Reciprocity may be understood as relation of action and reaction in the sense of
Hegel’s philosophical definition. Quoting
Kant, freedom and ethical necessities are reciprocally limited. In this contribution, a more mathematical than philosophical reflection about reciprocity as an ever-present dual property of everything was given. As a crystallographer, the author is familiar with the action of
Fourier transforms and the relation between a crystal lattice and its reciprocal lattice, already pointing to the duality between particles and waves. A generalization of the reciprocity term was stimulated by results of the famous
Information Relativity (IR) theory of
Suleiman with its proven physical manifestation of matter-wave duality, compared to the set-theoretical E-
Infinity theory developed by
El Naschie, where the zero set represents the pre-quantum particle, and the pre-quantum wave is assigned to the empty set boundary surrounding the pre-particle. Expectedly, the most irrational number
of the golden mean is involved in these thoughts, because this number is intimately connected with its inverse. An important role plays further
Hardy’s maximum quantum entanglement probability as the fifth power of
φ and its connection to the dark matter. Remembering, the eleven dimensions in
Witten’s M-theory may be decomposed into the
Lucas number L
5 = 11 =
φ−5 –
φ5. Reciprocity is indeed omnipresent in our world as piloting waves that accompany all observable earthen and cosmic matter. As a side effect of the IR theory some fundamental constants such as the gyromagnetic factor of the electron,
Sommerfeld’s fine-structure constant as well as the charge of the electron must be marginally changed caused by altered relativistic corrections. Consequences also arise for our vision about the evolution of life and consciousness.