## User Tag List

1. Originally Posted by ygolo
Consider the following 4-dimensional hypercube with MBTI types on each node:

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. You made my day... really... I'm not kidding you.

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

2. Rotating a hypercube:

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

Originally Posted by nemo
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.

Originally Posted by Gabe
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.

Originally Posted by Gabe
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)

Originally Posted by Gabe
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.

Originally Posted by runvardh
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?

Originally Posted by nightning
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. 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.

4. 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:

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.)

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.

6. 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.

7. Originally Posted by nemo
Rotating a hypercube:

How is this four dimensional?

8. 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.

9. 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.

#### Posting Permissions

• You may not post new threads
• You may not post replies
• You may not post attachments
• You may not edit your posts
•