The explanation was fairly well-stated; they did a good job at simplifying the concepts of the 0-10 dimensions. I think I'll borrow the book when my local library gets it.
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The explanation was fairly well-stated; they did a good job at simplifying the concepts of the 0-10 dimensions. I think I'll borrow the book when my local library gets it.
Which dimension has the cowboy world in it? Because I totally want to go there.
I've always had trouble thinking 4th dimensionally, now they have 10? Geez, I'm fucked.
Well it's not exactly the way of showing a 10th dimensional object, even though there is technically already one. What is it called again, Hypercube or something like that?
Hypercube in 10 dimensions:
H_10 = { (x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9, x_10) : 0 <= x_i <= 1 for each i=1,2,3,4,5,6,7,8,9,10 }
In fact, four dimensional mathematics is notoriously the hardest to manage. The list of geometrical dimensions in order of how hard it is to do geometry goes: 4, 3, 2, higher dimensions (hardest to not as hardest).Quote:
I've always had trouble thinking 4th dimensionally, now they have 10? Geez, I'm fucked.
Why are the low dimensions harder? This all stems from the following fact: If you have two curves which close up on themselves in an n-dimensional space, then as long as the dimension is > 4 you can fill in the middle of the curves with disks (pieces of a surface) so that the two pieces you filled in do not intersect. This gives you a way of chopping up a high dimensional space into more manageable pieces with simpler geometry. On the other hand, in dimension four the best you can do is fill in the curves with little surface pieces that intersect in finitely many points, so you loose the decomposition of your space. Things should be worse in dimension 3, since the best you can do is find surface pieces that intersect in some curves (think of two circles linked together in space), and in some instances things are worse (see the poincare conjecture), but on the whole we have more intuition for three dimensional spaces.
I'm having enough trouble enough with the first 3 dimensions. I'll pass, thanks.