Wednesday, May 17, 2017

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.


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.)


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…

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.)


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!

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