# Is there a relationship between $e$ and the sum of $n$-simplexes volumes?

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"?

Volume formula
http://en.wikipedia.org/wiki/Simplex#Geometric_properties

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– kennytm Jul 23 '10 at 17:58
The function e^x is the solution of functional equation exp(x+y)=exp(x)exp(y) s.t. exp'(0)=1. I wonder, if one can see that the generating function for simplex volumes satisfies this equation... – Grigory M Jul 23 '10 at 18:00
@Kenny I messed up the question, I meant e^x, is that what is confusing you? – Jonathan Fischoff Jul 23 '10 at 18:15
@Jon: No, I was asking the definition of "sum of simplexes volumes from n to infinity". – kennytm Jul 23 '10 at 19:09
Oh yes. But not neccarily unity. Depends on what x is. Like e^1.5i could be thought of as adding and subtracting oriented volumes that are not unity ... I think – Jonathan Fischoff Jul 23 '10 at 20:32

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.