And That Is The Rest Of The Story

Ask me anything   throw idea's my way!   Hello everyone, my name is Aidan Patterson, and I am the owner of my own thoughts and actions. My hobbies include Mathematics, Karate, Physics and anything music related. Never be afraid to tell me I am wrong, but be prepared to prove it, and may I say; good luck proving it to me! P.S. I'm secretly an activist! I'm a feminist, I fight for anything related to LGBTQIA rights, as well as for people of color, and those who don't have the privilege of abelism! Feel free to chat with me about that too :)

myartblogbitch:

EVERYONE WHO REBLOGS THIS POST BY AUGUST 20TH WILL GET A PIECE OF ART IN THERE INBOX BASED ON THEIR BLOG

Interesting idea :)

(via the-irrationals)

— 5 days ago with 45198 notes
Group Theory and Graph Theory

So, I’ve been gone for a while with school and work, but I’ve found some spare time and I wanted to bring to your attention an amazing connection between group theory: the study of abstract collections of objects and how they interact with when combined, and graph theory: the study of sets with an irreflexive relation R. Hopefully this will be the first of many posts to come this summer :)

A group is a set which is acted on by an operation. This operation takes two elements and combines them to make a third. There are some rules (Axioms) for the combining of elements, the first being that the combination of any two elements in the set creates another element that is in the set. This is called closure, and is very important. Another rule related to this one is that every element is required to have an inverse in the set, which gives us our next rule, that all groups require exactly one element that acts like the number 1 does for multiplication. Whenever anything is combined with this identity element, we get the other number back (x*1=x). the one thing that’s missing from this definition is the associative property; the idea that (x*y)*z=x*(y*z). This obviously isn’t the end of it, but we’ll come back to this later. 

Graphs are pretty cool as well. They are part of model theory (so are groups for that matter), and are used to describe situations in either a visual or abstract way.To get a graph, we take a finite nonempty set V and construct a relationship between the elements of V called R this relationship is symmetric, meaning if (u,v) is in R, then (v,u) is in R. We denote E the set of subsets of R that include all symmetric elements. If V is {a,b,c,d}, then R is {(a,b), (a,c), (a,d), (b,c), (b,d), (c,d), (d,c), (d,b), (d,a), (c,b), (c,a), (b,a)}, and E is {{(a,b), (b,a)}, {(a,c), (c,a)}, {(a,d), (d,a)}, {(b,c), (c,d)}, {(b,d), (d,b)}, {(c,d), (d,c)}}. The letters correspond to their functions, V is the set of vertices, the the generator elements, R is the set of all possible lines (Relationships) that can be drawn between the vertices, and E is the set of all edges on the graph. For any graph G, we call V the vertex set, and  call E the edge set. The order of a graph is the number of elements in V, and the size is the number of elements in E. We denote the vertex set of G as V(G) and the edge set as E(G). All graphs have to have Vertices by definition, but not all graphs have to have edge sets that contain elements, which is interesting. 

Here are some graphs!

How do we relate the two? Well, groups originally arose out of the study of permutations. Permutations are a kind of homomorphism called an automorphism, which is a bijective function from a set to the same set. Every group is isomorphic to a group of permutations by the application of Cayley’s theorem.

Let’s say we have a graph G of order p, and V(G)={a_1, a_2, a_3… a_p}. If we create an isomorphism, say B:V(G)→V(G) such that B(a_i)=a_i for i=1,2,3,…p. This is just the identity permutation, but other automorphisms exist which correspond to other permutations. Under the operation of simple combination of permutations, these automorphisms form a group supremely denoted “the group of G” and given the symbol A(G). To get a feel for this, consider a straight line with endpoints 1 and 3, with a third point 2 between them. Taking a permutations implies renaming the points in a way which yields the same kind of relationship. Since 1 is connected to 2, 1 needs to remain connected to 2, and likewise for 3. The only other permutation that makes sense is the permutation which exchanges the places of 1 and 3. So, we get the permutations {1,2,3}→{1,2,3} and {1,2,3]→{3,2,1}. These two form a group together under combination, and are the group of G if V(G)={1,2,3} and E(G)= {{(1,2),(2,1)}, {(2,3), (3,2)}}. Both the vertex and edge set work together to produce the group A(G).

It’s interesting to wonder if because edges and vertices are the objects which define A(G) that the symmetry group of a graph laid out in some efficient way has anything to do with A(G). It actually does, and you can read all about it here (This actually surprised me a little)!

The other way that group theory is related to graph theory is again through Cayley It’s almost like he was important or something. Groups can be thought of in terms of elements called generators. Generator elements can generate other elements in the group through successive combination with themselves, hence their name. If there is one generator the group is called cyclic because beginning at the identity element, after some number of combinations with the generator you arrive back at the identity element. You can have a group with more than one generator as well, in fact, here’s one

This one is in terms of what looks like rho (the P thing) and phi (the loopy thing). When you have generators like this, they often follow rules. For the one above, we can see that rho cubed never appears and that rho times rho squared is equal to the the identity element, so we can conclude that rho cubed is equal to the identity element. We also see that phi times phi is equal to the identity element, so phi squared is equal to the identity element as well. That’s all well and good, but can you prove that all groups can be written in terms of some number of generators (and not just the trivial one with all elements of G included)? I’ll do another post on that, but to bring this back to graph theory, I need to introduce cayley diagrams, which are a great way to come up with new groups!

This is a simple Cayley diagram (the one from the group shown before with a equal to phi, b equal to rho, and rho times phi equal to a times b squared). From the picture, you can see that the solid arrow means multiply by b, and the dashed arrow means multiply by a; you are allowed to add more types of arrows when you have more than two generators. An arrow head is used to show the direction of multiplication (which is always assumed to occur on the right side) unless the element is equal to its own inverse, which is the case with a, when a simple line is used.

How are these related to graphs? Each vertex represents a new element of G, so let V(G)={e,a,b,b2,ab,ab2}, and E(G) be equal to the set of all elements that are connected by a single straight line. You’ve now turned a set into a graph, and can now commence doing graphy things to it! 

These are just a few ways that group and graph theory are related, I didn’t even go into coloring theorems, topology, or Graph matrix theory. It’s interesting to think that it’s connections like this that have allowed us to make major breakthroughs in mathematics. Problems which seemed impossible in one theory have be translated over in a sense to problems in another area where they were more easily accessible. One immediate example comes to mind with the proof of the Taniyama-Shimura Conjecture by Andrew Wiles which proved Fermat’s Last Theorem. The problem aimed to connect two areas of mathematics, modular forms and elliptic curves, and in turn made a statement more accessible in number theory of all places. 

All mathematics is related somehow, you just have to find the right bridge. Happy Proofing :)

— 3 weeks ago with 28 notes
#mathematics  #maths  #math  #mathema  #physics  #group theory  #graph theory 

mathematica:

For all you know, dear follower, toroids are my secret fetish: Numberphile — Topology of a Twisted Torus

"Hi, I’m Carlos Séquin, and this is a bagel.”

[CJH]

— 1 month ago with 30 notes
Why Your First Course in Topology Will Probably Be Disappointing →

ryanandmath:

image

My professor had one of these stickers right on his office door. It’s a classic and simple topology joke, playing off the fact that topology is often regarded as the study of “nice” continuous deformations, such as stretching, shrinking, and twisting. The professor himself liked to describe…

— 1 month ago with 91 notes
ladieslovescience:

weakinteractions:

I officially got my PhD this weekend!

You go, girl!

You are an inspiration :’) Are you going on to study, or are you looking into something else?

ladieslovescience:

weakinteractions:

I officially got my PhD this weekend!

You go, girl!

You are an inspiration :’) Are you going on to study, or are you looking into something else?

(via can-i-leave-this-blank)

— 1 month ago with 73 notes

So I went to prom with my best friend over the weekend :)

— 1 month ago with 29 notes
#prom  #personal 
http://yunglavender.tumblr.com/post/88233962640/okay-this-was-sent-to-me-by-mohsina →

yunglavender:

okay this was sent to me by mohsina johnny-football (we need to hang out when i come back to auburn btw i miss ur face) and i was supposed to answer 11 questions and then make up 11 then tag 11 people so HERE GOES:

1) if you could visit any place, where would it be?

- probs (definitely)…

1) if you had to choose between being part of the x-men or part of the teen titans, which would you choose?

TEEN TITANS ALL THE WAY!!! Childhood would be complete

2) do you like that weird purple icing taste?

Surprisingly yes!

3) folk punk or new wave?

math animated GIF

4) if you could dye your hair any color without any social repercussions what color (or combination of colors) would you choose?

BRIGHT RED >:D Maybe blue though. IDK one of those is happening once I’m in university damn it.

5) if you could only save one thing from a flash flood, what would u save?

That book shelf full of books which are so old they can’t be replaced that’s sitting in my room. I love those books :)

6) is zac efron a douche to you too?

No word of a lie, I thought he was so attractive, and now I look back and question my life choices.

7) favorite high school musical song?

Bop to the top obviously :D

8) would you rather have a giant guinea pig or a very small elephant for a pet?

Gigantic Guinea pig because that would be amazing and I could ride it to school and have it devour my enemies.

9) will you eat chocolate chip cookie dough without fear of salmonella?

What is fear anyway? *Eats 3 gallons of cookie dough*

10) every time it rains, what do you think about?

OMG! I think about how rain is the opposite of when air bubbles float to the top of water. Something about that is so cool.

11) who do you think was right about the direction of mutantkind— professor xavier or magneto?

 Magneto all the way.

— 1 month ago with 6 notes
#yunglavender 
syekick-powers:

demigray:

insertintellectualusernamehere:

Finally >:T  I am a proud Demisexual! And I always have a hard time explaining it to other people, let alone myself some times? But this makes it really easy ^^ please reblog and share this.

This is neat; I like it.

FINALLY AN EDUCATIONAL GRAPHIC THAT USES THE DEFINITION OF BISEXUAL THAT I ACTUALLY IDENTIFY WITH

NO BUT THIS IS SO IMPORTANT! I actually didn’t know that there were other sexualities besides homosexual, heterosexual, and bisexual until I was 15. I didn’t know that you could be anything other than heterosexual until I was 13 ish. I hope that so many people see this, even if it is by accident.
I used to get this sort of panicked feeling when I couldn’t find the words to describe to people exactly how I was feeling, and who I was having those feelings towards. It seems easy to me now, but I remember feeling trapped constantly. I wanted to speak to people about my identity and sexuality, but I couldn’t work up the courage because of my age and the place that I lived. For a long time I just assumed that I was abnormal, and during this time, I would get very aggressive with people for no reason. I’m quite sure that I’m fine now, but even so, I lost time in the order of years of self loathing, and I’d hate to see anyone else go through that as well.
Please reblog this millions of times so that others can avoid the awkward “am I just weird?” stage of life. Disclaimer, you can play around with the categories! you can exhibit characteristics from two or more different sexual/romantic orientations at once, as well as mix and match the sexual and romantic orientations to suit you. This list does not cover gender identities (READ THIS), which you could also look up. It also may not be complete, and feel free to identify with none of them. Have a great life :)

syekick-powers:

demigray:

insertintellectualusernamehere:

Finally >:T
I am a proud Demisexual! And I always have a hard time explaining it to other people, let alone myself some times? But this makes it really easy ^^ please reblog and share this.

This is neat; I like it.

FINALLY AN EDUCATIONAL GRAPHIC THAT USES THE DEFINITION OF BISEXUAL THAT I ACTUALLY IDENTIFY WITH

NO BUT THIS IS SO IMPORTANT! I actually didn’t know that there were other sexualities besides homosexual, heterosexual, and bisexual until I was 15. I didn’t know that you could be anything other than heterosexual until I was 13 ish. I hope that so many people see this, even if it is by accident.

I used to get this sort of panicked feeling when I couldn’t find the words to describe to people exactly how I was feeling, and who I was having those feelings towards. It seems easy to me now, but I remember feeling trapped constantly. I wanted to speak to people about my identity and sexuality, but I couldn’t work up the courage because of my age and the place that I lived. For a long time I just assumed that I was abnormal, and during this time, I would get very aggressive with people for no reason. I’m quite sure that I’m fine now, but even so, I lost time in the order of years of self loathing, and I’d hate to see anyone else go through that as well.

Please reblog this millions of times so that others can avoid the awkward “am I just weird?” stage of life. Disclaimer, you can play around with the categories! you can exhibit characteristics from two or more different sexual/romantic orientations at once, as well as mix and match the sexual and romantic orientations to suit you. This list does not cover gender identities (READ THIS), which you could also look up. It also may not be complete, and feel free to identify with none of them. Have a great life :)

(Source: weneedtotalkaboutcastiel, via lesbianmakototachibana)

— 1 month ago with 129965 notes
#personal 
SOOOONNN

I only have 11 days left of school! I’ve done a lot of work with group theory this year, so I’ll get around to posting some of the cooler things I’ve learned :) HAPPY EXAMS I AM SO SORRY!

— 1 month ago with 10 notes
#personal 
I ♥ mathematics →

curiosamathematica:

Some days ago I noticed a brilliant bumper sticker, saying "I ♥ topology" where the heart was replaced by a topologically homeomorphic disk (●). Amused by the idea, I tried to work out some related versions for other mathematical subjects. Here they are:

  • For geometry, the obvious choice was a

GET ME A SHIRT WITH ALL OF THESE!!!

— 2 months ago with 1143 notes
Pi Plays Pokemon? →

onychoprion:

So, “Twitch Plays Pokemon” caused quite a stir on the interwebs a while back. Well, a while in internet time. But it gave me an idea.

Since there are less than 10 buttons required to play your average pokemon game, all the controls could be mapped to the digits 0-9. You know what’s great at…

Math and pokemon rock my socks off :D

— 2 months ago with 73 notes
allofthemath:

This, ladies and gentlemen and genderqueer folks, is Pascal’s tetrahedron, a three dimensional analogue of Pascal’s triangle, and it’s pretty freaking great.

YOU GO YOU AMAZING PERSON

allofthemath:

This, ladies and gentlemen and genderqueer folks, is Pascal’s tetrahedron, a three dimensional analogue of Pascal’s triangle, and it’s pretty freaking great.

YOU GO YOU AMAZING PERSON

— 2 months ago with 1544 notes
Youtube

I kind of want to start a youtube channel. Any ideas on what I should do? Should it be a vlog thing, or a math thing, or just a bunch of weird video’s or what? Tell me what you think! 
?

— 2 months ago with 11 notes
#Youtube  #blog  #vlog  #idk  #ideas 

imathematicus:

alevelmaths:

hey, so non-regular posting aside, it’s coming up very close to exam time, so just to remind you; if you (or anyone you know) needs a hand with a maths question or even a whole maths topic, anything, just hit my ask or submit box and i’ll do what i can :)

happy revising! (ahaha)

same for me.

cheers

HEY! Same goes for me, also I’m really good at doing essays, and critical commentary, so if you need to outsource for ideas, I’m always here :D

(via manifestlycovariant)

— 3 months ago with 42 notes
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