Dr. Greene,
I'm currently reading your book, Fabric of the Cosmos and I would like to first state that I am big fan and I am truly enjoying the read! I am just now reading your chapter on the multiple dimensions necessary for string theory and you made a statement that I wanted to comment on. In your description of how to "visualize" multiple dimensions you say that it's pretty difficult to visualize 6 dimensions and you haven't met anyone who could.
Well I certainly don't claim to, but I had to laugh a little bit because when I was a Freshman math major at Trinity University in San Antonio in 2001, I was taking basic undergraduate linear algebra and was floored by the idea that I couldn't visualize the inputs of my equations when they were bigger than 4 dimensions. I then thought about it a little more and I wrote my professor at the time Dr. Chapman a memo on how I imagined different dimensions higher than 3 or 4.
Assumption: Clearly, we live in a world of 3 visible spatial dimensions and 1 time dimension. Since man has taught History I'm pretty sure we have used the concept of a "time-line" to describe the changing of our 3 spatial dimensions across time. By doing so we took the mathematical R3 and converted its "infinite" size to an infinitesimal point and simply gave the point a "nudge" in the direction of time and the time-line was drawn. My assumption was that any time we were in an infinite 3 spatial dimension space we could then, for the purposes of visualization convert that space into an infinitesimal point.
Induction: I don't claim this to be a proof by induction but just the method at which I used to visualize higher dimensions. Once you shrunk the infinite world to the size of a point and struck it you drew a line. This was R4 and if you struck that line (much like a guitar string) it would vibrate showing you the 5th dimension or R5. You could imagine this as different possible futures (maybe even your multiverse picture). If you have now a membrane in that is R5, you can imagine that much like a drum and when you strike it, it vibrates like a drum showing you the 6th (R6) dimension or a new infinite space with 3 spatial dimensions within which each point represented a snapshot of our 3 dimensional space. So continuing on you can quickly see that R6 can then via the assumption be shrunk down to a point and the process repeated. Continuing you would see the pattern that dimensions that can be written as R(3n) where n is a whole number are "spaces" and by assumption equivalently points, R(3n+1) are lines similar to our time line, and R(3n+2) are membranes or planes.
Now this is for whole numbers of n. Clearly there exists a R(0) which is the infinitesimal points in our world. Since it can be written R(3n) is can then be assumed to also be an infinite space. Inside this space we can then start going backwards using n as negative integers. R(-1) is equivalent to R(3n+2) where n = -1. So it is a membrane. Further down is R(-2) which can be written R(3n+1) where n = -1 so R(-2) is a line and lastly R(3n) where n = -1 is a point in that space. Now you can then use this method of "visualization" just as I did in Linear Algebra to try to imagine what different dimensions "look" like. Clearly, this is only for the purposes of helping my 18 year-old self get through my freshman year, but I thought that now you could at least say you've met someone who tried to visualize not only 6 dimensions below ours but "n" dimensions above and below our ours.
I know you're a busy man and I don't want to take any more of your time, but thank you again for writing your book and keep up the good work, and with that I'll leave you with a poem your book has inspired.
I have written a beautiful proof
String theory is true! No goof!
The letters are quite small
Plancnk's constant they are tall
Just trust me, I'm telling the truth.
I'm currently reading your book, Fabric of the Cosmos and I would like to first state that I am big fan and I am truly enjoying the read! I am just now reading your chapter on the multiple dimensions necessary for string theory and you made a statement that I wanted to comment on. In your description of how to "visualize" multiple dimensions you say that it's pretty difficult to visualize 6 dimensions and you haven't met anyone who could.
Well I certainly don't claim to, but I had to laugh a little bit because when I was a Freshman math major at Trinity University in San Antonio in 2001, I was taking basic undergraduate linear algebra and was floored by the idea that I couldn't visualize the inputs of my equations when they were bigger than 4 dimensions. I then thought about it a little more and I wrote my professor at the time Dr. Chapman a memo on how I imagined different dimensions higher than 3 or 4.
Assumption: Clearly, we live in a world of 3 visible spatial dimensions and 1 time dimension. Since man has taught History I'm pretty sure we have used the concept of a "time-line" to describe the changing of our 3 spatial dimensions across time. By doing so we took the mathematical R3 and converted its "infinite" size to an infinitesimal point and simply gave the point a "nudge" in the direction of time and the time-line was drawn. My assumption was that any time we were in an infinite 3 spatial dimension space we could then, for the purposes of visualization convert that space into an infinitesimal point.
Induction: I don't claim this to be a proof by induction but just the method at which I used to visualize higher dimensions. Once you shrunk the infinite world to the size of a point and struck it you drew a line. This was R4 and if you struck that line (much like a guitar string) it would vibrate showing you the 5th dimension or R5. You could imagine this as different possible futures (maybe even your multiverse picture). If you have now a membrane in that is R5, you can imagine that much like a drum and when you strike it, it vibrates like a drum showing you the 6th (R6) dimension or a new infinite space with 3 spatial dimensions within which each point represented a snapshot of our 3 dimensional space. So continuing on you can quickly see that R6 can then via the assumption be shrunk down to a point and the process repeated. Continuing you would see the pattern that dimensions that can be written as R(3n) where n is a whole number are "spaces" and by assumption equivalently points, R(3n+1) are lines similar to our time line, and R(3n+2) are membranes or planes.
Now this is for whole numbers of n. Clearly there exists a R(0) which is the infinitesimal points in our world. Since it can be written R(3n) is can then be assumed to also be an infinite space. Inside this space we can then start going backwards using n as negative integers. R(-1) is equivalent to R(3n+2) where n = -1. So it is a membrane. Further down is R(-2) which can be written R(3n+1) where n = -1 so R(-2) is a line and lastly R(3n) where n = -1 is a point in that space. Now you can then use this method of "visualization" just as I did in Linear Algebra to try to imagine what different dimensions "look" like. Clearly, this is only for the purposes of helping my 18 year-old self get through my freshman year, but I thought that now you could at least say you've met someone who tried to visualize not only 6 dimensions below ours but "n" dimensions above and below our ours.
I know you're a busy man and I don't want to take any more of your time, but thank you again for writing your book and keep up the good work, and with that I'll leave you with a poem your book has inspired.
I have written a beautiful proof
String theory is true! No goof!
The letters are quite small
Plancnk's constant they are tall
Just trust me, I'm telling the truth.
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