Something I Should Have Said On March 14
Jun. 15th, 2019 08:28 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowIn high school geometry, people learn formulas. "The area of a circle is pi r squared". "The circumference of a circle is 2 pi r." As far as I know, they don't learn the real point, the magic of Euclidean geometry.
Here's the magic: the ratio between the area of a circle and the square of its radius is the same as the ratio between its circumference and its diameter, and that ratio is the same for all circles.
That's it. That's what those formulas are about. In a non-Euclidean space, they are false - false in all clauses.
I'll admit that I didn't see that that was the point until I was middle-aged, and I speak as a professional mathematician; it's no wonder layfolk never see it.
How shall they know, if they are not told?
Here's the magic: the ratio between the area of a circle and the square of its radius is the same as the ratio between its circumference and its diameter, and that ratio is the same for all circles.
That's it. That's what those formulas are about. In a non-Euclidean space, they are false - false in all clauses.
I'll admit that I didn't see that that was the point until I was middle-aged, and I speak as a professional mathematician; it's no wonder layfolk never see it.
How shall they know, if they are not told?
