Umbilic torus

The umbilic torus or umbilic bracelet is a single-edged 3-dimensional shape. The lone edge goes three times around the ring before returning to the starting point. The shape also has a single external face. A cross section of the surface forms a deltoid.

The umbilic torus occurs in the mathematical subject of singularity theory, in particular in the classification of umbilical points which are determined by real cubic forms ${\displaystyle ax^{3}+3bx^{2}y+3cxy^{2}+dy^{3}}$. The equivalence classes of such cubics form a three-dimensional real projective space and the subset of parabolic forms define a surface – the umbilic torus. Christopher Zeeman named this set the umbilic bracelet in 1976.

The torus is defined by the following set of parametric equations.

${\displaystyle x=\sin u\left(7+\cos \left({u \over 3}-2v\right)+2\cos \left({u \over 3}+v\right)\right)}$

${\displaystyle y=\cos u\left(7+\cos \left({u \over 3}-2v\right)+2\cos \left({u \over 3}+v\right)\right)}$

${\displaystyle z=\sin \left({u \over 3}-2v\right)+2\sin \left({u \over 3}+v\right)}$

${\displaystyle {\text{for }}-\pi \leq u\leq \pi ,\quad -\pi \leq v\leq \pi }$