> Mathematicians often visualize this problem in terms of spheres. You can think of each code word as a high-dimensional point at the center of a sphere. If an error-filled message (when represented as a high-dimensional point) lives inside a given sphere, you know that the code word at the sphere’s center was the intended message. You don’t want these spheres to overlap — otherwise, a received message might be interpreted in more than one way. But the spheres shouldn’t be too far apart, either. Packing the spheres tightly means you can communicate more efficiently.
I went to math prep school for 2 years, attended 12 hours of math class in agebra and analysis per week, which I think proves I've done more math than most people in the general population, and this makes no sense to me. It either lacks introduction required to understand the analogy, or I've become really dumb. I want to understand this based on what the article says, but I can't. I can't represent error-filled messages as high-dimensional points. It's easier for me to imagine what the intersection between 4D spheres would look like in geometry.
I found this for anyone interested in understanding 4D spheres without knowing too much math: https://baileysnyder.com/interactive-4d/4d-spheres/
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