Posts Tagged ‘ mathematics ’

Separating Fact from Fiction (Essay)

Date Written:

2012

Abstract:

An short exploration of the life and works of Hypatia of Alexandria through a first-hand account. Written for a History of Mathematics assignment at the University of Leeds.

Full Text:

Separating Fact from Fiction

The Mathematics of Grace

Today while I was studying, God spoke something to me. Often, God helps me understand things about Him by using mathematics. My brain is just wired that way. This is what He gave me today.

First, I’ll explain the math background. Suppose f is a function of time. So given some time, t, you can input it into f and get a result: f(t)=x Functions can have various properties, but we are going to say that this function is “memoryless.” That means for any time t1 that occurs before t2, f(t1) has no influence on f(t2).

Ok, enough about the pure math. How does that relate to grace? God’s grace works in mysterious and wonderful ways. As humans, we can offer forgiveness, but often that forgiveness is influenced by the past. God’s forgiveness is not the same. Because of grace, when we ask God for forgiveness, He doesn’t look back at the long list of our sins and make a final judgement. Through the awesome power of His grace, His forgiveness is always available if we just ask.

His grace is memoryless.

Thesis Draft 5

From here on, each draft of my thesis will be constructed using PCTeX.

Thesis Draft 5

Thesis Draft 4 (PCTeX)

Dr. Sharma asked me to rewrite what I had so far for my thesis using PCTeX. PCTeX is a software that uses TeX, a mathematical language, to produce documents. I have been learning the software and language as I go, so these drafts are truly works in progress.

Thesis Draft 4 (PCTeX)

Thesis Draft 3

This draft reflects the new organization of my thesis and main topics we aim to cover. Some of the topics have already been discussed, but I did not have time to add them to this draft of the paper. See blog posts for up-to-date information on what topics I have already researched.

Thesis Draft 3

Isomers of Alkanes

After studying C6H14, we became interested in the structure of carbon hydrogen compounds in general. I discovered, later, that molecules that are made up of only carbon and hydrogen atoms that contain no cycles are called alkanes. Isomers are compounds that have the same number of carbon and hydrogen atoms, but have different structures. Since hydrogen atoms do not add to the basic structure of alkanes, it is sufficient to study the underlying structure of the carbon atoms.

In connection to graph theory, studying the structure of the carbon atoms in alkane isomers is equivalent to studying the structure of nonisomorphic trees with no vertex of degree greater than four. The following graphs (representing trees, or the structure of carbon atoms in alkanes) are grouped vertically by the number of vertices.

The result that we began to notice while comparing the groups of isomers was that the beginning of a Fibonacci sequence appeared: 1, 1, 2, 3, 5.

Seeing that this pattern was developing, we tried to think of a reason why. But once I tested C7H16, the pattern fell apart. Instead of finding 8 isomers, as a Fibonacci sequence would produce, there are 9 (see below).

Isomers of C6H14

In researching the applications of graph theory, Dr. Sharma asked me to consider the graphs of isomers of Carbon-Hydrogen compounds. We derived the formula for possible molecules. If “c” is the number of Carbon atoms and “h” is the number of Hydrogen atoms, then h=2c+2

So if you are given 6 Carbon atoms, it is possible to form a C6H14 molecule. With these fixed number of atoms, there are several possible structures of the graphs. These graphs are called isomers.

In the graphs of Carbon-Hydrogen compounds, each Carbon is connected to 4 other atoms, but each Hydrogen can only connect to one atom. Therefore, since the graphs of molecules must be connected, Hydrogen atoms only connect to Carbon atoms. So the true underlying structure of Carbon-Hydrogen compounds comes from the construction of the Carbon atoms.

Therefore, drawing all non-isomorphic trees on 6 vertices, where the highest degree of any vertex is 4, creates the underlying structure of all possible C6H14 isomers. Below are pictures of the possible isomer constructions. The true construction of each isomer can be formed by attaching a Hydrogen atom to each Carbon atom until the degree of each Carbon atom is 4.

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