I've done the following construction. I expected to get the Clifford torus or another "Hopf torus", such as the ones we can see here: lobed Hopf tori.
Here is my construction. First I take the "inverse" Hopf map: $$ H^{-1}(q,t) = \frac{1}{\sqrt{2(1+q_1)}} \begin{pmatrix} -(1+q_1)\sin t \\ (1+q_1) \cos t \\ q_2 \cos t + q_3 \sin t \\ -q_2 \sin t + q_3 \cos t \end{pmatrix}, q \in S^2, t \in \mathbb{R}. $$
I generate all the great circles of $S^2$: $$ C_{\theta,\phi} = \begin{pmatrix} \cos \theta \sin \phi \\ \sin \theta \sin \phi \\ \cos \phi \end{pmatrix}. $$
Then I calculate the circles of $S^3$ obtained by the inverse Hopf map as follows: $$ [0, 2\pi] \ni t \mapsto H^{-1}(C_{\theta, \phi}, t) $$
Finally I apply the stereographic projection to these circles in $\mathbb{R}^4$ in order to map them to $\mathbb{R}^3$. And this is the picture I obtain:
Here is the same object but drawn in a different way:
Its not ugly but this does not look like a torus. And I've never seen such a picture elsewhere. What is it?
More importantly, I would like to know how to obtain the lobed Hopf tori (lobed Hopf tori). Is the way to get them close to my construction of am I totally in a wrong way ? In case I'm totally in a wrong way I would appreciate a couple of hints.
I absolutely don't remember how I got the strange shape in my question...
Well, here is how to get a Hopf torus.
The preimage of a point $p=(p_x,p_y,p_z)$ on the unit sphere $S^2$ by the Hopf map is the circle on $S^3$ with parameterization: $$ \begin{array}{ccc} \mathcal{C}_p \colon & (0,2\pi[ & \longrightarrow & S^3 \\ & \phi & \longmapsto & \mathcal{C}_p(\phi) \end{array} $$ where $$ \mathcal{C}_p(\phi) = \frac{1}{\sqrt{2(1+p_z)}} \begin{pmatrix} (1+p_z) \cos(\phi) \\ p_x \sin(\phi) - p_y \cos(\phi) \\ p_x \cos(\phi) + p_y \sin(\phi) \\ (1+p_z) \sin(\phi) \end{pmatrix}. $$
Now consider a spherical curve. That is, let $\Gamma$ be a function mapping an interval $I \subset \mathbb{R}$ to the unit sphere $S^2$. Then the Hopf cylinder corresponding to $\Gamma$ has parameterization $$ \begin{array}{ccc} H_\Gamma \colon & I \times (0,2\pi[ & \longrightarrow & S^3 \\ & (t, \phi) & \longmapsto & \mathcal{C}_{\Gamma(t)}(\phi) \end{array}. $$ When the spherical curve is closed, its corresponding Hopf cylinder is called a Hopf torus. Then we can get a beautiful 3D surface by stereographic projection. A "lobed Hopf torus" as the ones shown here is obtained by taking for a sinusoidal-like spherical curve, and the number of lobes corresponds to the number of sinusoidal waves.
For example, consider the tennis ball curve, given for a real constant $A$ and an integer constant $n$ by: $$ \Gamma(t) = \begin{pmatrix} \sin\bigl(\pi/2 - (\pi/2 - A) \cos(nt)\bigr) \cos\bigl(t + A \sin(2nt)\bigr) \\ \sin\bigl(\pi/2 - (\pi/2 - A) \cos(nt)\bigr) \sin\bigl(t + A \sin(2nt)\bigr) \\ \cos\bigl(\pi/2 - (\pi/2 - A) \cos(nt)\bigr) \end{pmatrix}, \quad t \in (0,2\pi[. $$ Then $\Gamma$ is $(2\pi/n)$-periodic and $n$ corresponds to the number of lobes of the Hopf torus. Here is the result:
Another example: take the closed curve $\Gamma \colon [0,2\pi] \to S^2$ defined by $$ \begin{cases} \Gamma_x \colon t \mapsto \sin\bigl(h\cos(nt)\bigr) \\ \Gamma_y \colon t \mapsto \cos(t)\cos\bigl(h\cos(nt)\bigr) \\ \Gamma_z \colon t \mapsto \sin(t)\cos\bigl(h\cos(nt)\bigr) \end{cases} $$ where $h$ is a real number and $n$ is an integer, corresponding to the number of lobes as before.