TC with Letters
September 3, 2017
By Watheq Al-Bnosha
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.
- $ 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.
- $B$ would be by itself because it has two holes on it and there is no other letter with two holes.
- $C \simeq E,F,G,H,I,J,K,L,M,N,S,T,U,V,W,X,Y,Z.$ 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.
- $ 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.
- $C \approx L, N, S, U, V, G, Z, W, \, and M.$ We do not have to choose a point with these letters because they are one piece only.
- $D \approx O.$
- $J \approx T, F, Y, \, and E.$
- $K \approx X$. Each one will be four pieces when we choose a point in the middle and cut it.
- $P \approx Q.$ Both have circles and a leg.
- $B, R,$ and $A$ are unique. These three letters are not going to have equal pieces when cutting them.
I think that part of the problem with the LaTeX here is that you wrote it in Word or in some other place that introduces format. Write directly in the blog field, it will look better and LaTeX will be displayed corrected. Be sure you do that in your next post.
ReplyDeleteHere, the font that the post is displayed, we will not have any problem with the Homotopy, but this font becomes complicated to look when we think about Homeomorphism because it looks like it is Times News Roman and when we remove a point from any one of the nodes we will have many segments. Also, in order for two things to be homeomorphism to each other, we should be able to map each segment with each other and should have same number of holes.
ReplyDelete