Editing Pentakis dodecahedron (section)

==Cartesian coordinates==

Let <math>\phi</math> be the [[golden ratio]]. The 12 points given by <math>(0, \pm 1, \pm \phi)</math> 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 <math>(\pm 1, \pm 1, \pm 1)</math> together with the points <math>(\pm\phi, \pm 1/\phi, 0)</math> and cyclic permutations of these coordinates. Multiplying all coordinates of the icosahedron by a factor of <math>(3\phi+12)/19\approx 0.887\,057\,998\,22</math> 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 <math>2/\phi</math>. Its faces are acute isosceles triangles with one angle of <math>\arccos((-8+9\phi)/18)\approx 68.618\,720\,931\,19^{\circ}</math> and two  of <math>\arccos((5-\phi)/6)\approx 55.690\,639\,534\,41^{\circ}</math>. The length ratio between the long and short edges of these triangles equals <math>(5-\phi)/3\approx 1.127\,322\,003\,75</math>.