Frieze Groups

                                                         

Humans love patterns. Finding and understanding patterns is something that has helped us stay alive throughout history. For example, if people go too long without drinking water then they get really sick and can potentially die. This happens over and over and becomes a clear pattern. Then you notice that when people drink an appropriate amount of water, people don't get as sick and don't die as easily. So the solution, also found through seeing a pattern, is to drink enough water. This is a kind of cheesy, over simplified example, I know. The point is that we see patterns all around us and can often use patterns to our advantage whether it be to make a scientific discovery, form a piece of art, or learn something new for ourselves.

In mathematics patterns are a hot topic. We learn about patterns in math clear back in elementary school with addition, subtraction, division, multiplication, shapes, reading, and colors. Recently I have been studying patterns that appear in Group Theory. These patterns are called Frieze Groups. The definition of frieze from Google says, "A frieze is a broad horizontal band of sculpted or painted decoration, especially on a wall near the ceiling." They Also often appear in jewelry. 
A frieze group, similar to a frieze, is a group of symmetries, the set of the geometric transformations that are made with translations and reflections that preserve the pattern. A frieze pattern is a two-dimensional design that repeats infinitely in both directions. There are seven types of frieze patterns, let's take a look at them.


So after throwing all of these facts/definitions at you, why should you care about frieze patterns? Well first off, they form beautiful patterns that show up all over in designs for architecture/furniture, wallpaper, jewelry, and art- who doesn't like to appreciate beautiful things? Second of all, they show up in practical scenarios as well. If you go and examine the tread on your tires, you should find that the patterns there are built from frieze patterns, this helps carry water away from the tires (note that $F_1$ and $F_3$ probably won't appear in this scenario as they would not carry water away from both halves of the tire equally). And thirdly, archeologists will often use frieze patterns to help them classify artifacts that they find, so knowing these patterns can help us connect with people all throughout time. And zoologists can even use frieze patterns to classify animals/ their movements.

Now that we have an idea about why these patterns matter, let's practice recognizing them. Below is a chart that helps me ask the right questions when I am trying to identify a pattern.  


To learn more about how mathematicians view and form frieze patterns, here is a great paper to read on the details involved.

To end, here are a few pictures I took of frieze patterns that I saw around me. Thanks for reading!


   


            


      



       


Work Cited:
Contemporary Abstract Algebra by Joseph A. Gallian, Ninth Edition
https://gifer.com/en/WaUg
https://en.wikipedia.org/wiki/Frieze_group
https://mathstat.slu.edu/escher/index.php/Frieze_Patterns
https://www.maa.org/sites/default/files/images/upload_library/4/vol1/architecture/Math/seven.html
http://mathcentral.uregina.ca/RR/database/RR.09.01/mcdonald1/
https://www.learner.org/series/learning-math-geometry/symmetry/part-c-translation-symmetry-and-frieze-patterns-60-minutes/
https://www.onlinemath4all.com/frieze-patterns-geometry.html
https://careertrend.com/info-8517788-main-tools-used-zoology.html

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The First Isomorphism Theorem Proof

~A proof for my Dad, with help from Ben, my teacher.

Definitions

Isomorphism: 

* To be an isomorphism the function must be a homomorphism and be surjective (everyone has a room) and injective (no one is sharing a room), this means it is a bijection.

Homomorphism:

We say that a function f from G to H is a group homomorphism if f preserves the group structures, which means $f(a\cdot b)=f(a)\times f(b)$ for every a and b in G. 

Surjective:

  • A function f from A to B is surjective if and only if () implies (there exists  such that ).

Injective:

  • A function f from A to B is injective if and only if ( such that ) implies ().

The First Isomorphism Theorem that I prove below:

Let f from G to H be a surjective homomorphism with kernel $K$. Because we know that $f(x)=f(y)$ for any $y\in Kx$ (elements in the same coset of the kernel have the same image under $f$), then we can define a map $\phi:G/K\to H$ by defining $\phi(Kg)=f(g)$. This map $\phi$ is always an isomorphism.


POOF TIME!!!

To prove this theorem we must show that $\phi: G/K\to H$ is injective and surjective, thus it is an isomorphism.

First, let $f:G\to H$ be a surjective homomorphism with kernel $K$. Also, let $\phi:G/K\to H$ be defined for all $gK\in G/K$ by $\phi(gK)=f(g)$.

Now let's show that $\phi$ is a function. 

This means we must show that for every $X\in G/K$ there exists a unique $h\in H$ such that $\phi(X)=h$.

To start let $X\in G/K$.

Next pick $g, j\in G$ such that $X=gK=jK$. 

Since there are different ways to represent the same coset, we have chosen two of them, and will now show that the choice $g$ or $j$ does not change the outcome of the rules given by $\phi$. (Similar to how some people know me as Haylee Zierow and some know me as Haylee Lau, but despite the different names I am the same person).

We will now show that $\phi(X)\in H$ and that $\phi(X)$ is unique.

We compute $$\phi(X)=\phi(gK)=f(g)\in H.$$

Also, $$\phi(X)=\phi(jK)=f(j)\in H.$$

This shows that $\phi(X)\in H$.

Now, since $gK=jK$, we know that $g\in jK$ (refer to problem $82$).

This implies that $g=jk$ for some $k\in K$ which implies that $$f(g)=f(jk)=f(j)f(k)=f(j)e_H=f(j).$$

Since $f(g)=f(j)$ it follows that $\phi(gK)=\phi(jK)$ because of how $\phi$ is defined, hence $\phi(X)$ is unique.

We have now shown that $\phi$ is a function. 



Next, with $gK, jK\in G/K$ we will show that $\phi(gKjK)=\phi(gK)\phi(jK)$.

We compute $$\phi(gKjK)=\phi((gj)K)=f(gj)=f(g)f(j)=\phi(gK)\phi(jK).$$ 

So $\phi$ is a group homomorphism.

Now suppose that $\phi(gK)=\phi(jK)$ for some $gK, jK\in G/K$.

Then we know that $f(g)=f(j)$ and we also know that $f(g)(f(j))^{-1}=f(gj^{-1})=e_H$. 

This shows that $gj^{-1}\in K$, so $gK=jK$, thus we know $\phi$ is injective.



Now suppose that $h\in H$.

Since $f$ is surjective, that means there exists $g\in G$ such that $f(g)=h$.

We then have $$\phi(gK)=f(g)=h.$$

Thus $\phi$ is surjective.

Therefore, $\phi:G/K\to H$ is an isomorphism.



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The Magic Cube


The Rubik's cube was invented in 1974 by Erno Rubik, a Hungarian who originally called this toy, the Magic Cube. His puzzle hit the market in 1980. As the cube grew in popularity it soon became one of the most sold puzzles in the world. At first people's focus on the cube was just to solve it, today more people are interested in how fast they can solve it. The record for the quickest solve is currently 3.47 seconds. The Rubik's cube is also studied in several different fields, for example: computer science, engineering, and mathematics.

Recently I went on the adventure of solving a Rubik's cube myself. I usually don't love puzzles, they make me cry, but this one was not too bad! I did find help along the way from this video by WIRED. Note, there are many ways to solve a Rubik's cube, and my solving was not for time, just for fun. My interest in Rubik's cubes stemmed from my Abstract Algebra class where we have been learning about groups. You may ask, "What is a group? Is that like a gathering of people or things?" Well, kid of, let's define it mathematically;

    A Group- To be a group, you must satisfy four conditions:
$1)$ Be closed under the operation. For example let's say we have a group $H$ (for Haylee😉) and let the group operation be multiplication. Let the elements $i,j \in H$. Since the operation is multiplication, then $i*j\in H$ as well.
$2)$ Associativity must exist. Let $i, j, k \in H$, then because of associativity $(i*j)*k = i*(j*k)$, we can regroup our elements.
$3)$ There is an identity element $e\in H$ such that $e*h = h*e = h$ for all $h\in H$.
$4)$ Inverses exist so for any $h \in H$ there exists $h^{-1}\in H$ such that $h*h^{-1} = h^{-1}*h = e$.

*Note that commutativity is not a group requirement, but if the elements in a group are commutative we say the group is Abelian (named after Niels Abel, a contributor to group theory, check out my earlier blog post on him to learn more! ). The group made from Rubik's cube permutations is non-Abelian, so commutativity does not exist.  

Now that we know what a group is, the next question we can ask is, what do groups have to do with a classic puzzle? I'm glad you asked!

All the fun of a Rubik's cube comes from messing up each of its perfect faces and then moving the cubies (no, I did not make up this word, it really is what people call the individual cubes on a Rubik's cube) back and forth until each face of the cube is perfect again. It is quite satisfying! In mathematics we would classify these movements as permutations. Here we will describe a permutation as any legal move where you rotate one of the six faces, $\{$F, B, L, R, U, D$\}$, of a $3X3X3$ cube  $90^{\circ}$ or one of its multiples. It turns out that the Rubik's cube has $43,252,003,274,489,856,00$ permutations! This means there are $43,252,003,274,489,856,00$ possible arrangements of a Rubik's cube. To see how this is calculated go to page two in this article. These permutations are what make up the Rubik's cube group. But do the movements of a Rubik's cube actually fit the definition of a group?

Before we determine whether a Rubik's cube and it's permutations satisfy the definition of a group, let's look at the make up of the cube and talk about how we refer to each of the pieces. 

A cube has $8$ corner cubies, each of them show three faces. It also has $12$ edge cubies, each of those showing two faces. And lastly, there are $6$ center cubies where you only see one face. We will also name the six faces of a Rubik's cube.
When you see an algorithm like, " FURU'R'F' " This would translate as a $90^{\circ}$ clockwise rotation of the F$=$Front Face, then U$=$Up Face, then R$=$Right Face, and then a $90^{\circ}$ rotation counter clockwise on the U'$=$Up Face, R'$=$Right Face, and finally F'$=$Front Face. Note, the ' means we are turning the face counter clockwise instead of the normal clockwise. If you see something like R^$2$ that means rotate the right face $90^{\circ}$ clockwise, twice, or $180^{\circ}$. 

Okay, now it's time to make the set of moves of a Rubik's cube into a group!


Let this group be called $R$ for Rubik's cube. Let's show it is closed under the operation of multiplication. Let $M_1$ and $M_2$ be two moves performed on the cube. Since we can make the move $M_1$ and the move $M_2$, then we can also make the move $M_1*M_2$, thus $R$ is closed under multiplication.

Now, let $e$ be the nothing move, meaning $e$ does not change the make up of the cube. Then a move, $M$ times $e$ would give you the same result that you got from just the move $M$, so $M*e = M$. The same idea applies if you did nothing, $(e)$, and then move $M$, you would have $e*M = M$. Thus $M=M*e = e*M = M$, thus $R$ has an identity.

Next let's say that you make a move $M$ and then you reverse that move by performing $M'$. This means you move $M*M'$, first moving $M$ then reversing all the steps of $M$, and the result is as if you did nothing. This would mean that $M*M' = e$, so $M'$ is the inverse of $M$. This idea also works with $M'*M = e$. Therefore, every element in $R$ has an inverse. To picture this better, think about the steps of putting on shoes. First you have bare feet $(e)$, then you put on your socks and then your shoes ($M$). To get back to bare feet $(e)$, you first have to take off your shoes and then your socks $(M')$, and voila, you have bare feet $(e)$ again! So $e = M*M' = M'*M = e$.

Lastly, we must show that the elements in $R$ are associative. Let $M_1, M_2, M_3 \in R$. Let's say that first you move $M_1$ then you move $M_2$ and then you move $M_3$. This would look like $M_1*(M_2*M_3)$ which is clearly the same as if you moved $(M_1*M_2)*M_3$.  Since we have $M_1*(M_2*M_3) = (M_1*M_2)*M_3$, we know that $R$ is associative.  

And now we know that $R$ is truly a group, woohoo!

So how does knowing that $R$ is a group help solve a Rubik's cube? Well, to solve a Rubik's cube most people follow algorithms, set patterns that you apply in certain situations to move the pieces of the cube. These moves are what we have already labeled as permutations and the permutations are what make up our group $R$.  Specifically, these algorithms are made from generators of $R$. This means that mathematically we would let $T$ be a subset of $R$ and we would say that $T$ generates $R$ if $R = \langle T \rangle$ (if $T$ spans $R$, meaning the elements of $T$ can be used to get everything that is in $R$). Here we will let $T= \{D, U, L, R, F, B\}$, our faces from above, so $R = \langle D, U, L, R, F, B\rangle$. Knowing this, you can create your algorithm and you are good to go!

At this point we have successfully touched how the world of Group Theory can relate to a Rubik's cube and seen a little how people can begin to solve a Rubik's cube using mathematics.

       Thanks bunches,
        ~Haylee Jo Lau   



Works Cited:

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True to His Standards in Politics, Religion, and Math

 Augustin Louis Cauchy

Aug. 21, 1789-May 23, 1857

In Abstract Algebra we work with permutations a lot. What is a permutation you may ask? A permutation is essentially when you have a set of items and you change the order of the items, you then have created a permutation. In an earlier blog post we worked with permutations of a square. Here is a more specific definition of permutations from my text book, Contemporary Abstract Algebra by Joseph A. Gallian-

    "A permutation of a set $A$ is a function from $A$ to $A$ that is both one-to-one and onto. A permutation group of a set $A$ is a set of permutations of $A$ that forms a group under function composition."

Why am I interested in permutations? Well first off they are super cool, secondly, it is because this guy, Augustin Cauchy was interested in permutations too! Cauchy was one of the pioneers of analysis and the theory of permutation groups. He, like many mathematicians, added to several different areas of mathematics and physics including studying differential equations, infinite series and their convergence and divergence, determinants, calculus, probability, and optics.

Cauchy lived through the French Revaluation and when he was very young his parents feared for their safety and health as they had very little food and resources. However things got better as time passed. Laplace and Lagrange both visited Cauchy's home and encouraged his studies. In 1807 he graduated from the Ecole Polytechnique. He then attended an engineering school and eventually, in 1810, worked as an engineer for Napoleons' English invasion fleet at Cherbourg while continuing to do mathematical research on the side.

Around 1813 Laplace and Lagrange had convinced Cauchy to focus solely on mathematics. He came back to Paris and spent time teaching and doing research for years. Though Cauchy produced many important papers for mathematics, he had a way of offending his colleagues... he was very Catholic and came off as self-righteous and judgmental. Even poor, kind Mr. Abel was offended by this man! Abel exclaimed-"Cauchy is mad, and there is no way of being on good terms with him, although at present he is the only man who knows how mathematics should be treated. What he does is excellent, but very confused…"

In 1830 when Charles X was exiled and Louis-Philippe took the throne, Cauchy also went into exile since he refused to take an oath of allegiance. Because of this he lost all of his positions and chairs that he held in Paris. He began to teach in Turin and eventually attempted to teach the grandson of Charles X. This was an interesting experience I'm sure... Most articles reveal that the grandson was not the best of students and Cauchy was not the best of teachers, so the two made a less than stellar pair to say the least. 

Finally, in 1838 Cauchy returned to Paris and gained back his seat at the Academy, the Ecole Polytechnique, but was not allowed to teach as he still refused to take the oath of allegiance. By 1848 when Louis-Philippe was overthrown, Cauchy regained his chair at Sorbonne and held that position till he died. Some of his last words were, "Men pass away, but their deeds abide." Cauchy's works certainly abide today, and we are so thankful for that.


    Thanks bunches,

      -Haylee Jo Lau


Works Cited:

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A Revolution of Her Own

 Sophie Germain

Do you know how Archimedes died? He died by being speared to death by a Roman soldier. He had ignored the soldier because he was so intrigued by a geometric figure drawn in the sand. If you are like Sophie Germain, you too are wondering what was so interesting about a shape in the sand that you would lose your life over it? This was the story that first inspired Sophie to start studying mathematics.

Germain was born in Paris on April 1, 1776. Her life was surrounded by revolutions. The year she was born the American Revolution started and thirteen years later a revolution started in her own country. 

She was thirteen years old and stuck in her home since the revolts around their land made it unsafe to be out. There she started exploring her father's library which was where she found Archimedes story. She continued searching her father's books and began studying mathematics on her own, studying works by Newton and Euler in Latin and Greek. Germain had to study at night by candle light to hide from her parents.  Her family was middle class so her education had been limited and on top of that, social norms claimed that the studies Germain was carrying out were inappropriate for females. In an attempt to stop Germain before she got too carried away, her parents tried everything they could to stop her, but she fought back with just as much zeal till her parents decided that it could not be helped, their daughter was going to study mathematics one way or another.

In 1794, when  Sophie was about eighteen years old, the Ecole Polytechnique was founded in Paris. It was meant to train mathematicians and scientist for the country but did not accept women to study there. Despite that, Germain was able to get ahold of lecture notes and study the materials taught there. She eventually took the pseudonym M. LeBlanc and sent her work on analysis to a teacher whos lectures she was especially fond of, J.L. Lagrange.  Lagrange was so impressed, he insisted on meeting the author of the well done paper. When he found out that M. LeBlanc was really Sophie Germain, he looked past the fact that she was a female and decided to be her mentor, recognizing her capabilities in math. And just like that, with her new escort, she had been introduced into the formal society of mathematics.

Through the years Germain continued to do work with Lagrange and worked with Gauss (who also did not know she was a woman originally) and Jean-Baptiste- Joseph-Fourier and other mathematicians. And eventually, in 1816 on her third attempt, she won a prize from the French Academy of Sciences. She was the only one to have entered the contest originally.
With her paper Memoir on the Vibrations of Elastic Plates she not only won the prize but had been accepted to the Academy of Sciences. She would finally go and study mathematics as an equal collaborator and be able to refine her proofs in number theory.

Sophie Germain died on June 27, 1831 at the age of 55. She had been fighting breast cancer. Before she died, Gauss, an earlier mentor of hers, had petitioned the University of Gottingen to give her an honorary degree. However, she had passed away before they could get it to her.  Still, Germain was a revolutionary woman who worked hard to do what she loved despite the obstacles that loomed over her. And she made it! She is now a recognized and appreciated mathematician. Today some of her most important work is recognized in the theory of elasticity and in number theory.

Here is an example of some of the work she accomplished. If both p and $2p+1$ are prime, then $p$ is a Sophie Germain prime. The first few such primes are $2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113,$ and $131$. Around 1825 Sophie Germain proved that the first case of Fermat's last theorem is true for odd Germain primes. 

*My favorite quote by Sophie Germain, "Algebra is but written geometry and geometry is but figured algebra."


     Thanks bunches,
      -Haylee Jo Lau

Works Cited:
Contemporary Abstract Algebra, Joseph A. Gallian, Ninth Edition

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Graphs, Tables, and Groups- Oh My!

 Arthur Cayley

Last week we learned about Sylvester, this week let's look at one of his dearest friends who also had a liking of mathematics.

Sylvester commented, "Mr. Cayley, of whom it may be so truly said, whether the matter he takes in hand be great or small, "nihil tetigit quod non ornavit," which translates to "He touched nothing without embellishing it". Indeed Cayley had a knack for mathematics and many things. Born on August 16, 1821 in England, Cayley ended up writing enough papers on mathematics to create thirteen volumes, about 600 pages each, of just his work. He also had an interest in astronomy, mechanics, painting, traveling, architecture, and hiking. He taught at John Hopinks University for a bit and was a lawyer for fourteen years- the whole time writing papers for mathematics. Eventually he became a professor of mathematics at Cambridge University (also getting married the same year) until he died on January 26, 1895. There he not only taught mathematics and carried out research, but he successfully pushed for women to be accepted into Cambridge as well. 

Due to Cayley's amount of work mentioned above, it is not hard to believe that the man has contributed to many fields of mathematics- he helped come up with the theory of invariants, he worked with linear algebra, geometry, he invented Cayley graphs, Cayley tables, and worked quite a bit more with groups, the list could go on. For right now let's focus specifically on groups and Cayley tables as Abstract Algebra is all about groups. This will be fun!

Imagine you just got a brand new car (Santa was extra giving this year...).


It is in beautiful condition, but in awhile will need new tires. You buy tire 1, tire 2, tire 3, and tire 4. There is no specific order you have to put the tires on even though they are numbered. How many different combinations can you come up with? (*You should be able to put together 8 different combinations!)

While doing this, you may notice that you cannot move just one tire. If you want to switch one tire that means you are going to have to move at least one other tire. One other option is to move nothing and leave everything in its original place. 

Now that you have thought about this for a bit, let's look at something very similar with just a plain square. 


Here each vertex is labeled, we have a vertex set! Let's call our new set $V$ and let's say that $V =$ {$1, 2, 3, 4$}. We also have a set of edges, call this set $E$ and say our set $E =$ {{$1,2$}, {$2,3$}, {$3,4$}, {$4,1$}}. We now have a graph where $G = (V, E)$. How many different ways can we move these vertices? Well, squares are made from right angles and right angles are $90^{\circ}$right? So let's just rotate this square clockwise a couple times and see what we get.

      $R_{0}$ gives,              $R_{90}$ gives,             $R_{180}$ gives,              $R_{270}$ gives,                                                                                     
                                                    

      $H$ gives,                 $V$ gives,               $D$ gives,                $D'$ gives, 

                                                    
The top row is what we should get when shifting everything clockwise $90^{\circ}$ several times and includes what happens when we leave everything in its original spot, like what we mentioned earlier. The bottom row has first a horizontal flip, a vertical flip, a diagonal flip and then an off diagonal flip.
These eight different movements make up a specific group called $D_4$. And actually each movement that makes up $D_4$ is considered a function. So what happens when we combine two of these movements? Is that allowed? Well, when we think of these movements as functions we can then use function composition. Function composition works like so, let $f$ and $g$ be functions so then $f \circ g$ means "$g$ followed by $f$". Using our combination of movements let's apply this same idea. Let's take $HR_{90}$ which means "$R_{90}$ followed by $H$". This new combination of movements will give us $HR_{90} = D$. We can make more movements by combining our original eight from $D_4$! So then how many combinations of movements can we make? We can actually put all of this into a table called a Cayley table believe it or not, named after Arthur Cayley, the man who invented it.







In Abstract Algebra we can use Cayley Tables to see how groups interact with each other and to see if it could possibly be a group since there are specific criteria that need to be met for something to be a group.

And now you have explored a small part of Arthur Cayley's work with groups and gained a basic understanding of what we look at in Abstract Algebra. If you are still interested in groups and Cayley Tables, here is a link to a cite that goes a little more in depth on the subject.


    Thanks bunches,
      -Haylee Jo Lau
(P.S. Here is a cool video my Abstract Algebra teacher shared with our class to get us excited about groups, hope you enjoy!)




Works Cited:
Contemporary Abstract Algebra, Joseph A. Gallian, Ninth Edition
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Mathematics is the Music of Reason- J.J.S.

 James Joseph Sylvester


In my experience, many people picture mathematicians as cold, isolated, geniuses. Though Sylvester was an extremely capable man, he got things wrong and was far from cold and isolated. One student said about him,

    Sylvester’s methods! He had none. “Three lectures will be delivered on a New Universal Algebra,” he would say; then, “The course must be extended to twelve.” It did last all the rest of that year. The following year the course was to be Substitutions-Théorie, by Netto. We all got the text. He lectured about three times, following the text closely and stopping sharp at the end of the hour. Then he began to think about matrices again. “I must give one lecture a week on those,” he said. He could not confine himself to the hour, nor to the one lecture a week. Two weeks were passed, and Netto was forgotten entirely and never mentioned again. Statements like the following were not unfrequent in his lectures: “I haven’t proved this, but I am as sure as I can be of anything that it must be so. From this it will follow, etc.” At the next lecture it turned out that what he was so sure of was false. Never mind, he kept on forever guessing and trying, and presently a wonderful discovery followed, then another and another. Afterward he would go back and work it all over again, and surprise us with all sorts of side lights. He then made another leap in the dark, more treasures were discovered, and so on forever.
    -Ellery W. Davis

James Joseph Sylvester, was born on 3 September 1814 and grew up to become a bright, passionate man. He died on 15 March 1897, but not before leaving his mark on the world of mathematics. His major mathematical contributions were in  the theory of equations, matrix theory, determinant theory, and invariant theory (which was discovered with Cayley). Sylvester loved poetry and his proofs were often described as, "flowery and eloquent", and were said to match his sensitive and enthusiastic personality. 

We will often see terms that Sylvester coined such as matrix, invariant, discriminant, commutant, and more. There have also been ideas named after Sylvester such as Sylvester's determinant identity, the Sylvester-Gallai theorem, Sylvester Matrix, Sylvester's theorem and the list can go on. Taking a look at Sylvester's work, we can see his finger prints on Abstract Algebra today. And Sylvester's contributions to things like invariant theory  are a branch of Abstract Algebra. Another reminder that the math we study now was built from the shoulders of those who proceeded us.

In 1837 Sylvester earned second place in the Mathematical Tripos Examination but did not graduate with a degree as he was Jewish and refused to subscribe to the Thirty-Nine Articles of the Church of England, which was required to graduate at the time. However, Sylvester continued to spread his influence despite the roads that were closed to him because of his religious beliefs.

For three years Sylvester was the chair of natural philosophy at the University of London, then in 1841 he finally was able to receive his B.A. and  his M.A. from Trinity College, Dublin. By the time Sylvester was 27 he applied to be the chair of mathematics in the University of Virginia in Charlottesville in the United States. In 1843 he left the U.S. and returned to England. There he got into actuary work and decided to study law. This happened to be a very lucky choice on his part as it was what the mathematician Arthur Cayley was doing as well. They met, and though very different in many ways, became great friends who would end up doing lots of research together. Among Sylvester's many accomplishments in 1876 he was appointed to a prestigious position at the Johns Hopkins University, he founded the American Journal of Mathematics, and held a teaching position at Oxford from 1884 until he died.

What did I learn from studying Sylvester? It is okay to not get things right the first time... keep trying, you never know how much your contributions can benefit others.

Thanks bunches!
    -Haylee Jo Lau




Contemporary Abstract Algebra, Joseph A. Gallian, Ninth Edition













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Poor, Smart, and Handsome, the Tale of a Mathematician...

Niels Henrik Abel
 

     

In Abstract Algebra, when we have a group where $ab = ba$ for all choices of group elements $a$ and $b$, we say the group is commutative, or Abelian, named in honor of Niels Abel.

Niels Abel was a young, Norwegian mathematician. He was born August 5, 1802 in Finnoy, Norway and passed away on April 6, 1829 in Froland, at the young age of 26. He died from tuberculosis. It has been said though that, "He [Abel] has left mathematicians something to keep them busy for five hundred years." - Charles Hermite

By the age of 16 Abel began reading mathematical works by Newton, Gauss, Lagrange, and Euler-introduced to him by his teacher Bernt Holmboe, who saw great potential in the young man.

Despite his father passing away in 1820, leaving his family in a worse financial state than they were before, Abel attended the University of Christiania in 1821.  This was thanks to Holmboe who was able to get Abel a scholarship. A year later Abel graduated. He continued to study mathematics on his own and through traveling to collaborate with others. 

Though still very poor and ill, often only able to afford one meal a day, Abel worked hard to print his work and send it to universities and other mathematicians.

Through his adventures, Abel was able to pick up friends who encouraged him in his work and helped out a bit financially. Among those was Bernt Holmboe who got him started in mathematics, a proffessor from college- Christopher Hansteen and his wife who cared for Abel like he was family, Christine Kemp who was Abel's fiancé, and August Leopold Crelle the creator of Crelle's Journal, a journal for mathematical papers. He was Abel's friend, helped print his work, and helped to get him an academic position working in Berlin, but Abel had passed away before the good news could reach him. 

At the time, few people recognized the influence and importance of Abel's work. However, today is a different story. In fact, in 2002 Norway created a prize called the Abel prize for mathematics in his honor. It is considered the equivalent to to a Nobel Prize. 

Among the many mathematical topics Abel studied and worked on, Abel's Impossibility Theorem stuck out to me the most. It states that,
    "The general algebraic equation with one unknown of degree greater than $4$ is insoluble in radicals, i.e. there does not exist a formula, which expresses the roots of a general equation of degree greater than four in terms of the coefficients involving the operations of addition, subtraction, multiplication, division, raising to a natural degree, and extraction of roots of natural degree."

To see a proof with some author notes to go along with the theorem, here is a link that walks you through the process http://www.math.caltech.edu/~jimlb/abel.pdf . Look at pg. 7-13.

So now we know, if anyone tries to ask us to solve an equation with a polynomial of degree five or higher, it may be a bit trickier than pulling out our quadratic formula. Thank you Abel! 


*Favorite Quote From Niels Abel, "Everyone wants to teach and no one wants to learn."

Thanks Bunches!
    -Haylee Jo Lau


Works Cited:
Contemporary Abstract Algebra, Joseph A. Gallian, Ninth Edition

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