10
$\begingroup$

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

  • 0
    Do you mean [$\sum_{n=0}^\infty \text{volume of unit-}n\text{-simplex}$](http://latex.codecogs.com/gif.latex?%5Csum_%7Bn%3D0%7D%5E%5Cinfty%20%5Ctext%7Bvolume%20of%20unit-%7Dn%5Ctext%7B-simplex%7D)?2010-07-23
  • 0
    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...2010-07-23
  • 0
    @Kenny I messed up the question, I meant e^x, is that what is confusing you?2010-07-23
  • 0
    @Jon: No, I was asking the definition of "sum of simplexes volumes from n to infinity".2010-07-23
  • 0
    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 think2010-07-23
  • 0
    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

1 Answers 1

4

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.

  • 0
    I think understand the basic idea. The relationship between the border of the simplex and its volume, is such it can phrased in a way that satisfy's the same functional equation that equation that e satisfies, mainly that it's own derivative? Is that close?2010-07-26
  • 1
    @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