# I Is there a geometrical derivation of e

1. Jan 12, 2017

### DaTario

Hi All,

Mr. James Grime from Numberphile channel has said () that the Euler´s number e has basically nothing to do with geometry.
I would like to know if there is any derivation of e based on geometrical arguments.

Best Regards,

DaTario

2. Jan 12, 2017

### Staff: Mentor

None that I've ever seen...

3. Jan 12, 2017

### DaTario

Thank you, Mark44.

4. Jan 12, 2017

### Staff: Mentor

Probably not what you're looking for, as it is no construction method (as there can't be any with compass and ruler). But you can define $e$ to be the right boundary of the area under the standard hyperbola with its left boundary $1$ that equals $1$.
$$1 = \int_1^e \frac{1}{x}dx$$

5. Jan 12, 2017

### DaTario

Yes, fresh_42,

This kind of geometrical association seems to be more inclined to the calculus itself.
Like e is the value of a base $b$ such that $b^{i \theta}$ corresponds to the complete unit circle as $\theta$ goes from $0$ to $2\pi$.

6. Jan 12, 2017

### Staff: Mentor

How would you answer your question if it was $\pi$? This is clearly a geometric number, but cannot be constructed either. It's the relation between diameter and circumference of a circle or the area of a circle to its squared radius. Circles and hyperbolas are closely related, so I find the definition above quite similar.

7. Jan 12, 2017

### DaTario

Radius, length, areas (total area), curvature are kind of anatomic parts of geometrical objects. Taking a function and doing a definite integral from a to b is a method which can generate almost anynumber, don´t you agree?

8. Jan 12, 2017

### DaTario

Take all squares having integer sides, take the inverse of these areas and sum. Finally multiply by 6 and voi la $\pi ^2$.

9. Jan 12, 2017

### DaTario

Besides, the position of the hyperbola in relation to the cartesian system plays an important role.

But the axis in this case are the assimptote...

10. Jan 12, 2017

### Staff: Mentor

No, I don't. The integral was simply shorter to write than to make a picture of the area. You can also generate any number as the area of a circle which is also an integral.

11. Jan 12, 2017

### DaTario

In the case of circle one can speak of total area. Regarding the hyperbola, it doesn´t happen. I see some difference, but I agree that there is some geometry involved here.

12. Jan 12, 2017

### DaTario

What is your opinion with respect to the example of the squares above?

13. Jan 12, 2017

### Staff: Mentor

What does this have to do with e? Wasn't that what you first asked about?

14. Jan 12, 2017

### DaTario

Sorry Mark44.

15. Jan 12, 2017

### DaTario

The importance of my question was in determining what is a geometrically based defnition. The use of $\pi$ was an exercise,

16. Jan 12, 2017

### Staff: Mentor

Here you can find some more "natural" geometric identities involving $e$
https://en.wikipedia.org/wiki/Logarithmic_spiral

I admit, $e$ is kind of hidden in comparison to $\pi$ and I wouldn't call it a geometric definition as you required. But at least it's not completely out of business.

17. Jan 12, 2017

### DaTario

Ok, but the identities on this web page seem to be related to the logarithm as an operation which does not select, in principle, a prefered base.

18. Jan 12, 2017

### Staff: Mentor

The first definition they give, for a logarithmic curve, is $r = ae^{b\theta}$. The next equation they give is for $\theta$, in terms of the natural log function, $\ln$. The parametric equations are given in terms of exponential functions involving e.

19. Jan 12, 2017

### DaTario

ok, but the parameter b tells us that any base could in principle be adopted, isn´t it?

20. Jan 13, 2017

### Staff: Mentor

The parameter doesn't have anything to do with it, as far as I can see. You can change an exponential expression in one base to any other base.

$$e^x = b^{x\log_b(e)}, b > 0, b \ne 1$$
However, in the wiki article that fresh_42 linked to, the base was e.