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
Menger sponge itself is the limit of this process after an infinite number of iterations. The "n"th stage of the Menger sponge, "M", is made up of 20 smaller cubes, each with a side length of (1/3). The total volume of "M" is thus (20/27). The total surface area of "M" is given by the expression 2(20/9) + 4(8/9). Therefore the construction's volume approaches zero while its surface area increases without bound. Yet any chosen surface in the construction will be thoroughly punctured as the construction continues, so that the limit is neither a solid nor a surface; it has a topological