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
Volume element formula_25. In coordinates, where formula_27 is the determinant of the metric tensor "g" written in the coordinate system. A simple example of a volume element can be explored by considering a two-dimensional surface embedded in "n"-dimensional Euclidean space. Such a volume element is sometimes called an "area element". Consider a subset formula_28 and a mapping function thus defining a surface embedded in formula_30. In two dimensions, volume is just area, and a volume element gives a way to determine the area of parts of the surface. Thus a volume element is an expression of the form that allows one to