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David Darling's Newsletter #19


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March 4, 2004


Contents

1. Meanderings
2. Higher Dimensions
3. Bookends


1. Meanderings

There's just a hint of spring in the air here in Minnesota – the snow and ice are finally melting, squirrels are out digging up the acorns they stashed away last autumn (how on Earth do they remember where they put them?), and there's a general feeling of waking up after the long winter hibernation. A couple of weeks ago I had my first taste of snow-shoeing thanks to my friend John, who lives nearby. Now I can add this to my list of strange and unique experiences, which include accidentally standing in front of the exposed neutron beam from a nuclear reactor (still awaiting the consequences!), having my name in lights on Broadway, encountering a giant crab on the south coast of England, meeting a Vulcan in Yellowstone Park, and being abducted by aliens. Only one of these isn't true (yes, really); I'll tell you which next time.

Amazing data continue to pour in from the Mars rovers and from the orbiting Mars Express. We now know, beyond doubt, that at the Opportunity rover site there was once enough water to support life as we know it. This is really the first time that we've had proof of long-standing liquid water anywhere beyond Earth and it's a crucial step forward in the search for extraterrestrial life. Anyhow, more on this and all the other new discoveries from the Red Planet next month. In this issue I want to chat a bit about a subject that's always fascinated me, ever since as a young lad I read a book called Flatland by Edwin Abbott.


2. Higher Dimensions

We live – or seem to live – in a world of three dimensions: up-down, forward-backward, and side-to-side. The key thing about these is that they lie at right angles to one another. So, what could possibly be meant by a fourth dimension? Can you imagine another direction that's at right angles to the three we're familiar with? How can you think in four (let alone five, six, or more dimensions) when you only have a three-dimensional brain? Well, plenty of books have been written and a number of schemes devised to nudge our imaginations into thinking four-dimensionally. One of the oldest and best is the book I just mentioned – Flatland written in 1884 by Edwin A. Abbott, an English clergyman, under the pseudonym "A. Square". As the title suggests it's all about an imaginary two-dimensional world in which the inhabitants have no concept of up and down.

Flatlanders appear to each other as mere points or lines (try looking along the edge of a sheet of paper to get the idea). But from our privileged, three-dimensional perspective we can look down on Flatland and see that its people are "really" a variety of shapes, including straight lines (females), narrow isosceles triangles (soldiers and workmen), equilateral triangles (lower middle-class men), squares and pentagons (professional men, including the author of the tale), hexagons and other regular polygons with still more sides (the nobility), and circles (priests). The whole thing is a satire as well as a scientific romance and Abbott uses these geometrical distinctions, especially the appearance of Flatland females and the working class, as a commentary on the discrimination against women, the rigid class system, and the lack of tolerance for "irregularity" that was prevalent in Victorian Britain. In a dream, A. Square visits the one-dimensional world of Lineland where he tries, without success, to persuade the king of Lineland that there really is such a thing as a second dimension. In turn, the incredulous Mr. Square is told of three-dimensional space by a sphere who moves slowly through the plane of Flatland, growing and shrinking as his cross-section changes in size. Now step up a dimension. Imagine what would happen if a hypersphere – a four-dimensional sphere – were to cut through our 3-d space. It would first appear as a tiny bubble, grow to a sphere of maximum size, and then shrink again. What we saw as a growing and shrinking sphere would really be cross-sections of the hypersphere as it passed through our universe.

Higher dimensions were all the rage in Victorian times. Spiritualists latched on to the idea of the fourth dimension as a home for the spirits of the departed. And why not? It would neatly explain how ghosts could pass through walls, disappear and reappear at will, and see what was invisible to mere three-dimensional mortals. Some scientists lent their weight to these claims, often after being duped by clever conjuring tricks. There was the astronomer Karl Zöllner, for instance, who wrote about the four-dimensional spirit world in his Transcendental Physics (1881) after attending séances by Henry Slade, the fraudulent American medium. Recently, an ingenious person suggested to me that the oddly changing shapes sometimes reported in connection with UFOs could be the result of us seeing various cross sections of these objects as they intersected our reality from a higher plane. A good plot for a science fiction story, at least – and who knows?

For most us, analogy is the closest we're likely to come to mentally capturing the fourth dimension. Here's another example. A sketch of a cube is a 2-d perspective of a real cube, right? So a real cube can be thought of as a perspective of a 4-d cube, or tesseract. At a movie, a 2-d picture represents a 3-d world, whereas if you were to watch the action on the set live, in three-dimensions, this would be like a screen projection in four dimensions. H. G. Wells dabbled in the fourth dimension in The Time Machine (1895), and also in The Invisible Man (1897), in which the central character drinks a potion "involving four dimensions," and in "The Plattner Story" (1876), in which the hero of the tale, Gottfried Plattner, is hurled into the fourth dimension by a school chemistry experiment that goes wrong and comes back with all his internal organs switched around. Just as we could lift a right-handed 2-d glove out of Flatland, flip it over in our third dimension, and put it back as a left-handed glove, so a four-dimensional being could flip over one of our 3-d gloves or shoes, or, as in Plattner's case, our entire body, and make it the mirror image of the original. Could you train yourself to see in four dimensions? Some people think so. Charles Hinton, an eccentric English-born mathematician, became obsessed with the problem. Early in his career he taught for a few years at Uppingham School. Another teacher there was Howard Candler, who was a friend of Edwin Abbott, so there may have some cross-fertilization of ideas. Anyhow, in the early 1880s Hinton published a series of pamphlets starting with "What is the Fourth Dimension?" and "A Plane World", which were reprinted in the two-volume Scientific Romances (1884). And he became convinced that it was possible to learn to see in four dimensions. He built an amazing set of little colored cubes to represent the various cross sections of a tesseract (a name he may actually have coined) – versions were sold commercially – and then he memorized the cubes and all their possible orientations so that he could (he claimed) mentally reconstruct the fourth dimension. I said he was eccentric. At the time he was teaching in England, he married Mary Everest Boole, the eldest daughter of George Boole, the founder of mathematical logic. Unfortunately, he also married a Maud Wheldon and ended up being tried at the Old Bailey for bigamy. After serving a day in prison, he fled with his (first) family to Japan, taught there for some years, and then took up a post at Princeton where he built a species of cannon for automatically pitching baseballs. Crazy guy!

One thing that people don't give a lot of consideration to is that it's almost as difficult to think in two dimensions as in four. Try to imagine what Flatlanders would actually see. Abbott is aware that he cheats a bit in his description of this. In the preface to the second edition, he gives a lengthy but not-too-convincing reply to the objection, raised by some readers, that a Flatlander, "seeing a Line, sees something that must be thick to the eye as well as long to the eye (otherwise it would not be visible...)." No matter how hard we try we can't imagine being able to see a line of zero thickness!

Of course, higher dimensions are very fashionable in physics these days. But the interest goes way back to an early attempt to unify gravity with electromagnetism. In 1919 Theodor Kaluza pointed out that if you extend general relativity theory (Einstein's theory of gravity) from the normal 4-d spacetime to a 5-d spacetime, the equations can be separated out into ordinary 4-d gravitation plus an extra set, which is equivalent to Maxwell's equations for the electromagnetic field, plus an additional field known as the dilaton. So electromagnetism is explained as a manifestation of curvature in a fourth dimension of space, in the same way that gravitation is explained in Einstein's theory as a manifestation of curvature in the first three. In 1926 Oskar Klein suggested that the reason the extra spatial dimension isn't seen is that it's "compact" – curled up like a ball with a fantastically small radius. Then, in the 1980s and '90s, the Kaluza-Klein theory was revived and is now embedded in modern string theory. But that's another story.


3. Bookends

More about higher, lower, and fractional dimensions, as well as infinity, surreal numbers, impossible figures, Carrollian conundrums, and goodness-knows-what-else, in my new book The Universal Book of Mathematics, to be published by Wiley in August. I'll soon be setting up some pages on my website with puzzles, paradoxes, and other mathematical curiosities to amuse you and perhaps even tempt you into buying the book. You have been warned! Meanwhile, I heartily recommend getting hold of a copy of Abbott's Flatland from the library or bookstore (I believe Princeton UP do a reprint).


Until next time,
Best wishes,
David Darling