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On a side note - let's have a test of imagination. Try and make a verbal description of how would you imagine a place with fractional value of number of dimensions. Let's say, instead of mainstream 3 dimensions let it be 3.14 dimensions.
kurtwaters:
My first thought was do you mean the three familiar dimensions plus another .14 dimension or a small bit added to the 3 familiar dimensions. I chose the latter.
So I imagine a world of 3.14/3 or three 1.046.... dimensions.
My mind then approaches the problem in this manner:
a point (no dimensions ) would lengthen a little tiny bit.
a line ( one dimension) widen a little tiny bit a plane (two dimensions) would be raised a little tiny bit
a cube ( three dimensions) would cast a funny shadow.
so the 3.14 dimensional world would be a weird, blurry world as though I had the wrong glasses prescription. I would be able to feel the edge of a finite plane. And any point I tried to make would expand. I will continue to imagine.
Jee-Host:
If I may, can I suggest a couple of known conceptions that may add some more perspective to this task of imagining you decided to undertake? Feel free not to read them if you want to do this completely on your own. I'll make the text white so you won't accidentally read it if you don't want to. You're probably familiar with 4-dimensional imaginary hypercube (aka tesseract). It's kind of a common conception, there are even videos of 2D and 3D 'shadows' of tesseract as a computer model. I think observing the way it behaves both mathematically and in terms of theoretical physics can be a decent starting point for this imagination challenge. Another good conception is geometry. Most common Euclid's geometry bases on the fact that sum on triangle angles is always 180 degrees. But there are approaches that challenge this premise. Riemann and Lobachevsky went opposite ways on this and ended up having planes looking like sphere and distinct saddle-like form respectively. I get the impression that you think that you can imagine 3.14 dimensions more accurately than you already did. Thus I'm not going to give my take on it yet to not influence your effort. Please tell me if(when) you'd like me to do it.
Mike:
I came upon your puzzle late at night, wrote my initial reaction in this e-mail and then went to bed. As I lie awake waiting for sleep to come I began to realize problems. How does one actually define a dimension? A line can only be 2, right? A point always 0, no? Then I fell asleep.
In the morning I decided I needed help and before I opened this e-mail I discovered Hausdorff dimensions. Now I have read your e-mail and the cleverly hidden clues and I am imagining a fractional dimension space that would curve and twist. From the point of view of an observer inside this space he would believe he was walking a straight line, but someone outside would observe him walking in curves and would the curves be continuous? And on and on it goes, my attempting to understand.
So,now, I am ready to hear your take on it. My major in college was Math, but I had to abandon that study because I had to work to make money to pay the bills because I had a wife and two kids by then. I couldn't do both at the same time. My kids are now grown, I divorced the first wife and married another and I am, so to speak, going back to school.