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MBTIc Math thread

Nocapszy

no clinkz 'til brooklyn
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If I'm reading you correctly, it is a matter of combinatorics. I may have interpreted you incorrectly but I agree with Urchin.

Think of this way, you have 3 options for a shirt (first slot), 3 options for pants (second slot), and three options for a tie (third slot).
This analogy doesn't allow you to use all three shirts and no pants or no ties. I mean... you could, but then why bother with clothes?

You have three independent choices and three things to choose from for each choice. So yes, it is 3^3.
Yeah that's what I was thinking.
But the order-independent version can be thought to be equivalent to the following:

You have a giant(infinite) vat of ping-pong balls numbered 1 to 3, and choose 3 ping pong balls from there. You subsequently forgot which order you chose them. How many distinguishable possibilities are there?

It is 10, like Urchin mentioned,
What if you got two 3 balls? Or 3 twos? Then it would be 27 again.
but how do you do it more generally? With choosing r balls from n types of balls? That is an interesting question, imo.

The short answer is it is C(n+r-1,r), where C is the choose function.

I tried typing the "star" and "bar" explanation but I didn't like the formating.

So, here is a link.

Nocapszy, if you like these types of questions you'll like Discrete Math (which is the mathematics behind computer science).
I actually have been toying with things like this (binary mostly though) and you're right. I do like that kind of math, if any. It's a really layered math - fun for Ne and/or Ti.
The following source has explanations of the basics.
Discrete Math Project - Permutations with Repetition

Also note, that these basics come in four flavors:
1)permutations without replacement
2)combinations without replacement
3)permutations with replacement
4)combinations with replacement

You'll find that transforming your counting problem into one of these is the most likely source of answers. That along with generating functions will fulfill most people's combinatorics type questions.

Thank you.
 

Urchin

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This analogy doesn't allow you to use all three shirts and no pants or no ties. I mean... you could, but then why bother with clothes?

This is the same logic you used, but possibly more obfuscated. What he's saying is you worked the problem thus: You have "spot one," "spot two," and "spot three," then there are three choices for each spot. 3^3.

However, this will cause the set {3, 3, 1} to be counted as different from the set {1, 3, 3} and {3, 1, 3}. These are permutations.

You're looking, I think, for a combination. This would cause the three sets I mentioned to be counted as the same thing because they all have two 3s and one 1. In this case you have:

333, 332, 331, 322, 321, 311, 222, 221, 211, 111

Which is 10 combinations.
 

Nadir

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Even simpler 27-set analogy (though I'm not sure if it works...):

You have three sets of three coins. The coins are made of gold, silver, and copper, and each coin represents a value of 100, 10, or 1. Let's say you have a purse which can take 3 coins at a time. There are 27 different possibilities for you involving depositing your money.

For the 10-set example, consider all coins in a single set to have equal worth.
Then you have the following possibilities:

1. Gold, Gold, Gold
2. Gold, Gold, Silver
3. Gold, Silver, Silver
4. Silver, Silver, Silver
5. Silver, Silver, Copper
6. Silver, Copper, Copper
7. Copper, Copper, Copper
8. Copper, Copper, Gold
9. Copper, Gold, Gold
10. Gold, Silver, Copper
 

ygolo

My termites win
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The last one is a good analogy.

Counting permutations would be counting the distinguishable ways of putting the coins in the purse.

But counting combinations would be counting the distinguishable states when looking in the purse after the coins got all jumbled up.

You'd have no way of knowing which way the coins went in if you only saw them in the purse afterwards.

I hope the difference between combinations and permutations makes sense now.
 

ygolo

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Group Theory, Symmetry, Super-Symmetry, and M-Theory

Since the combinatorics related discussion on this thread has waned...

I was wondering what you guys knew about symmetry groups, their relation to symmetry in general, supersymmetry, and M-Theory.

I believe understanding lie algebra and gauge theories are an important part of understanding these things.

There is a lot of hefty math in there, and String Theory appeals to many people who aren't mathematically inclined. So if there is a way to transform the math to some pictures or some other form of intuitive understanding, I think it would go a long way to actually understanding string theory (as opposed to just being aware of it).

Also, I like knowing the "skeletal rigorous framework" for a mathematical ideas. By this I mean, the minimum set of definitions and theorems needed to understand a mathematical idea. But I haven't figured out what it is for M-Theory yet.

Anyone care to collaborate on this thread to create an informal, multi-type friendly, math-flavored exposition?
 

Nocapszy

no clinkz 'til brooklyn
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This is the same logic you used, but possibly more obfuscated. What he's saying is you worked the problem thus: You have "spot one," "spot two," and "spot three," then there are three choices for each spot. 3^3.
It's not quite the same. The permutations you bring up are considered different possibilities (I was figuring it out for a game I play, but then I just liked the math part and wanted to show you guys)

However, this will cause the set {3, 3, 1} to be counted as different from the set {1, 3, 3} and {3, 1, 3}. These are permutations.
Yeah. Each permutation counts as a different possibility in the game. I see what you're saying though and also how it would be confused with a match for ygolos points.

You're looking, I think, for a combination. This would cause the three sets I mentioned to be counted as the same thing because they all have two 3s and one 1. In this case you have:
No. You had it right a second ago.


Edit: The game actually does go by combination, and not by the permutations. The way my friend explained it though, it sounded the other way. One way or another, my question here was posed searching for all combinations and incarnations of that combination as separate.
 

ygolo

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MBTI Types, Hypercubes, and Hamiltonian Paths

Consider the following 4-dimensional hypercube with MBTI types on each node:

attachment.php


A Hamiltonian Path is (informally) a way to visit each node/vertex in a graph without picking up your pencil.

Can you find all Hamiltonian Paths on the MBTI hyper-cube?

One such path was used in my post on MBTI superlatives.

How many different such Paths are there?
 

nemo

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nemo, do you have any clear picture/understanding of Stone-Weierstrass or the Baire Category theorems?

Not really. I haven't really taken a proper analysis course yet -- the one I took this semester was an undergraduate intro to analysis/set theory combined course, and a lot of the philosophy and historical context was integrated into the material. But I had a blast with it.

Next fall is when I start the joint undergraduate/graduate real & complex analysis sequence. I'll be starting abstract algebra and topology at the same time. I'd take it now, but my school is sufficiently small that those upper-level sequences are only offered once a year.

Ironically, the one place where I have encountered the Stone-Weierstrauss Theorem was in a numerical analysis course. But we didn't get into the theoretical aspects enough to really give me any kind of understanding of it.
 

Gabe

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What in the world is a hyper cube?
Jumping in late here, but here we go: I only know a bit of the highly theoretical stuff. I like the gradient devinition, and the idea of potential functions and conservative fields (whichever theorum that stuff is part of).
Mostly in terms of how it applies to physics and the real world.
Oh, and I still have trouble with imaginary numbers. What is the deal with those?
 

runvardh

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Instinctual Variant
sx/so
square n^2
cube n^3
hypercube n^4
 

nightning

ish red no longer *sad*
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Consider the following 4-dimensional hypercube with MBTI types on each node:

attachment.php


A Hamiltonian Path is (informally) a way to visit each node/vertex in a graph without picking up your pencil.

Can you find all Hamiltonian Paths on the MBTI hyper-cube?

One such path was used in my post on MBTI superlatives.

How many different such Paths are there?

OH WOW! Hypercube! Of course! I'm so stupid! 4 dimensional axis... *repeatedly whacks self on the head* Thanks ygolo... for bringing me clarity in life. :worthy: You made my day... really... I'm not kidding you.

That explains everything... *stares at the image with a happy grin on her face*
 

ygolo

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The thread came back to life...I didn't notice.

Ironically, the one place where I have encountered the Stone-Weierstrauss Theorem was in a numerical analysis course. But we didn't get into the theoretical aspects enough to really give me any kind of understanding of it.

Unfortunately, the same was the case with me.

What in the world is a hyper cube?

Usually, it is the 4-dimensional object that has the same relationship to the cube that a cube has to the square.

Jumping in late here, but here we go: I only know a bit of the highly theoretical stuff. I like the gradient devinition, and the idea of potential functions and conservative fields (whichever theorum that stuff is part of).
Mostly in terms of how it applies to physics and the real world.

Stokes' Theorem is the relevant one I believe. It is rather helpful for finding out if a vector field is conservative. (Green's theorem is the 2 dimensional analouge)

Oh, and I still have trouble with imaginary numbers. What is the deal with those?

Just think of complex numbers as a pair of real numbers. You can do all the 2-D vector addition stuff you do with a pair of real numbers with a complex numbers. In addition, you have extended definitions of sine, cosine, the exponential function, and a definition of multiplication of complex numbers that preserve the relation... e^(i*x)=cos(x)+i*sin(x), e^(i*pi)+1=0, and i^2=1.

One great use of imaginary numbers is in measuring space-time, "distances" (often called intervals). Time in this sense is an "imaginary" dimension.

Given two coordinates in space-time (x1,y1,z1,t1) and (x2,y2,z2,t2) the interval between these coordinates is sqrt([x1-x2]^2+[y1-y2]^2+[x1-x2]^2-[t1-t2]^2)

As you can see, the negative in front of the time dimension lends naturally to thinking of time as an "imaginary" space dimension.


square n^2
cube n^3
hypercube n^4

Are you answering the number of Hamiltonian paths question somehow? Or stating the number of verteces if n=2?

OH WOW! Hypercube! Of course! I'm so stupid! 4 dimensional axis... *repeatedly whacks self on the head* Thanks ygolo... for bringing me clarity in life. :worthy: You made my day... really... I'm not kidding you.

That explains everything... *stares at the image with a happy grin on her face*

I'm sure I'm not the first one to think of it. But I'm glad it makes things clearer for you somehow.
 

nemo

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So as I've said elsewhere, a huge interest of mine is in dynamics and number theory, especially topics where these two areas overlap. I've been playing with some ideas lately, and they're of pretty general interest so I thought my fellow amateur mathematicians here might enjoy them as well and revive this great thread a bit.

First: A picture of a "golden ratio" polar plot. Each point is rotated "golden ratio revolutions". The function is defined as F(n) = (n^(1/2), 2*pi*GoldenRatio*n) in polar coordinates -- i.e. (radius, angle) -- where n is a natural number. This is supposedly the most "optimal" arrangment of the points in the plot in the sense that the distances between all of the points are maximized but still bounded by the successively increasing radius. (Can you prove this?) This is supposedly why it conspicuously resembles something natural; like, say, a sun flower. This is the first 3000 points:

attachment.php


Second: I made a detail of the picture with 500 points (see attached; too big to show explicitly). Each number corresponds to the particular value of n of the point. Look at the numbers that begin to converge on the red line -- does a sequence pop out at you? Can you find the relationship(s) between that sequence and the "revolutions" of the plot? Can you prove the sequence converges on that line?

(It's possible to solve this using just basic algebra, but knowing some linear algebra will cut a great deal out.)

*Warning, spoilers/partial solution ahead*

The sequence you should be able to find is an extremely simple linear difference equation (the discrete analogue of a differential equation). In fact, it's usually the first difference equation you're taught in a dynamics class (think of an Italian guy who studied rabit reproduction!) What's so interesting to me is that the polar plot visually "linearizes" the sequence so well, and the particular value of the "revolutions" has something to do with the eigenvalues of a related dynamical system.

*End Spoilers*

Challenge:
One of the problems given to me in my first number theory class was to come up with a way of finding numbers that are square numbers (of the form j^2) that are also triangular (of the form k[k+1]/2). Here are the first 10 square & triangular numbers (found by brute force using matlab):

0
1
36
1225
41616
1413721
48024900
1631432881
55420693056
1882672131025

Can you find a linear difference equation relating the square/trianglar numbers? (Small hint: start by looking at the square-roots of the numbers above.) If you can, is it possible to find a specific formula giving the n-th square & triangular number? Would the "polar plot" "linearize" the dynamical system; and would the choice of the "_____ revolutions" for the plot have the same relationship to the difference equation as above? Do you see any interesting convergence patterns?

Fun, fun, fun. I'm currently in the process of generalizing this for a possible senior project, but I'll have to see what my advisor thinks first. Oh, and if I'm not too lazy I may post my solutions in a week or so (if there's any interest).

Edit: Oh, and I'll provide Mathematica/Matlab code too. Just PM me.
 

rhinosaur

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I'm no mathematician but I think this stuff is pretty cool, too. I read something once about prime number theory, where if you plotted everything a certain way, patterns would emerge.

And pi! Carl Sagan hit on that one in the end of "Contact."

Group theory is also very useful, interesting and beautiful.

Stats are interesting. I use Gaussian and Lorentzian curves a lot in chemistry, but I don't think they're inherently beautiful. Just a convenient way to describe stuff.

Spherical harmonics are neat. That we can predict the behavior of molecules so precisely with elegant math is very cool.
 

nemo

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They are 3-dimensional cross sections.

It's impossible for a rigid polytope that's only 3-dimensional to move like that.

Edit: so there has to be a "hidden axis" that allows it to turn itself inside out like that.
 

Nocapszy

no clinkz 'til brooklyn
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I can understand about the hidden axis, and I was actually figuring this with a friend the other day, but I'm not convinced that this graphic does a good job displaying it.

I won't say there isn't concept embedded in the gif though.

Edit: Perhaps it does.

I would probably have preferred two of the sides 'crashing into' one another and turning into a single side, both being replaced by those of the other dimension.
 
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