Permutoassociahedron

Okay, so imagine you have a bunch of really cool shapes called polygons. You can draw these polygons on paper and color them in any way you like. Now, let's say you have three of these polygons and you want to put them together in a cool way so that they make a bigger shape.

The permutoassociahedron is a really fancy name for a cool way you can put polygons together. It's like a puzzle, but instead of putting the pieces together any old way, you have to follow some rules.

To build a permutoassociahedron, you have to follow these three important rules:

1. You can only put polygons together if they have the same number of sides.

2. You can only join polygons together along their sides (the straight lines that make up their edges).

3. Each time you join two polygons together, you have to create a new polygon that has the same number of sides as the two polygons you joined.

So, let's say we have three polygons: one with three sides (a triangle), one with four sides (a square), and one with five sides (a pentagon).

First, we can join the triangle and the square together along one of their sides. This creates a new polygon with seven sides (because the triangle had three sides and the square had four sides).

Next, we take the new polygon with seven sides and join it to the pentagon along another side. This creates yet another new polygon with nine sides (because the seven-sided polygon and the pentagon both had odd numbers of sides).

So, we've managed to put these three polygons together in a cool way following the rules of the permutoassociahedron.

In real life, permutoassociahedrons can have hundreds or even thousands of polygons all joined together in cool and complicated ways. Mathematicians study and create these shapes to help understand different mathematical concepts. Cool, right?!