The formula for the quantum amplitude of the Veneziano dual resonance model is shown to be formally analogous to the dimensionality of a K-theoretical fractal quotient manifold of the non-commutative geometrical type. Subsequently this analogy is used to deduce the ordinary energy of the quantum particle and the dark energy of the quantum wave. The results agree completely with cosmological measurements. Even more surprisingly the sum of both energy expressions turned out to be exactly equal to Einstein’s iconic formula E = mc². Consequently Einstein’s formula makes no distinction between ordinary and dark energy.