Tunnelled pentagonal rotunda

Tunnelled pentagonal rotunda
	(3D model); (OFF file)
Rank	3
Type	Quasi-convex Stewart toroid
Elements
Faces	5 pentagons, 5 squares, 5+5+5+5+5 triangles
Edges	60
Vertices	10+5+5+5
Measures (edge length 1)
Central density	0
Related polytopes
Convex hull	Pentagonal rotunda
Abstract & topological properties
Flag count	240
Euler characteristic	0
Orientable	Yes
Genus	1
Properties
Symmetry	H2×I, order 10
Convex	No

Vertex coordinates[edit | edit source]

The vertex coordinates for a tunnelled pentagonal rotunda with edge length 1 can be given as:

$\left(0,\,\sqrt{\frac{5+\sqrt{5}}{10}},\,\sqrt{\frac{5+2\sqrt{5}}{5}}\right)$ ,
$\left(\pm\frac{1}{2},\,-\sqrt{\frac{5+2\sqrt{5}}{20}},\,\sqrt{\frac{5+2\sqrt{5}}{5}}\right)$ ,
$\left(\pm\frac{1+\sqrt{5}}{4},\,\sqrt{\frac{5-\sqrt{5}}{40}},\,\sqrt{\frac{5+2\sqrt{5}}{5}}\right)$ ,
$\left(0,\,-\sqrt{\frac{5+2\sqrt{5}}{5}},\,\sqrt{\frac{5+\sqrt{5}}{10}}\right)$ ,
$\left(\pm\frac{1+\sqrt{5}}{4},\,\sqrt{\frac{25+11\sqrt{5}}{40}},\,\sqrt{\frac{5+\sqrt{5}}{10}}\right)$ ,
$\left(\pm\frac{3+\sqrt{5}}{4},\,-\sqrt{\frac{5+\sqrt{5}}{40}},\,\sqrt{\frac{5+\sqrt{5}}{10}}\right)$ ,
$\left(\pm\frac{1}{2},\,-\sqrt{\frac{5+2\sqrt{5}}{20}},\,\sqrt{\frac{5-\sqrt{5}}{10}}\right)$ ,
$\left(\pm\frac{1+\sqrt{5}}{4},\,\sqrt{\frac{5-\sqrt{5}}{40}},\,\sqrt{\frac{5-\sqrt{5}}{10}}\right)$ ,
$\left(0,\,\sqrt{\frac{5+\sqrt{5}}{10}},\,\sqrt{\frac{5-\sqrt{5}}{10}}\right)$ ,
$\left(\pm\frac{1}{2},\,\pm\frac{\sqrt{5+2\sqrt{5}}}{2},\,0\right)$ ,
$\left(\pm\frac{3+\sqrt{5}}{4},\,\pm\sqrt{\frac{5+\sqrt{5}}{8}},\,0\right)$ ,
$\left(\pm\frac{1+\sqrt{5}}{2},\,0,\,0\right)$ .

The two elongations of the tunnelled pentagonal rotunda differ in the shape of their tunnels.

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.^[1] It cannot be gyroelongated.

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

Stewart, Bonnie (1964). Adventures Amoung the Toroids (2 ed.). ISBN 0686-119 36-3.