While playing around in Blender, I recently stumbled across a certain shape. The shape is found by taking the volume shared between two identical horn tori rotated at right angles to each other:
The shape itself looks like this (Assuming lemma 1 below turns out to be true. Otherwise it may look somewhat different):
Onto the question: First of all, I would like to know if this shape has a name, and, if so, what it's name is. I searched on Google for quite a while, but I couldn't find anything with any of the vague terms I could think up. Second, I'm curious to know what the volume of this shape is.
I believe the surface of the shape is identical to the surface found by integrating the translation of one circle around another circle perpendicular to the first [lemma 1]. If one instead translates a disk, the volume of the integrated shape is equal to the volume of a cylinder with length equal to the diameter of the circle (i.e., $V_1 = \frac{d^3}{4}$). The "double-horn-torus" should have equal volume to this shape, minus the volume of two of the "circular cones" [lemma 2] (It seems to be a revolution of the area under a circle [lemma 3], so $V_2 = \frac{\pi \left(4-\pi \right)}{6}r^3$).
From this, I deduce that the volume is $V = V_1-V_2 = d^3\left(\frac{12 - 4\pi + \pi^2}{48}\right)$, where $d$ is the major length.
This is based upon some pretty flimsy lemmas, though. Especially lemma 1, which is just a theory with purely visual evidence; I have no idea if the "double-horn-torus" actually does have the same surface as the revolved circle. If anyone could give the correct volume (or something better than a guess linking the two surfaces, if they are indeed the same), I would be much obliged.