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.

Related concepts

the Lp norm ball shape changes as p goes from 1 to 2 to infinity

How does the shape of a ball change as we measure distance differently?

the volume of a unit ball approaches zero as dimensions increase

How does a coastline's length change with different measuring sticks?

Area of a circle

Why does a pizza slice get smaller as you cut it into more pieces?

Norm (mathematics)

L∞ norm equals max absolute value

Fundamental group

How can we measure the "holes" in shapes?

L1 vs L2 regularization: L1 gives sparsity (feature selection), L2 gives small weights

L1 regularization: L1 = L2 + sparsity; L2 regularization: L2 = L1 + small weights

Swipe through 100 ML concepts daily

Open Pocket Polymath