A Page of My Notes

[AngelFish]

If I knew wherw I could find it I'd post my noyes on relativity in relation to newtons laws.

That would be cool

Relativity is what happens when you modify Newtonian physics a bit to handle the Lorentz transform and work in Minkowski Space. I.E.: Make a few adjustments to Newton and special relativity comes out without too much fuss. General Relativity is a bit more difficult.

[Deep Thought]

I take notes either when it's for a grade or if I'm somehow inspired by the topic. We were talking about perspective and 4D hypercubes in Psych class today, so I drew a couple of 5-dimensional cubes, made some notes on it, and figured out the math behind it Good thing we're not turning in our notebooks.

[ruler501]

I take notes where I summarize the lefture into about half a page depending on what it is it might go to a whole page. My teachers hate it and tell me to take "C-Notes" and occasionally grade on that as I have no idea how to take them I usually fail that

[Xeda112358]

I was at a math conference when it finally clicked for me what a hyper cube was. Once I figured it out, I came up with a way to draw them and I was applying graph theory to it

[Deep Thought]

Our Psych teacher gave us a quick video showing how to draw a single hypercube, so I kept drawing them over and over in different positions until I figured out what it was

[ruler501]

I was at a math conference when it finally clicked for me what a hyper cube was. Once I figured it out, I came up with a way to draw them and I was applying graph theory to it

Id love to understand that. It completely confuses me. I cant seem to visualize >3dimensional objects

[Deep Thought]

I was at a math conference when it finally clicked for me what a hyper cube was. Once I figured it out, I came up with a way to draw them and I was applying graph theory to it

Id love to understand that. It completely confuses me. I cant seem to visualize >3dimensional objects

No one can really visualize them (for obvious reasons), but if you think about it enough you can figure out how it relates to cubes and lower-dimensional stuff.

[Xeda112358]

When dealing with hypercubes, a cube is not necessarily 3 dimensional. Here are a few examples:
a 0 dimensional cube is a single point
a 1 dimensional cube is a line segment
a 2 dimensional cube is a square
a 3 dimensional cube is a cube as we know it

Now let us look at the number of vertices here:
0 dimensions has 1
1 dimensions has 2
2 dimensions has 4
3 dimensions has 8

notice a pattern? There is more of a pattern, though. I will give you a hint: Pascals triangle

EDIT: Also, the number of edges from each vertex is equal to the dimension number

[Deep Thought]

And if you count the number of sides (yes, I was that bored), they look something like this:

Dimensions	Sides
0	0
1	1
2	4
3	12
4	32
5	80
n	n2n–1

More generally, if you draw a rectangle of n dimensions, you can see distinctly 2n rectangles of lower dimensions as the "faces" of the new structure:

Dimensions	Subrectangles (not sure how to put that)
0	null
1	2 points
2	4 lines
3	6 squares (faces)
4	8 cubes
5	10 hypercubes
n	2n

It gets really interesting—draw a hypercube, and you can actually count eight 3-dimensional cubes forming its "faces," in a way

[Xeda112358]

n^2n–1

n2^n–1 *cough* ?

I wonder if a program should be made to draw hypercubes? 

[aeTIos]

Do it. Now!!

[Xeda112358]

Actually, Qwerty.55 said he was on IRC... not sure how far he will get, but that seems like it will be cool

[phenomist]

More generally, the number of k-dimensional subcomponents in a n-dimensional hypercube is 2^n-k*_nC_k.

With the gamma function we can easily analyze non-integer dimension hypercubes

[ruler501]

More generally, the number of k-dimensional subcomponents in a n-dimensional hypercube is 2^n-k*_nC_k.

With the gamma function we can easily analyze non-integer dimension hypercubes

how do you get non integer dimensions?

[phenomist]

Well, the gamma function is a real extension of the factorial function used to define combinations, so I suppose it can be done. Only theoretical for now, though things like fractional derivatives are applicable in fields like fluid dynamics.