Is there a geometrical derivation of e

[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

 

[Mark44]

None that I've ever seen...

 

[DaTario]

Thank you, Mark44.

 

[fresh_42]

I would like to know if there is any derivation of e based on geometrical arguments.

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 $1$.
$$ 1 = \int_1^e \frac{1}{x}dx$$

 

[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 $2\pi$.

 

[fresh_42]

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.

 

[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?

 

[DaTario]

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

 

[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...

 

[fresh_42]

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?

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.

 

[DaTario]

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.View attachment 111493

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.

 

[DaTario]

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

 

[Mark44]

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

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

 

[DaTario]

Sorry Mark44.

 

[DaTario]

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

 

[fresh_42]

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

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.

 

[DaTario]

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.

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 preferred base.

 

[Mark44]

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 preferred base.

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.

 

[DaTario]

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

 

[Mark44]

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

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.

 

[DaTario]

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.

Dear Mark, we are speaking of e appearing as a singular real number in a geometrical context. Statements where e can be exchanged by any other positive real number does not seem to help.
We may say that it was only a matter of the author´s preference.

 

[Mark44]

Dear Mark, we are speaking of e appearing as a singular real number in a geometrical context. Statements where e can be exchanged by any other positive real number does not seem to help.
We may say that it was only a matter of the author´s preference.

The number e is the nearly universally preferred base in mathematics topics.

The original question was whether there is any geometric presentation of the natural number e. The answer appears to be, no.

Can we end this thread?

 

[micromass]

The answer appears to be, no.

The answer is yes, but the OP is not very happy with this answer, it appears.

 

[DaTario]

I am satisfied with the contributions. It is perhaps the case that the question per se is not so well formulated. I am sorry. You may end the thread. I also believe the answer to the OP is yes.