Duality becomes apparent when you work the 3 x 3 magic square. Self-duality marks some numbers; others become dual to opposite number pairs. Dual nature spills over into the Platonic solids. We’ve already discussed some ways on Revivingantiquity.com that explains how the same property gets transferred to the five Platonic solids. The 3 x 3 magic square builds these five regular polyhedrons in so many ways.
Duality in specific pairs exists between four of the five Platonic solids. These fur are pictured above. The tetrahedron, not pictured is self-dual. Connecting mid face to a corner can only create an up-side-down tetrahedron. The pictured four figures are also dual in reverse. However, when the tetrahedron is reversed, it recreates itself smaller and up-side-down. Therefore a tetrahedron is unchangeable into a dual figure. It is self-dual, while the other four can reverse their identity and are truly dual.
Of the five solids, two become dual to each other. One can be inscribed in the other corner to mid-face. In this manner, a dodecahedron can be placed inside an icosahedron; and an octahedron (higher figure) can be inscribed inside a cube. The reverse can also happen.
Duality in the magic 3 x 3 square: Taking numbers three at the time that do not cross the center, they are dual in reverse direction. With their dual figure, they always total 1,110. Here are a some examples: 816 + 294 = 1,110. Or, 438 + 672 = 1110. So what makes a self dual number so special? Any three digit number in a straight line that crosses the central five, when added to itself backwards, total 1,110. Self dual numbers are their own compliments. They do not need an opposite number placed backwards like my two previous examples. In a mirrored image, they are self dual. Here are some examples of self-duality with numbers on this magic square; 951 + 159 = 1,110. Or. 456 + 654 = 1,110. Internal link: Geometrical Mind
Leave a Reply