What curve is this?

Question

This is my earring (see the image please) and my question is: Does this curve have a name? If it does, which one?

Regards! And thank you.

Answer 1

It depends on how you want to describe it. One way is to project it onto a plane and see it as $26$ circles, each of which passes through the origin. Let $r$ be the radius. Each one has the form $x^2-2xx_c+y^2-2yy_c=0$ with $(x_c,y_c)$ the center of the circle and $x_c^2+y_c^2=r^2$. If the circles are equally spaced and pass through the origin, we have $(x_c,y_c)_n=(r \cos \frac {n\pi }{13},r \sin \frac {n\pi }{13})$ for $n$ ranging from $0$ through $25$.

Another view would be as a single curve that travels rapidly around the circle while the circle rotates more slowly around the origin. I would do this as the center of the circle is at $(r \cos \pi t, r \sin \pi t)$ and relative to that the point on the circle is at $(r \cos (26 \pi t - \pi), r \sin (26 \pi t - \pi))$ where the $-\pi $ represents that we start out at the origin. The total is then $(r \cos \pi t+r \cos (26 \pi t - \pi), r \sin \pi t+r \sin (26 \pi t - \pi))$

Here is a plot from Alpha

This is still in two dimensions. If you want to try to capture the $z$ variation, that is probably also sinusoidal. Maybe it is over the range $\pm \frac r6$. In that case it would be $(r \cos \pi t+r \cos (26 \pi t - \pi), r \sin \pi t+r \sin (26 \pi t - \pi), \frac r6 \sin (26 \pi t))$ but I couldn't get a nice 3D plot out of Alpha. Maybe somebody with Mathematica can do so.

Answer 2

Your earring is an example of fibers in the Hopf fibration. Watch this. Also check this.

A video of Niles talking about the Hopf fibration can be found here.

Answer 3

@Ross; I don't think the picture you have is thesame as that in the question. This is clear by mere looking at it. The circles don't pass through any origin and in particular, they don't intersect. Each circle is a fiber over a point on $S^2$.

This is a picture from Hopf fibration. In fact, it is a torus which is a fiber over the circle by the map $f$ : $S^3$ $\rightarrow$ $S^2$. Any two of the circles you see in the picture are Hopf linked . It is amazing the the circles are not just topological circles but geometric circles.

For more details, watch Niles talk on link. If you enjoyed that and wants to see more, take a look at modular fibration.