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2024-10-09 16:00:09

There is a geometric thought experiment that is often used to demonstrate the counterintuitive shape of high-dimensional phenomena. This article is an interactive visual journey into the construct in the thought experiment, and the mathematics behind it.

We start with a 4×44 \times 4 4 × 4 square. There are four blue circles, with a radius of one, packed into the box. One in each corner. At the center of the box is a red circle. The red circle is as large as it can be, without overlapping the blue circles. Drag the slider to add a third dimension. On this and all other 3D diagrams, you can drag the image to rotate it, and scroll to zoom.

When extending the construct to 3D, many things happen. All the circles are now spheres, the red sphere is larger while the blue spheres aren’t, and there are eight spheres while there were only four circles.

The dimensional extension is decomposed into three phases. First, the circles and square turn into spheres and a cube. Next, the central ball grows, and everything else shifts so that it ends in the center. Finally, four new spheres appear, completing the 3D version of the construct.