Coxeter complex
Imagine you have a bunch of mirrors that you can arrange in different shapes. Now, let's say you want to play a game with these mirrors. The goal of the game is to create beautiful patterns by reflecting light off the mirrors.
To start the game, you arrange the mirrors in a certain shape, let's say a triangle. Then, you shine a light on one of the mirrors. This light will bounce off the mirror and hit another mirror, reflecting again and again until it stops. As it bounces off each mirror, it creates a path of light.
Now, let's make things more interesting. Instead of just using flat mirrors, what if you could use special mirrors that are actually made out of triangles? These special mirrors are called Coxeter mirrors, named after a mathematician called H.S.M Coxeter.
Using these Coxeter mirrors, you can create something called a Coxeter complex. The Coxeter complex is like a big puzzle made out of different shapes, where each shape is a mirror. The goal is to arrange the mirrors in a way that creates the most beautiful pattern when the light bounces off them.
But here's the fun part. You can make all sorts of different Coxeter complexes depending on how you arrange the mirrors. You can make a complex with just one mirror, or with many mirrors arranged in a grid. You can even make more complex shapes like stars or polygons.
When you shine a light on a Coxeter complex, the light will bounce off the mirrors in all sorts of directions and create a beautiful, symmetrical pattern. This pattern is called a reflection pattern because the light reflects off the mirrors and forms a repeating design.
Coxeter complexes may seem like a game, but they are actually really important in math. Mathematicians study Coxeter complexes to understand symmetry and geometry. These complexes can help us understand how shapes fit together and how they can create amazing patterns.
So, next time you see a triangle or a star-shaped pattern, remember that it could be a reflection pattern from a Coxeter complex, created by bouncing light off special mirrors made of triangles.
To start the game, you arrange the mirrors in a certain shape, let's say a triangle. Then, you shine a light on one of the mirrors. This light will bounce off the mirror and hit another mirror, reflecting again and again until it stops. As it bounces off each mirror, it creates a path of light.
Now, let's make things more interesting. Instead of just using flat mirrors, what if you could use special mirrors that are actually made out of triangles? These special mirrors are called Coxeter mirrors, named after a mathematician called H.S.M Coxeter.
Using these Coxeter mirrors, you can create something called a Coxeter complex. The Coxeter complex is like a big puzzle made out of different shapes, where each shape is a mirror. The goal is to arrange the mirrors in a way that creates the most beautiful pattern when the light bounces off them.
But here's the fun part. You can make all sorts of different Coxeter complexes depending on how you arrange the mirrors. You can make a complex with just one mirror, or with many mirrors arranged in a grid. You can even make more complex shapes like stars or polygons.
When you shine a light on a Coxeter complex, the light will bounce off the mirrors in all sorts of directions and create a beautiful, symmetrical pattern. This pattern is called a reflection pattern because the light reflects off the mirrors and forms a repeating design.
Coxeter complexes may seem like a game, but they are actually really important in math. Mathematicians study Coxeter complexes to understand symmetry and geometry. These complexes can help us understand how shapes fit together and how they can create amazing patterns.
So, next time you see a triangle or a star-shaped pattern, remember that it could be a reflection pattern from a Coxeter complex, created by bouncing light off special mirrors made of triangles.
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