Two Mathematical Examples

Suppose we are in some nice \(D\) -dimensional space, and we are considering objects \(A_{\ell}\) of dimension \(\ell=0, \ldots, D\) . Then there is a duality of \(A_{\ell}\) with \({B}_{\tilde \ell}\) , where \(\tilde{\ell} = D-\ell\) . In other words, the duality takes objects of dimension \(\ell\) to objects of dimension \(\tilde{\ell}\) . For example, for \(D=2\) , a 0-dimensional object (a point) transforms into a 2-dimensional object (such as a triangle) and vice versa, and lines are dual to lines as seen below. This is a manifestation of “Poincaré” duality.