Homotopic and Homeomorphic Letters

TC with Letters

September 3, 2017
By Watheq Al-Bnosha

Introduction

Homotopy and Homeomorphism apply to many objects and shapes. In this paper, we are going to apply homotopy and homeomorphism to letters and study which letters are homotopic, homeomorphic, and/ or neither. We are going to put in consideration the way a letter is written on different computers and handwriting.

The English alphabet are $ A, B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z. $

Homotopic Letters

In homotopy, dimensions do not matter. What matters is that we cannot glue or cut the shape of the letters or any other shape in general. But instead, we can compress and stretch the shapes. We are going to define the English alphabet into three sections depending on the holes they have.

1. $ A \simeq D, P, O, R, $ and $Q$. Because all of them have holes and if we compress or stretch them, they’ll look like the same.
2. $B$ would be by itself because it has two holes on it and there is no other letter with two holes.
3. $C \simeq E,F,G,H,I,J,K,L,M,N,S,T,U,V,W,X,Y,Z.$ Because if we compress or stretch them, they’ll look like the same.

Homeomorphic Letters

In homeomorphism, we have only 1-dimension. Also, we cannot compress or stretch letters We are going to use a technique to define the letters that are homeo to each other and the letters that are not homeo. In case there is a letter that is suspicious whether is homeo or not, we are going to choose a point in the letter and cut it. If the letter has the same points as the other one, then it is homeo, if it doesn’t then it is not homeo. We are going to put the English letters into seven sections depending on their dimensions.

1. $ H \approx I$. Because if we choose a point in a letter $H$ and cut it, it will have the same amount of pieces as the letter $I$ when we choose to a point and cut it.  
2. $C \approx L, N, S, U, V, G, Z, W, \, and M.$ We do not have to choose a point with these letters because they are one piece only. 
3. $D \approx O.$ 
4. $J \approx T, F, Y, \, and E.$ 
5. $K \approx X$. Each one will be four pieces when we choose a point in the middle and cut it. 
6. $P \approx Q.$ Both have circles and a leg.  
7. $B, R,$ and $A$ are unique. These three letters are not going to have equal pieces when cutting them.