Grassmann graph

A Grassmann graph is a way to draw pictures that help us understand how different things relate to each other. It's kind of like a map, except instead of showing us how to get from one place to another, it shows us how different types of spaces and objects are connected.

Imagine you have a bunch of different types of spaces - maybe some are really big and some are really small, maybe some are flat and some are curved. Now imagine you want to see how these spaces relate to each other. You could try drawing pictures of each space and how they're connected, but that could get really confusing really fast.

That's where the Grassmann graph comes in. Instead of drawing pictures of each individual space, we draw lines and dots that represent whole families of spaces. The dots are called "vertices," and they represent something called "Grassmannians" - collections of subspaces within a bigger space. The lines connecting the vertices represent "adjacencies," which tell us which Grassmannians are related to which other ones.

It might sound complicated, but here's an example to help make it clearer. Imagine you have a big room with a bunch of furniture in it. Some of the pieces of furniture are chairs, some are tables, and some are maybe even big couches. Now imagine you want to represent all the different ways you can arrange these pieces of furniture in the room. You could draw pictures of each individual arrangement, but that would be a lot of pictures. Instead, you could draw a Grassmann graph with three vertices - one for chairs, one for tables, and one for couches. The lines between them would represent different combinations of these furniture types, like "a table and two chairs" or "a couch and three tables."

So in short, a Grassmann graph is a way of drawing pictures that help us understand how different types of mathematical spaces and objects are connected to each other. It's like a map, but for mathematicians!