Let ${\displaystyle \phi }$  be the golden ratio. The 12 points given by ${\displaystyle (0,\pm 1,\pm \phi )}$  and cyclic permutations of these coordinates are the vertices of a regular icosahedron. Its dual regular dodecahedron, whose edges intersect those of the icosahedron at right angles, has as vertices the points ${\displaystyle (\pm 1,\pm 1,\pm 1)}$  together with the points ${\displaystyle (\pm \phi ,\pm 1/\phi ,0)}$  and cyclic permutations of these coordinates. Multiplying all coordinates of the icosahedron by a factor of ${\displaystyle (3\phi +12)/19\approx 0.887\,057\,998\,22}$  gives a slightly smaller icosahedron. The 12 vertices of this icosahedron, together with the vertices of the dodecahedron, are the vertices of a pentakis dodecahedron centered at the origin. The length of its long edges equals ${\displaystyle 2/\phi }$ . Its faces are acute isosceles triangles with one angle of ${\displaystyle \arccos((-8+9\phi )/18)\approx 68.618\,720\,931\,19^{\circ }}$  and two of ${\displaystyle \arccos((5-\phi )/6)\approx 55.690\,639\,534\,41^{\circ }}$ . The length ratio between the long and short edges of these triangles equals ${\displaystyle (5-\phi )/3\approx 1.127\,322\,003\,75}$ .

 
The pentakis dodecahedron in a model of buckminsterfullerene: each (spherical) surface segment represents a carbon atom, and if all are replaced with planar faces, a pentakis dodecahedron is produced. Equivalently, a truncated icosahedron is a model of buckminsterfullerene, with each vertex representing a carbon atom.

The pentakis dodecahedron is also a model of some icosahedrally symmetric viruses, such as Adeno-associated virus. These have 60 symmetry related capsid proteins, which combine to make the 60 symmetrical faces of a pentakis dodecahedron.

The pentakis dodecahedron has three symmetry positions, two on vertices, and one on a midedge:

A concave pentakis dodecahedron replaces the pentagonal faces of a dodecahedron with inverted pyramids.