Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 infinity. Does anyone know of a stronger relationship beyond, "they sort of look similar"?

Here are some links:
Volume formula

Taylor Series

share|improve this question
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

1 Answer 1

up vote 4 down vote accepted

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.

share|improve this answer
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? –  Jonathan Fischoff Jul 26 '10 at 18:26
@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). –  Grigory M Jul 27 '10 at 5:52

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.