Here's an interesting thing I thought up a while ago that I thought would be interesting for all of you. I would like some answers if possible, please!
Take a cos wave - a standard cos wave. Rotate it around the y axis, and you get a 3d cos wave, like ripples in a pond. Now, take a sound wave in action, and graph it as it moves (as it would travel in space), rotated around the y axis. Here's what you get:
So cool! Now, theoretically, based on this rotation idea, you can rotate this 3d wave around the Z axis, and get a 4 - dimensional Sin wave. below is a 3 - dimensional projection of a 4 - dimensional sine wave created in grapher:
I guess this might be related to the finding of higher dimensional standards - a point is 0d. A moving point makes a line, 1d. Moving line, square, 2d. A moving square, cube, 3d. A moving cube, 4d, hypercube, etc.
I guess you could continue to find even higher dimensional waves... you just probably couldn't graph them easily. :D
Welp, sorry - guess I'm not done. Area of a square is L^2, Volume of a cube is L^3, Volume (I don't know what else to call it - what is 4D volume called) is L^4. Area of circle is πR^2, Volume of a sphere is 4/3πR^3, and Volume (again, Idk what to call it) of a so called "hypersphere" is π^2r^4 all over 2. Now, using the fundamental theorem of calculus, we can find the area under a cos wave from the points 0 to 2π (one phase) with respect to x:
Using the power rule and evaluating from 0 to 2π, we get 4π as the area under this wave from 0 to 2π. (Not too good with LaTeX, so I didn't show all of my steps) Now, my question is how to find the area (or Volume, rather) underneath the wave rotated around the y axis, not as it fluctuates, but at a single point in time - the volume under the ripples to the XZ plane.
This would definitely involve multivariable calculus and a ton of time... this would be so complex. thinking in 2d, we find the area under a curve by thinking of an infinite amount of rectangles that we fit under it - for a 3d wave, we'd have to find an infinite amount of rectangular prisms to fit underneath... would this qualify as multivariable calculus? I honestly don't have a clue :D
Then, what about the area under a 4d sine wave? we'd be dealing with an infinite number of hyper - prisms fit under a hyper curve...
My head hurts. I honestly don't know how to approach this. Is something like this possible in even higher dimensions as well? I mean, we can do it in 2D, it is theoretically possible in 3D, but in things like 4D, is it even feasible to try? And of what use would it be if it was possible?
Any suggestions or ideas would be greatly appreciated - this is a pretty difficult problem!