Want an example of the type of useless miscellanea that keeps me up at night until I’m forced to get out of bed and spend an hour or two surfing the web for an answer?
Okay.
What is the volume of a k-dimensional sphere?
I found it, believe it or not. You can find anything on the internet.
For even-k dimensions, the answer is …
For odd-k dimensions, the answer is …
Just don’t expect me to prove them. If I try to, there’s a very good chance my brain will percolate into an oatmeal-like substance and leak out my nostrils and ear canals.
However, I got to thinking: what is volume, exactly? Isn’t it a three-dimensional extension of two-dimension area, which itself is a two-dimensional extension of one-dimensional length? So is volume of a k-dimensional object here just a semantic shortcut? No matter what the extended dimension, we’ll still call it “volume”? I can live with that. But what does “volume of a k-dimensional object” mean? I know we can’t visualize objects greater than three dimensions, but can we understand what volume of a k-dimensional object is without necessarily visualizing it?
Imagine a box on a table. You slide it across the five or six foot length of the table over a period of sixty seconds. Step back and think about what you just observed. If you can hold the image of that box at every single interval over those five or six feet, the path the box took in its continuous entirety, then you are visualizing four dimensions. Visualizing the interior of that box over the sixty-second path it took gives you a representation of four-dimensional volume.
But how about when k is ten, eleven, twelve, or ninety or a million?
Oatmeal time!
Volume by its nature is 3 dimensional. If not, we would call what is defined as "area" as volume but for 2 dimensions. Now, can the 3 dimensions of volume be any 3 dimensions or must they be Height, Length and Width? I say volume is the strict measure of SPACE based on a given Height, Length and Width. To discuss additional dimensions requires different terminology. FWIW.
ReplyDeleteUncle
Word.
ReplyDeleteBut to avoid an infinity of terms, you'd just need to make up one, like "vorve", to refer to "volume" of a higher-dimensional object.
"Say, Tom, what's the vorve of a 26-dimensional string?"
Also, re-reading my post, I'm wondering for the first time why there are two entirely different equations based on whether the number of dimensions is odd or even.
Strange. Something else to research...
hi...i am doing a project on the volume of n-dimension spheres using monte carlo method....and infact i have to write a program for it in c++ language....i found ur article to be really helpful since it contains just enough details to help u visualize...gud work !!!......i would really love it if u could tell me the source from where u took this information of ur article....and any other relevant sources....thanks !!!!
ReplyDeleteGlad you stopped by, but please be warned: I am an amateur's amateur when it comes to this stuff. I did take college courses in higher math, but that was almost 20 years ago, and sadly I've forgotten most of it.
ReplyDeleteI don't have the source for the exact formula I posted, but if you go to wikipedia (I know, I know) and search for "sphere," there are similar formulae about half-way down the article.
Good luck with your project!
PS - an excellent source for higher dimensional ideas that's not too difficult to digest is Rudy Rucker's The Fourth Dimension. That's where I got most of the other stuff in my post from. I bought the book in the early 90s and still crack it open every now and then.
ReplyDeletekk thanks mate !!!....i wl chek out d book ....and i hv already gone thru all available articles on wikipedia probably lolz...anyways thnx...cheers !!
ReplyDelete