Tunnelled pentagonal rotunda

The  is a quasi-convex Stewart toroid. It can be made by excavating a pentagonal rotunda by a pentagonal antiprism and a pentagonal cupola.

Vertex coordinates
The vertex coordinates for a tunnelled pentagonal rotunda with edge length 1 can be given as:
 * $$\left(0,\,\sqrt{\frac{5+\sqrt{5}}{10}},\,\sqrt{\frac{5+2\sqrt{5}}{5}}\right)$$,
 * $$\left(\pm\frac{1}{2},\,-\sqrt{\frac{5+2\sqrt{5}}{20}},\,\sqrt{\frac{5+2\sqrt{5}}{5}}\right)$$,
 * $$\left(\pm\frac{1+\sqrt{5}}{4},\,\sqrt{\frac{5-\sqrt{5}}{40}},\,\sqrt{\frac{5+2\sqrt{5}}{5}}\right)$$,
 * $$\left(0,\,-\sqrt{\frac{5+2\sqrt{5}}{5}},\,\sqrt{\frac{5+\sqrt{5}}{10}}\right)$$,
 * $$\left(\pm\frac{1+\sqrt{5}}{4},\,\sqrt{\frac{25+11\sqrt{5}}{40}},\,\sqrt{\frac{5+\sqrt{5}}{10}}\right)$$,
 * $$\left(\pm\frac{3+\sqrt{5}}{4},\,-\sqrt{\frac{5+\sqrt{5}}{40}},\,\sqrt{\frac{5+\sqrt{5}}{10}}\right)$$,
 * $$\left(\pm\frac{1}{2},\,-\sqrt{\frac{5+2\sqrt{5}}{20}},\,\sqrt{\frac{5-\sqrt{5}}{10}}\right)$$,
 * $$\left(\pm\frac{1+\sqrt{5}}{4},\,\sqrt{\frac{5-\sqrt{5}}{40}},\,\sqrt{\frac{5-\sqrt{5}}{10}}\right)$$,
 * $$\left(0,\,\sqrt{\frac{5+\sqrt{5}}{10}},\,\sqrt{\frac{5-\sqrt{5}}{10}}\right)$$,
 * $$\left(\pm\frac{1}{2},\,\pm\frac{\sqrt{5+2\sqrt{5}}}{2},\,0\right)$$,
 * $$\left(\pm\frac{3+\sqrt{5}}{4},\,\pm\sqrt{\frac{5+\sqrt{5}}{8}},\,0\right)$$,
 * $$\left(\pm\frac{1+\sqrt{5}}{2},\,0,\,0\right)$$.

Related polytopes
The can be elongated in two ways to form quasi-convex Stewart toroids, both with the elongated pentagonal rotunda as their convex hull. It cannot be gyroelongated.

12 tunnelled pentagonal rotundae are used in the construction of the Webb toroid.