![]() The 3D rotation is a simple extension of the 2D rotation, as long as we choose the axis of rotation to lie in the 3rd-dimension (the z-direction). The fixed point at the origin becomes a fixed line along the z-axis. The 3D rotation The 3D rotation is not much more complicated. It should end up as (0, 1): going from pointing right along the x-axis to pointing up along the y. You can check your intuition by thinking about rotating a point (1, 0) by ninety degrees counter-clockwise. It can be written as a 2x2 matrix that transforms point (x, y) to point (x’, y’). The 2D rotation describes how a points on a plane change position when you rotate around the origin. The 2D rotation The familiar 2D rotation in the xy plane. 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. 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. Here’s how I might try to explain it intuitively: imagine a car wheel spinning about its axle. 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. 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. For instance, The Fourth dimension: Toward a geometry of higher reality (1984) submitted February 2022. Occasionally, an article about the Fourth Dimension pops up on Hacker News.
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