# Shawn Huesman

### From www.norsemathology.org

## Contents |

# Early Life

I was born and raised in Bellevue, Kentucky and I have lived on the same street nearly my whole life in 4 different houses. I went to Grandview Elementary School and Bellevue High School. Some part of me wanted to be an architect growing up which I believe was a result from spending a lot of time playing with Legos and playing the video game Minecraft. Another part of me wanted to do nothing but play video games all day every day.

## Early Thoughts on Mathematics

In elementary school, math was considered a class among the others - science, 'gym', reading, social studies and art. I feared 'gym' the most, always thought I was going to die while running for some reason. After that was science, I thought it would increase in complexity super fast and I would not be able to catch up - I think this was a result of media's depiction of chemistry. But math felt *plain* to me. It was not easy for me, but it also was not something I thought was very hard - or that I thought about much at all. *Math was another set of instructions given by teachers that I had to copy * so I would not get punished. I did not want to have to stay after school, miss recess, not be able to go to another class, or any number of possibilities.

One thing I did often like, however, was drawing. It felt like limitless possibility. I could make anything I wanted on my page in any way I wanted. I remember looking forward to days in which we could draw anything in art. It made my loss of autonomy a bit more tolerable.

## Thoughts on Mathematics Through High School

Mathematics grew a reputation in middle school and high school. It was no longer the ugly ignored step child like it was in elementary school, * math was a living, fire-breathing, dragon. * It was intimidating, but if you 'knew it' you were respected. You were deemed 'smart' by your peers if you 'understood' math, if others believed you were capable of slaying the dragon. Also, it was clear that many faculty were content feeding the schools' pet dragon and sharpening its' teeth by echoing its' importance without reason. * You need to understand math to get a good job and go to college *

While this dragon was amuck, I discovered something incredible as I moved up through middle school and to high school: after I got finished with my work, I could do anything I wanted as other people were working. That meant I could draw - or think about anything I wanted to. It meant I had a chance at autonomy during school. This was my motivator to understand things and complete assignments as fast as possible. And it worked well. I remember getting all awards for each subject I was in each year, getting the highest subject test scores, and valedictorian. I put off the appearance of slaying the dragon, and all of its' children.

The reality was, however, that I convinced the dragon to play dead for me. I never truly understood any of the actual math or science, or their implications on the universe. I saw the patterns and played the game to get the autonomy I wanted. I did not feel bad at all about this, it made me upset that I had to jump through others' hoops just to do and think what I wanted without punishment.

## My High School English Teacher: Mr. Rauckhorst

My English teacher, Mr. Rauckhorst, showed me that anything can be magical when seen from another perspective - including learning. Whenever you were around him, it felt like he was trying to teach you something - without even speaking. He encouraged me to look at the world around me and break it apart, question what I thought about it, and when I thought I had a decent answer, to question even more. Everyone I knew loved him. He questioned authority and played by his own rulebook, something that is not heavily valued in society, which is a major reason he was fired/requested to leave during my last year at high school.

so much depends

upon

a red wheel

barrow

glazed with rain

water

beside the white

chickens

Mr. Rauckhorst encouraged us to participate in activities that scared us. I went to the Hugh O' Brian Leadership program because he said it could be a great experience - and it was. It was such a great experience I worked the hardest I could to get into another program similar to it - the Kentucky Governors' Scholar Program, which I believe ultimately led me to pursue college. He was a huge supporter in my math journey because he encouraged me to look for meaning and magic even in what seems to be painfully ordinary.

A vivid example I remember is encouraging us to critically think about the poem "The Red Wheelbarrow" by William Carlos Williams.

We were encouraged to write down everything we thought about this poem - from the way it looks to the way it feels to how the stanzas are shaped to derive meaning. This poem can, of course, be considered subjective. But so can * everything else. * Mr. Rauckhorst encouraged us to critically think about subjectivity - to challenge it, to be comfortable with it.

# Thoughts on Mathematics In College

My first semester at NKU, I took MAT 109: Algebra for College Students. It was hard for me - staying up until 3:00AM in the morning many nights hard. I pattern-matched in high school which means I did not understand most of what was going on and that I actually had to understand the concept in order to move fast enough to get through the course. I can remember that in this course, I actually learned how the cartesian graph worked. I learned the relationship between the x and y axis and how equations could explain that relationship. In all of the years showing and talking about the patterns of this graph in high school, I never understood anything past the algorithms teachers told us such as *'move twice across and once up to get to the spot on the line then write it down on your paper'*

During my first semester, mathematics felt fruitless - especially compared to the other field I was studying: Computer Science. I was taking INF 120: Elementary Programming. It was my first experience in programming and it felt easy for me - all the rules immediately made sense and had application, unlike my math class. But I still kept spirit that Math would make more sense soon.

The summer after my first semester (in the Spring), I did research with my physics professor because he was offering, and I learned from experience (and Mr. Rauckhorst) to take opportunities outside of my comfort zone. The research was concerning Josephson Junction Weak Links for Superconducting materials. I wish I could tell you what that meant. The majority of my 'research' was watching people do things in a physics lab and being amazed at the human capacity to do what was going on in this lab, which I understood very little of. I do remember, though, doing a mapping process of a 'probe' that I believed would go into liquid nitrogen which involved testing which point on one end created an electric charge to the other end. What I found interesting about this was that only one point on each side corresponded (electricity flowed through) to only one point on the other end. I remember testing every point just to be sure. This helped me understand what a one-to-one mapping was. Also, from this experience, I was exposed to a lot of physics equations that involved very intense math. I understood nearly none of it, but I was able to see that the intangible qualities of mathematics helped these physicists do work and create interesting physical things.

During my second semester, I took MAT 119 - Pre-Calculus Mathematics and another programming class. I was able to apply a lot of what I learned in my algebra class and there were a few mentions of possible applications of what we learned. It felt like I was getting closer to understanding what math meant, but I still felt like an observer of mathematics.

In my third semester, I took MAT 129 - Calculus I. I can remember being nervous my first day in class. There was a lot of build-up to this class. From how people spoke about calculus, I prepared that it would be the most difficult class I would take in my life. This was not the case. In this class I learned that the symbols of math were just to communicate ideas. The meaning is the meaning we place on them, and that they could be anything. Even more than that, I understood that learning math was first and foremost about the idea and not what is on the paper or whiteboard or stone tablet. Math started to feel *smooth* and *interesting.*

In my fourth semester, I took MAT 229 - Calculus II. Calculus II was magical to me because it was a large collection of extremely interesting math ideas such as integration techniques, summations and series, and polar coordinates. It felt as if all the unfruitful work of algebra and precalculus had paid off. But, in those moments and now, I can understand how what I learned in algebra is similarly beautiful.

I took Dr. Chris Christensen for Calculus I and Calculus II and I am extremely thankful for him. He taught us in a steady pace. It made me feel that math is not going anywhere and that the focus was on understanding the meaning behind the math, not to climb an imaginary ladder of success to 'know' the patterns of math so you can prove you are more worthy than another. He also went over concepts multiple times and spoke of them with the same slow and steady importance each time. This gave me comfort in that it was okay not to 'get it' the first time.

## Intersection of Mathematics and Computer Science: Dr. Ward

On the Computer Science side of my fourth semester, I was taking CSC 364 - Data Structures and Algorithms with Dr. Jeffrey Ward. This class was a blatant and beautiful intersection of mathematics and computer science. Dr. Ward showed us how series, exponents, logarithms, and algebra applied to how we think about computation. I suspect he put a bit more emphasis on the math and often used the last minutes of class to expand on the math of algorithms and algorithmic time complexity.

Dr. Ward taught a beautiful, core part of computer science: algorithms, and highlighted the mathematic bones. Paul Lockhart described in "A Mathematician's Lament" describes math as a bird; and the ongoing process of hiding, or torturing the bird in K-12 education. In Dr. Ward's class, birds flew in the classroom, landed on your desk, and sang you a song.

I was fortunate to have the opportunity to do research with Dr. Ward during my third summer. We looked at video games and board games to see how they worked - and thought about possibilities, implementations, and implications of artificial intelligence in video games. Once again, mathematics was a highlight. I remember that we played the board game '7 Wonders Duel' together and talked about how we could use math to make an artificial intelligence program to play the game - and possible ways to make the game more complex. We also played a similar online game named 'Seventee' ^{1} that had an AI, but it was easily beatable.

Undoubtedly, Dr. Ward was passionate about games nearly as much as he was passionate about teaching. It is a rare moment you get to experience someone talk about something that they believe is as exciting and important as life itself. I was able to experience that during my research with Dr. Ward. He spoke about games and game systems with a near tangible intensity. This reminded me of how I experienced games when I was younger. But now, I was able to understand that behind the curtain, game systems are math, and games are about learning - and especially about learning math concepts.

He helped me see mathematics as a part of life; a creative outlet much like how I viewed paper and pencil in my K-12 education.

# Recursion & Self-Reference

Self-reference is used to denote an entity or idea that refers to itself whereas recursion is typically referred to as a computational problem solving tool.

## Recursion: Problem Solving Technique

Recursion is typically used to describe a scenario when solving problems with computer algorithms in which a problem can be broken into smaller sub-problems to be solved using the same set of rules by referencing itself. The word comes from the Latin root * recurrere * which means to 'run back.' This evolved into the Latin root * recurs-* which means 'returned.'

Consider the Fibonacci Sequence. The sequence begins with the values 0 and 1 and each consecutive value is the result of adding the last two values before it.

It can be represented as the following self-referencing equation:

And, even further it can be defined as the following *recurrence relation*:

This show the necessary conditions for the function to 'begin' and then continue infinitely by referencing itself. This is similar to 'proof by induction.' A proof by induction is built on the logic that * we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the step).* ^{[a]}

We can use this recurrence relation to create code. Use T(0) and T(1) as 'base cases' and then call the function repeatedly until it can break down into the base cases. For example, here is some Python code:

def F(n):

if n == 0: return 0

if n == 1: return 1

else: return F(n-1) + F(n-2)

This is a working solution, but it is not very efficient. Lets say we have n=3000, how many times do we calculate the 3rd element? Every single time after the base cases until the end of the solution. The solution to this is called Dynamic Programming which involves storing previous values so they do not need to be computed each time. This means the solution is no longer recursive, but thinking about recursion and mathematical induction are often extremely helpful in coming up with dynamic programming solutions. Recurrence relations are helpful in coming up with the time complexity / computational cost. Divide-and-conquer algorithms are another example of using recursion to come up with programmable solutions. These algorithms work by dividing problems into subproblems and are a subset of dynamic programming algorithms.

## Self-Reference in Mathematic History

Mathematic induction as a valid proof technique has shaped the way we view mathematics. However, it took a while to be considered proper mathematics as there was resistance to recursive functions as being considered a part of mathematics. This resistance was led primarily by Bertrand Russell, author of *Principia Mathematica* ^{[b]}. In Mathematics and the Metaphysicians^{2}, Russell writes:

One of the chief triumphs of modern mathematics consists in having discovered what mathematics really is, ... All pure mathematics – Arithmetic, Analysis, and Geometry– is built by combinations of the primitive ideas of logic

Russel had a monistic ^{3} philosophy of mathematics (as apposed to dualist or pluralist). This meant he believed that any 'abstract particulars' were not actual mathematics ^{[c]}. This included recursive functions. So, recursion held no place in his book, *Principia Mathematica*, which was aimed to be something of a textbook on the foundations of mathematics.

However, many believed that Russel logic was wrong and that mathematics was *incomplete* without these abstract structures, such as sets, geometric figures, and classes (which all involve recursion in some way). It was not until Kurt Gödel proved that the understanding of mathematics as presented in *Principia Mathematicia* was incomplete by his Incompleteness Theorem.

Gödel's Incompleteness Theorem(s), as paraphrased in the book *Gödel, Escher, Bach*^{[d]} is:

All consistent axiomatic formulations of number theory include undecidable propositions.

Without going into much detail, this means that, in number theory, for example, all theories have a sentence that roughly translates to "This theory is consistent" since statements in number theory are references of number theory itself (in which sentences are translated into numbers). This is a paradox; a self-referential and self-contradictory statement.

Consider the following 'Liar's Paradox':

I am lying.

Or another statement of a similar vein:

This statement is not true.

This had ground-breaking implications in mathematics. For instance, consider this snippet from *Gödel, Escher, Bach*:

Gödel's Theorem has an electrifying effect upon logicians, mathematicians, and philosophers interested in the foundations of mathematics, for it showed that no fixed system, no matter how complicated could represent the complexity of whole numbers: 0, 1, 2, 3, . . .

## Fractals

"beautiful, damn hard, increasingly useful. That's fractals." - Benoit Mandelbrot^{4}

This is a playful description of fractals but what * are * they? Mandelbrot, the mathematician who first used the term (in 1975)^{[e]}, defines them as *
"a shape made of parts similar to the whole in some way." *

You may have seen fractals before in nature.

### The Koch Snowflake

What do fractals have to do with mathematics? Let's look at the creation of a fractal: the Koch Snowflake.

Step 1: Draw an equilateral triangle.

Step 2: Divide the length of each side by 3.

Step 3: Draw an equilateral triangle using the side length of the middle of the 3 divided side lengths as the base.

Step 4: Repeat steps 2-3ad infinitum

We can think of a few things about this fractal.

For one, can we measure the perimeter of the Koch snowflake as a whole? Sure. We can find the perimeter by multiplying the number of sides by the length of each side.

For each iteration, we multiply the number of sides in the Koch snowflake by 4. Our starting condition, or step, is adding the first triangle which has 3 sides. Therefore, we can define N number of sides in the ath iteration of the fractal process as:

As we 'move' through the fractal, the length of each added triangle side is 1/3 the length of the side of the triangle before it. Again, we start with a triangle, so we can call that triangle's length 'x'. Thus we can define side length S for the ath iteration as:

Now the perimeter... and remember we are looking for the total perimeter of the Koch snowflake fractal:

(please let me know of any math mistakes)

This is a geometric series in which increases by 4/3rds on each iteration. In geometric series notation of the defined geometric constant 'r', we know that since |r| > 1, that this series diverges. This means the koch snowflake perimeter is infinite.

Alright, we could convince ourselves of that result not being surprising. Let's see what happens if we try to calculate the area.

The area for any equilateral triangle with side length 's' is (s^2 * sqrt(3)) / 4. Each new triangle in the koch snowflake generation starts out with a side length s/3 then s/3*3 then s/3*3*3, thus giving the closed equation s/3^n for the nth iteration. Also, using the logic for calculating the number of sides, we can find that the number of triangles per iteration is 3 * 4^(n-1). We can put these facts together to find the area:

(again, please let me know of math or logic errors)

What does this mean? The final result indicates that the area of the whole koch snowflake fractal is finite. Specifically, it is equal to 8/5 times the area of the starting triangle.

It is interesting that this 'structure' can have a infinite perimeter but a finite area.

Another note of interest is how much the structure depends upon a its' starting triangle. This finite entity determines the infinite structure of the fractal and has a core relationship to its' properties, or at least to its' area.

### The Mandelbrot Set

Do all fractals have finite beginnings?

Lets look at another fractal referred to as the Mandelbrot Set.

The formal mathematical definition of the Mandelbrot set is as follows:

*The set of complex numbers c for which the function f(z) = z^2 + c does not diverge when iterated from z=0 *^{5}

I do not have the understanding of complex numbers yet as to attempt to do any math on the area or perimeter of this shape, but we could analyze the beginning of this fractal.

If we look at the Mandelbrot set in 2-dimensional Euclidean space, we can see that it starts out kind of like a line.

But if we look at it in 3-dimensional space, we see that the 'line' has some unique properties.

This 'line' is a function described as *the* logistic map. The function is outlined as follows:

This function has roots in chaos theory, and it can also be used to describe grown of animal populations such as rabbits ^{6}.

And, as you can see, the results of this function are caused by a finite value. I am unsure, however, how the properties of that starting finite value affect the properties of the logistic graph and thus affect the properties of the Mandelbrot set fractal.

I encourage you to look at this fun website where you can zoom into the Mandelbrot set.

# Citations

Linked Citations:

[1], sevenee.mattle.online/.

[2], Mathematics and the Metaphysicians - Mysticism and Logic - Bertrand Russell, users.drew.edu/~jlenz/br-ml-ch5.html.

[3], “Monism.” Wikipedia, Wikimedia Foundation, 16 Mar. 2021, en.wikipedia.org/wiki/Monism.

[4], Mandelbrot, Benoit. "24/7 Lecture on Fractals". 2006 Ig Nobel Awards. Improbable Research.

[5] Wikipedia contributors. (2021, April 18). Mandelbrot set. Wikipedia. en.wikipedia.org/wiki/Mandelbrot_set

[6] Introduction to Chaos and the Logistic Map

Unlinked Citations:

[a] Concrete Mathematics, by Ronald L. Graham et al., Addison Wesley, 2011, pp. 3–3.

[b] Whitehead, Alfred North. Principia Mathematica. Cambridge University Press, 1997.

[c] Landini, Gregory. “Gödel’s Incompleteness Platonism Exempts Principia Mathematica.”

[d] Hofstadter, Douglas R. Gödel, Escher, Bach: an Eternal Golden Braid. Basic Books, 1999.

[e] Mandelbrot, B. B. (1982). The Fractal Geometry of Nature (2nd prt. ed.). Times Books.