Systoles of surfaces

Okay, so imagine you have a playdough ball, and you smash it down onto a flat surface. The playdough gets squished out in all directions, right? But it has to go somewhere, so it forms little mounds and valleys on the surface of the table.

Now, let's say you want to measure these mounds and valleys. You could use a ruler and measure their height above the table, or you could use a measuring tape and measure how long they are.

But what if you wanted to measure both the height and length of ALL the mounds and valleys on the table at once? That would be pretty tricky, right?

Well, scientists and mathematicians have come up with a way to do just that! They use something called "systoles" to measure the size and shape of the mounds and valleys on a surface.

Imagine you have a piece of paper that you crumple up into a ball. The surface of the ball now has little mounds and valleys all over it, kind of like the playdough on the table. The systoles of this surface are like the shortest possible paths you could draw on the surface, going up and down over the mounds and valleys.

These systoles help us understand how big the mounds and valleys are, how many there are, and how they're arranged on the surface. We can use this information to learn more about the properties of the surface, like how it stretches or bends.

So basically, systoles help us measure and understand the shape and size of the bumps and valleys on a surface, by finding the shortest possible paths over all of them at once. It's like a secret code that helps us unlock the mysteries of surfaces!