# Tunnelled pentagonal rotunda

Tunnelled pentagonal rotunda Rank3
TypeQuasi-convex Stewart toroid
Elements
Faces5 pentagons, 5 squares, 5+5+5+5+5 triangles
Edges60
Vertices10+5+5+5
Measures (edge length 1)
Central density0
Related polytopes
Convex hullPentagonal rotunda
Abstract & topological properties
Flag count240
Euler characteristic0
OrientableYes
Genus1
Properties
SymmetryH2×I, order 10
ConvexNo

The tunnelled pentagonal rotunda 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 tunnelled pentagonal rotunda 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.