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Occasionally, an article about the Fourth Dimension pops up on Hacker News. For instance, The Fourth dimension: Toward a geometry of higher reality (1984) submitted February 2022.

In the spirit of the classic Flatland, such articles usually try to give an intuitive sense of higher-dimensional space through 2D or 3D analogy, eschewing a more mathematical approach. Here I’ll try to do the opposite. I’ll use a little math to show something I find very unintuitive about 4D that rarely comes up: that an object in 4D can spin around two perpendicular axes at the same time.

Here’s how I might try to explain it intuitively: imagine a car wheel spinning about its axle. In 4D you could ram a second axle long-ways through the tire, start spinning the wheel around that axis, and still drive the car smoothly down the road.

This analogy is not great – we’ll see why at the end – so let’s step through the dimensions and do some math to sort this out.

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