skynugurq7

2022-07-10

The coordinates of an arc of a circle of length $\frac{2pi}{p}$ are an algebraic number, and when $p$ is a Fermat prime you can find it in terms of square roots.

Gauss said that the method applied to a lot more curves than the circle. Will you please tell if you know any worked examples of this (finding the algebraic points on other curves)?

Gauss said that the method applied to a lot more curves than the circle. Will you please tell if you know any worked examples of this (finding the algebraic points on other curves)?

Alexis Fields

Beginner2022-07-11Added 14 answers

Apparently the same exercise can be done for the lemniscate with the same result. We have this theorem

If the lemniscate can be divided in n parts with ruler and compass, then n is a power of two times a product of distinct Fermat primes.

The main difficulty, when compared to the better known theorem about the circle, appears to be the shift from circular functions (sin, cos) to elliptic functions. For instance one requires some sort of addition theorem for these functions.

This is only one more curve, but one that can be associated to the important elliptic integral $\int \frac{dt}{\sqrt{1-{t}^{4}}}$, making an appearance as the arc-length of the lemniscate. I'm guessing there is a wide class of curves that are associated to elliptics integrals this way, but I doubt that any of them would naturally be as interesting as the circle or the lemniscate.

If the lemniscate can be divided in n parts with ruler and compass, then n is a power of two times a product of distinct Fermat primes.

The main difficulty, when compared to the better known theorem about the circle, appears to be the shift from circular functions (sin, cos) to elliptic functions. For instance one requires some sort of addition theorem for these functions.

This is only one more curve, but one that can be associated to the important elliptic integral $\int \frac{dt}{\sqrt{1-{t}^{4}}}$, making an appearance as the arc-length of the lemniscate. I'm guessing there is a wide class of curves that are associated to elliptics integrals this way, but I doubt that any of them would naturally be as interesting as the circle or the lemniscate.

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