Platonic solid to that of a sphere of the same size (i.e. either the same surface area or the same volume.) The dodecahedron, on the other hand, has the smallest angular defect, the largest vertex solid angle, and it fills out its circumscribed sphere the most. Every polyhedron has a dual (or "polar") polyhedron with faces and vertices interchanged. The dual of every Platonic solid is another Platonic solid, so that we can arrange the five solids into dual pairs. If a polyhedron has Schläfli symbol {"p", "q"}, then its dual has the symbol {"q", "p"}. Indeed, every combinatorial property of one
Steinmetz solid observation that the tricylinder can be resampled by the cube with the vertices where 3 edges meet (s. diagram) and 6 curved pyramids (the triangles are parts of cylinder surfaces). The volume and the surface area of the curved triangles can be determined by similar considerations as it is done for the bicylinder above. The volume of a tricylinder is and the surface area is With four cylinders, with axes connecting the vertices of a tetrahedron to the corresponding points on the other side of the solid, the volume is With six cylinders, with axes parallel to the diagonals of