It is not difficult for a class to find the relative areas of the cube and the inscribed tetrahedron and octahedron. If _s_ is the side of the cube, these areas are 6_s_^2, (1/2)_s_^2[sqrt]3, and _s_^2[sqrt]3; that is, the area of the octahedron is twice that of the tetrahedron inscribed in the cube. Somewhat related to the preceding paragraph is the fact that the edges of the five regular solids are incommensurable with the radius of the circumscribed sphere. This fact seems to have been known to the Greeks, perhaps to Theaetetus (_ca._ 400 B.C.) and Aristaeus (_ca._ 300 B.C.), both of whom wrote on incommensurables. Just as we may produce the sides of a regular polygon and form a regular cross polygon or stellar polygon, so we may have stellar polyhedrons. Kepler, the great astronomer, constructed some of these solids in 1619, and Poinsot, a French mathematician, carried the constructions so far in 1801 that several of these stellar polyhedrons are known as Poinsot solids. There is a very extensive literature upon this subject. The following table may be of some service in assigning problems in mensuration in connection with the regular polyhedrons, although some of the formulas are too difficult for beginners to prove. In the table _e_ = edge of the polyhedron, _r_ = radius of circumscribed sphere, _r'_ = radius of inscribed sphere, _a_ = total area, _v_ = volume. ========================================================== NUMBER | | | OF FACES| 4 | 6 | 8 --------+-----------------+--------------+---------------- _r_ | _e_[sqrt](3/8) |(_e_/2)[sqrt]3| _e_[sqrt](1/2) | | | _r'_ | _e_[sqrt](1/24) | _e_/2 | _e_[sqrt](1/6) | | | _a_ | _e_^2[sqrt]3 | 6_e_^2 | 2_e_^2[sqrt]3 | | | _v_ |(_e_^3/12)[sqrt]2| _e_^3 |(_e_^3/3)[sqrt]2 ---------------------------------------------------------- ======================================================================== NUMBER | | OF FACES| 12 | 20 --------+----------------------------------+---------------------------- _r_ |(_e_/4)[sqrt]3([sqrt]5 + 1) |_e_[sqrt]((5 + [sqrt]5)/8) | | _r'_ |(_e_/2)[sqrt]((25 + 11[sqrt]5)/10)|(_e_[sqrt]3)/12([sqrt]5 + 3) | | _a_ |3_e_^2[sqrt](5(5 + 2[sqrt]5)) | (5_e_^2)[sqrt]3 | | _v_ |((_e_^3)/4)(15 + 7[sqrt]5) |((5_e_^3)/12)([sqrt]5 + 3) ------------------------------------------------------------------------