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When I look at the Taylor series for $e^x$ and the volume formula for oriented simplexes, it makes $e^x$ look like it is, at least almost, the sum of simplexes volumes from $n$ to $\infty$. Does anyone know of a stronger relationship beyond, "they sort of look similar"?

Here are some links:
Volume formula
http://en.wikipedia.org/wiki/Simplex#Geometric_properties

Taylor Series
http://en.wikipedia.org/wiki/E_%28mathematical_constant%29#Complex_numbers

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    This seems like the sort of thing that might be mentioned at the blog "The n-category cafe", although I wouldn't know how to begin looking for it there.2010-07-24

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The answer is, it's just a fact “cone over a simplex is a simplex” rewritten in terms of the generating function:

observe that because n-simplex is a cone over (n-1)-simplex $\frac{\partial}{\partial x}vol(\text{n-simplex w. edge x}) = vol(\text{(n-1)-simplex w. edge x})$; in other words $e(x):=\sum_n vol\text{(n-simplex w. edge x)}$ satisfies an equvation $e'(x)=e(x)$. So $e(x)=Ce^x$ -- and C=1 because e(0)=1.

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    @Jonathan Yes, something like this (I'd say "n-dimensional simplex is constructed from (n-1)-dimensional in such way that..."). In combinatorics such things happen quite often: you write down a generating function for something and then observe that it satisfies some simple differential equation (coming from reccurence relation on that something); and when you're solving differential equation you often encounter something like e^x (because it satisfies f'=f, indeed).2010-07-27