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When I got my first teaching job, at SUNY Geneseo, I began working out these connections in my lectures for a course on higher geometry. Philosophers and mystics meditate upon it; physicists and mathematicians calculate with it. One of these higher dimensions is time, another higher dimension is the direction m which space is curved, and still another higher dimension may lead toward some utterly different universes existing parallel to our own. By combining the three mutually perpendicular types of motion, one can trace out any possible curve in our space. This fact is illustrated in two dimensions by a toy that was popular a few years ago, the Etch-A-Sketch.

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When you measure a quantum system you are said to “rotate the state vector,” an abstract line of precisely defined length and orientation that represents the system’s state in a set of coordinate systems that constitute a Hilbert space. Or are they no more than convenient fictions used by physicists to simplify their calculations?

It is with such deep ontological questions that Rucker concerns himself throughout this book and especially in the final chapters.

Copyright © 1984, 2014 by Rudy Rucker 2nd edition, entitled The Fourth Dimension: Toward A Geometry of Higher Reality, Dover Publications, 2014 1st edition, entitled The Fourth Dimension: A Guided Tour of the Higher Universes, Houghton Mifflin, 1984 Reading note: Extended quotes or storylets appear throughout the paperback edition as sidebar material at the edges of the page.

In the online and ebook editions, the sidebar bits are in the main text, set off by horizontal lines.

My Fourth Dimension has been one of my best-selling books, with translations appearing in France, Israel, Italy, Japan, Korea, Germany, the Netherlands, Portugal, and spain.

I’m very pleased that Dover is republishing it in paperback and ebook editions.

By now, good old A Square feels like a friend of mine, or perhaps an eccentric cousin.

Eventually I went so far as to write a novel called Spaceland, featuring a man named Joe Cube.

He begins in our normal world, encounters a four-dimensional being, and travels into higher space. There are happy exceptions, of course, and at least one teacher of mathematics, Lewis Carroll, wrote immortal fantasy fiction.

It was an interesting challenge for me to visualize what he saw there. Eric Temple Bell not only wrote colorfully about mathematics for laymen, but under the pseudonym John Taine he churned out a raft of science fiction novels.

Rucker’s major mathematical interests are transfinite sets (he has a doctorate in mathematical logic from Rutgers) and spaces of higher dimensions.

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