the L1 unit ball is a diamond shape and the L2 unit ball is a circle

Why does a ball look different in various dimensions?

Imagine you're playing with clay. In 1D, you stretch a piece of clay into a line segment. In 2D, you flatten it into a disk. In 3D, you mold it into a sphere. What happens when you try to play with clay in 4D?

As you increase the dimension, the shape of the ball stretches out more. In 1D, it's a line. In 2D, it's a disk. In 3D, it's a sphere. In 4D, it becomes a hyperball, which looks like a diamond when projected into 3D space.

Example

If you tried to create a 4D ball with clay in 3D space, you'd get a shape that looks like a diamond when viewed from the side.

Remember this

A ball's shape changes with dimensions because it stretches differently in each dimension. In 4D, a ball's projection in 3D looks like a diamond.

Text adapted from Wikipedia, licensed under CC BY-SA 4.0.

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