Why are numbers 10, and 26 dimensions singled out in string theory?
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The simple answer is this: because that's what it takes to get answers that are not "silly." [See source.]
Here's one example. M theory. Witten had to add one more, for eleven total, to get rid of the silly answer that there were five, not one, string theories.
BTW, were you aware that when one adds a fifth dimension to the general theory of relativity, the Maxwell equations appear? That's right. Gravity and EM force in one set of equations by invoking one extra spatial dimension. That's in "Cosmos" which I recall you've read three times now.
Here's one example. M theory. Witten had to add one more, for eleven total, to get rid of the silly answer that there were five, not one, string theories.
BTW, were you aware that when one adds a fifth dimension to the general theory of relativity, the Maxwell equations appear? That's right. Gravity and EM force in one set of equations by invoking one extra spatial dimension. That's in "Cosmos" which I recall you've read three times now.
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The mathematical theory of modular functions is where this is all explained. But no one has the slightest understanding why 10 and 26 dimensions are singled out as the dimensions in which strings vibrate.