  We discuss the geometry of Laplacian eigenfunctions $-\Delta \phi = \lambda
\phi$ on compact manifolds $(M,g)$ and combinatorial graphs $G=(V,E)$. The
'dual' geometry of Laplacian eigenfunctions is well understood on
$\mathbb{T}^d$ (identified with $\mathbb{Z}^d$) and $\mathbb{R}^n$ (which is
self-dual). The dual geometry is of tremendous role in various fields of pure
and applied mathematics. The purpose of our paper is to point out a notion of
similarity between eigenfunctions that allows to reconstruct that geometry. Our
measure of 'similarity' $ \alpha(\phi_{\lambda}, \phi_{\mu})$ between
eigenfunctions $\phi_{\lambda}$ and $\phi_{\mu}$ is given by a global average
of local correlations $$ \alpha(\phi_{\lambda}, \phi_{\mu})^2 = \|
\phi_{\lambda} \phi_{\mu} \|_{L^2}^{-2}\int_{M}{ \left( \int_{M}{ p(t,x,y)(
\phi_{\lambda}(y) - \phi_{\lambda}(x))( \phi_{\mu}(y) - \phi_{\mu}(x)) dy}
\right)^2 dx},$$ where $p(t,x,y)$ is the classical heat kernel and $e^{-t
\lambda} + e^{-t \mu} = 1$. This notion recovers all classical notions of
duality but is equally applicable to other (rough) geometries and graphs; many
numerical examples in different continuous and discrete settings illustrate the
result.
