I have found that the topics of infinity and the fourth dimension really do *always* pique students’ interest. When I had a few moments left at the end of a class period, I could sometimes casually remark about one of these topics, and I immediately had the attention of the entire classroom.

One of my projects has been writing an introductory book on polyhedra, and I’m in the middle of a proposal to have the book published. A draft chapter was on the fourth spatial dimension, so I thought I’d share it here. As much as I love the topic, I rather surprised myself by looking at my blog index and finding I’d never talked about it before….

What follows is the draft chapter, slightly edited as a stand-alone series. Enjoy!

“What is the fourth dimension?

No, it’s not time.

Well, maybe it is if you’re studying physics. Even then, we have certain intuitive ideas about how time works.

For example, if I imagine that it’s 9:17 a.m. in San Francisco, then I know *what time it is* in any other city in the world. In New York City, it must be 12:17 p.m., since I add three hours to convert to Eastern Standard Time.

This assumption is just fine for getting along in daily life — and as far as most people are concerned, this way of thinking about time is *right.* But in fact, it works only because in our daily lives, we move around fairly slowly — at least compared to the speed of light.

To understand what happens when particles do move close to the speed of light, you need to study *special relativity* — and here, our ordinary intuitions about time are no longer valid.

But we’re interested in a fourth *spatial* dimension. How is this possible? Where is it? We are so used to living in a three-dimensional world, the idea of a fourth spatial dimension seems rather fantastic.

In 1884, Edwin Abbott’s delightful novella *Flatland* was published (Abbott, Edwin A. *Flatland: A Romance in Many Dimensions.* New York: Dover Thrift Edition. 1992). The protagonist was none other than A Square, an inquisitive four-sided being living in a purely two-dimensional world called *Flatland.*

He was chosen as the Flatlander to receive a visit from a Sphere on the eve of the Third Millenium. This was an unnerving visit for A Square, since the Sphere kept suggesting he consider the direction upward, but there was no upward for A Square. There was North, South, East, and West, but A Square just couldn’t fathom this direction, “upward.”

Finally, the Sphere lifted A Square out of his two-dimensional world to show him the glory of Space. Quite a revelation!

We are in the same predicament as A Square when it comes to contemplating a fourth spatial dimension. Yes, we can look forward, backward, to our left and right, up and down, but nowhere else. It doesn’t seem that there is enough room for a fourth dimension. Where would it be?

In a typical high school geometry class, you would likely have been introduced to points, lines, and planes — and perhaps were told that points are 0-dimensional, lines are 1-dimensional, planes are 2-dimensional, and that all objects of these types live in a 3-dimensional space. But while there were infinitely many points, lines, and planes, there was only *one* 3-dimensional space.

And while we do not encounter a fourth spatial dimension on a daily basis, we don’t actually encounter points, lines, or planes, either. Can we actually see a point if it has no length? How could we possibly see a line if it has no width? It would be invisible. These geometrical ideas are in fact mathematical abstractions — and once we enter the world of mathematical abstraction, our universe becomes almost unimaginably vast. A. R. Forsyth wrote almost one hundred years ago (A. R. Forsyth, *Geometry of Four Dimensions, *Cambridge University Press, New York, 1930, p. vii.):

Mathematically, there is no impassable bar against adventure into spaces of more than three dimensions of experience.

To get a handle on how to imagine the fourth dimension, we’ll look at one of the most popular and well-know denizens of that rarefied world — the hypercube, also known as a tesseract.”

What follows is an extended example of “thinking by analogy,” which would make this post much longer than I usually allow myself. So that’s where we’ll start next week!

Separate cells of the 5-cell (pentachoron).

Separate cells of the 8-cell (tesseract).

Separate cells of the 16-cell (hexadecachoron).

Separate cells of the 24-cell (icositetrachoron).

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