Loewner order

Have you ever played with those Russian dolls where there is a big doll and then a smaller one goes into it and then an even smaller one goes into that one and so on? Well, sometimes mathematicians like to play a similar game called the "Loewner order."

The Loewner order is a way of comparing different shapes, kind of like how you might compare different dolls in a set. But instead of just looking at the size, mathematicians look at the shape of the objects.

Imagine you have two different shapes, let's say a circle and a square. The Loewner order tells you which one is "bigger" or "smaller" in terms of how each shape can be transformed or squished into the other.

For example, if you took a square and slowly squished it into a circle, the Loewner order would say the circle is bigger because the square can fit inside the circle. But if you tried to squish a circle into a square, you would find it impossible because the circle has curved edges and the square has straight edges.

The Loewner order is a way for mathematicians to compare shapes in a more precise way. And just like different dolls in a set, there can be lots of different shapes and sizes to compare!