# Volume of intersection between two horn tori

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.

• What is lemma 1? Also, can't you be somewhat sure of the shape via intersection in your program? Another thing of course will be finding the formulas. Right off the bat it seems to me that the volume should depend on more than one parameter but I could be wrong. – GPerez Jun 19 '15 at 9:44
• @GPerez The lemmas are labeled in square brackets. Lemma 1 is that "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". Also, since the two horn tori are identical, the only operation that can be done on the shape is a full scale operation, and you only need one parameter to represent the scale. – Ethan MacBrough Jun 19 '15 at 18:42