For the final blog post, I was not sure of what I wanted to investigate and research more. That is, until I came about the Brouwer Fixed Point Theorem. The name caught my attention and I had not heard of it, so it was a perfect time to learn!

Suppose two disks of the same size, one red and one blue (picture below), that are placed so that the red disk is exactly on the top of the blue disk. Then, we can take the red disc and stretch, shrink, rotate, fold it in any way as long as it is not cut or torn. Then, placing it back on top of the blue disk so nothing hangs on. Brouwer's Theorem states that there must be at least ONE point, a fixed point, on the red disk that is exactly above its initial position on the blue disk.

Suppose two disks of the same size, one red and one blue (picture below), that are placed so that the red disk is exactly on the top of the blue disk. Then, we can take the red disc and stretch, shrink, rotate, fold it in any way as long as it is not cut or torn. Then, placing it back on top of the blue disk so nothing hangs on. Brouwer's Theorem states that there must be at least ONE point, a fixed point, on the red disk that is exactly above its initial position on the blue disk.

**WHAT?! HOW CAN THAT BE?!**I'm sure I am not the only one who was thinking that, so I decided to investigate further.

The best explanation that I found to make sense in my mind was a reference to maps and the existence of a You Are Here point on maps typically found in malls. However, not many people associate the Brouwer Fixed Point Theorem with maps. In effect, the function that takes place in the mall as the input and returns the image on the map as an output is continuous, so it must have a fixed point, meaning that where you are located is the fixed point that is indicated by a

*You Are Here*point. To help understand this better, consider the existence of a display similar to this, but outside. Our location cannot be accurately represented on a map, so there would be no

*You Are Here*point. A map outside of the mall would violate the condition that the function is mapped to itself, which results in not having the fixed point that is necessary.

This theorem I found to be very interesting. However, there is still so much more that can be learned about it. I have gained a brief understanding of the theorem itself and what it states.