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
Surface energy mass of liquid by an amount, δA, a quantity of work, γδA, is needed (where γ is the surface energy density of the liquid). However, such a method cannot be used to measure the surface energy of a solid because stretching of a solid membrane induces elastic energy in the bulk in addition to increasing the surface energy. The surface energy of a solid is usually measured at high temperatures. At such temperatures the solid creeps and even though the surface area changes, the volume remains approximately constant. If γ is the surface energy density of a cylindrical rod of