's thought: "'Run Through the Jungle'!")  It's one of those movies where the older, uglier, and/or more annoying actors tend to meet grisly fates, and the young, pretty, highly paid actors do not. There's one guy I knew was destined to get picked apart by pterodactyls or something from the first moment I saw him.

Well, so, kind of a stupid movie.* But, as you may know, I'm a sucker for movies about little people running from giant monsters, and this one is pretty awesome in that department.

One really cool aspect is the hint that all these weird creatures are coming out of gigantic caverns beneath the earth's surface, where MUTOs have apparently been thriving for millions of years. Skull Island is set in the same universe as that Godzilla movie that came out in 2014 (also stupid, but also quite enjoyable), and it seems likely that we're looking at appearances by Mothra, Rodan, and Ghidora in the near future. Bring on the MUTOs! All monsters attack! I'm giddy with excitement!

But back to the hollow earth thing. Since my earliest childhood, I've known deep down in my heart that the whole earth-is-just-melted-rock-until-you-get-to-China theory is false. I mean, no one has actually been down there, have they? It's much more likely that there are massive caverns inhabited by gigantic prehistoric creatures and forested with huge mushrooms. Otherwise, the planet would be mostly wasted space, and, if there's one thing we know about Nature, it's that she hates for things to go to waste.

So I was very interested to read some of the amazing hollow earth theories recounted by Ryan Harvey over at Black Gate in some of his Pellucidar posts. Clearly, I'm not alone in my deep-seated convictions. But the most interesting, I think, is the theory of Cyrus Teed, an amateur scientist who founded a religious sect (Koreshanity) in the belief that we are already living on the inside of the world. From his Cellular Cosmogony:

The sun is an invisible electromagnetic battery revolving in the universe's center on a 24-year cycle. Our visible sun is only a reflection, as is the moon, with the stars reflecting off seven mercurial discs that float in the sphere's center. Inside the earth there are three separate atmospheres: the first composed of oxygen and nitrogen and closest to the earth; the second, a hydrogen atmosphere above it; the third, an aboron atmosphere at the center. The earth's shell is one hundred miles thick and has seventeen layers. The outer seven are metallic with a gold rind on the outermost layer, the middle five are mineral and the five inward are geologic strata. Inside the shell there is life, outside a void.**

Teed established a commune in Florida in 1894, which finally fizzled out in the 1960s. The place is now a state historical site. Strangely enough, soon after reading Mr. Harvey's post, I met a professor who lives near the site and takes his students there on occasion. So when he started talking about this theory that we live on the inside of the earth, I actually knew what he was talking about and could respond intelligently. It's called social networking, people. "You see?" I told my wife. "Reading weird stuff on the Internet isn't just wasting time after all!"

But it's strange, isn't it, how many "alternative" scientific theories (hollow earth, Atlantis, spiritualism, etc.) of the turn of the century gave birth to subgenres of fantastic literature? One wonders what theories Burroughs was familiar with in creating Pellucidar. At any rate, he was apparently unfamiliar with the shell theorem, first proved by Isaac Newton, which states that, at any given point in a spherically symmetric distribution of mass, only the mass closer to the center than the point contributes to the gravitational force at that point; all other mass can be ignored because its gravity cancels itself out, so to speak. The upshot is that, inside a perfectly hollow spherical shell, there would be no gravity at all; if, as in Pellucidar, there were a massive sun-like body at the center of the hollow, everything would fall into that body and burn. Of course, there would be nothing to keep such a body in its place at the center.

Speaking of Pellucidar and stupid movies, I recently watched At the Earth's Core, an Amicus production starring Peter Cushing and Doug McClure, with my kids. ("Hi! I'm Doug McClure! You may remember me from such films as At the Earth's Core!") Not so great, but the kids loved it. We also recently watched The Valley of the Gwangi, a Ray Harryhausen film about cowboys trying to capture dinosaurs for their wild west show in Mexico. It has same basic plot as King Kong (another modern-myth candidate) and ends with an allosaurus stalking a cowboy, his girlfriend, and a boy named Lope through an empty cathedral, which is pretty awesome.

Kong-derived stories seem always to feature some kind of dark journey upriver to primordial beginnings, which we saw as a key element in the Heart of Darkness myth. And so we're brought back to the primordial beginnings of this post, that is, Apocalypse Now.

* At one point, a search party happens upon a letter that a guy who got eaten was writing to his family. They make a big deal about how they're going to see that his widow gets his things. It's a solemn moment, but I couldn't help but imagine how that would go: "Ma'am, I'm very sorry to inform you that your husband was killed in action. Well, no, actually it wasn't in Vietnam. No, he survived that. What happened was, we were sent on this special mission to a secret unexplored island inhabited by prehistoric monsters, where he was eaten by a giant lizard creature. I'm so sorry for your loss."

** Actually, I imagine that there's probably prehistoric creatures on the outside.

Oodles of Doodles

I have an art show coming up in September and am working to make a few quick pieces to fill out the exhibit. I've gotten two done in one week, which must be some kind of record for me. (Lately, anyway; in my tumultuous youth I used to dash paintings off in an hour or so.)

My art shows generally attract a few of my math students, so I've been under some pains to make mathematical art, dangerous though this is, beauty and truth being distinct transcendentals, &c., &c. A while back a student sent me a YouTube video about topological doodles. Being both a doodler and a student of topology, I added one technique contained therein to my Long Meeting Repertoire. Begin with a squiggle, turn each crossing into a twist, and, Voila!, you have a surface with boundary. In the fullness of time, this led to the advent of...the Chicken Man:

Chicken Man, 5" x 7", watercolor and ink on hot-pressed paper.

Born from a drawing completed on my mother-in-law's kitchen table. Is he a chicken? A man? An indeterminate embryo? A once-punctured non-orientable surface of genus 10?

He is, at any rate, inspired by Paul Klee and the ideas expressed in his highly mathematical Pedagogical Sketchbook. I regard Klee as one of the great artistic geniuses of all time; that I might link the present discussion back to the main topic of my blog, perhaps it would be apposite to recall that time I received the following disapprobation from the famous sci-fi author and political commentator John C. Wright for my opinion:

You sound very reasonable, but then I looked up the works of Paul Klee, and they were so abominably ugly and deranged that I admit the gulf between us can never be crossed. Artistically, it is the same as trying to reason out the sound of one hand clapping: you are looking at something meant to hinder the ability to admire art, to blind the eye and benumb the brain. It is garbage, pure and simple, and even simple drawings by comic book artists or commercial artists doing magazine covers show far more skill, sanity, proportion, color, composition, and execution. [...]

Why do you vainly tell me to ignore the evidence of my eyes, which I trust, and believe the conclusions of your judgment, which even the limited experience of this exchange proves is execrable.

Had you pointed me to some painting that was merely odd or incomprehensible to me, my reaction would be different. I could continue to give you the benefit of the doubt. Instead you have pointed me to the most absurd, ghastly, and disproportion bits of ugly lunacy imaginable, pieces that make me physically sick to look at, and call them good work. [...]

I am an artist. I know what art is because I can do it. I also know when I am looking at something far better than I do because I lack the skill, and I can see the garbage you like and I know I could draw as well with my left foot after my foot was run over by a tractor and I was pumped so full of painkillers that the lower half of my brain was sloshing.

It is also obvious that no further comment is needed. Why do you think words can make me see beauty where there is nothing but filth?

I like to look that over every so often, as a message addressed to me personally by a famous person, and a perfect specimen of some art form of which the name escapes me. Good thing I'm so indefatigably ebullient, or it might have hurt my feelings. But, as my father-in-law says, opinions are like, well, you know. Everybody's got one.

My Long Meeting Doodle Repertoire is quite extensive, and consists solely of mathematical explorations. That way, when my colleagues look over and see me doodling during an important meeting, they'll just think I'm a brilliant mathematician working out some new theory. Actually, I have to draw so I can listen attentively and recall what I hear, a fact that my fifth grade math teacher didn't fully appreciate. These days my sketches consist mainly of Platonic solids and the like, as I've been developing a graduate course on regular polytopes. Here's a few sticky notes:

My aunt bought me these "Dr." sticky notes for graduation, incidentally; I don't buy them for myself.

Why, you ask, am I showing you doodled-upon sticky notes from my office? Perhaps, my friend, you are looking for one of those "sensible" blogs.

Anyway, inspired by such, I spent the first part of this week super-doodling a sketch of the rhombic triacontahedron:

Triacontahedron, 5" x 7", watercolor and ink on hot-pressed paper.

The triacontahedron is a zonohedron, which, well, I won't bore you with the details. It's formed from thirty golden rhombi and is the basis of that cool lampshade you see in coffee shops sometimes. The coloration is determined by five intersecting cubes contained in the solid. The layout shamelessly mimics Leonardo da Vinci's polyhedron illustrations for Luca Pacioli's Da Divina Proportione. Pacioli is one of my mathematical ancestors – I am "descended" from him, advisor to student, down through the centuries, along with about two hundred thousand other mathematicians – so this is something of an homage to him, as well as to H. S. M. Coxeter, from whose book Regular Polytopes I first learned of zonohedra.

My multicolored sketches were the wonder of my graduate algebraic topology class, so perhaps I'll continue to produce a few more watercolors along the same lines, for old times' sake.

Tower and Fish

Midnight painting madness continues apace. I am happy to report that I have almost one half of the cover (whether the front or the back I don't yet know) of The King of Nightspore's Crown more or less completed:

I say midnight painting, though of a truth much of this was produced on weekends, mostly while "napping" the baby by bouncing her little chair with my foot. She wakes up the instant I stop this motion, and it takes quite a bit of coordination to keep the bounces out of the picture. And yet somehow I've never been able to dribble a basketball.

At any rate, here we have an abstracted version of the pseudospherical Tower of Bel reaching up into the stratosphere against an Enochite skyline, with the Leviathan that symbolizes both primeval Chaos and the all-powerful State swimming into a brackish hemlath swamp. Some aspects remain to be touched up, but I like how it's going so far.

In case you've forgotten, here's the original sketch of the cover in toto:

The pigment is somehow mixed with the identical purple dresses of Cora and Clarice, the resurrection of the mummy Xaltotun, the beheading of the vampire Lucy Westenra, the squashing of the witch Gagool beneath a heavy door, the revelation of Pip's benefactor on a storm-tossed night, and the horrible spontaneous combustion of the rag-and-bottle merchant Krook. As you can see, I paint very, very slowly.

I am tentatively to have another art show this summer. My friend who runs the gallery, a forward-thinking MFA and art instructor at the local college, is always just a tiny bit disheartened by my staid attention to naturalism, my addiction to illustration, and my meticulous planning. So I hope to complete a few more abstract and spontaneous pictures before now and then to gladden his spirit. To that end, I'm working on the Chicken Man:

He was originally drawn to please my four-year-old daughter; the ghostly image of Margo, her orange dinosaurian crony, may be seen through the Chicken Man's right leg, on the next page of my sketch book. Why he's called the Chicken Man I don't know. Perhaps because it's a hard world for little things. His pathos fills me with sad tenderness.

His body is formed from turning a random squiggle into a surface by converting the crossings into shaded twists, forming (in topological terms) a punctured surface. This particular surface happens not to be orientable, as an examination of his right hip suffices to indicate. It follows that he's not a Seifert surface, though I wouldn't tell him this to his face. His genus is nine. Well, ten, if you count his little toe-loop.

Man, the obscure geometry and topology references just keep coming tonight. Maybe the pressure is starting to get to me.

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.