Fractal spatio-temporal scale-free messaging: Amplitude modulation of self-executable carriers given by the Weierstrass function's components

Abstract

In communication systems, the circumstances and capabilities of senders and receivers cannot be known/assumed beforehand so as to design optimal semantic transference strategies. Regardless of the recipient (plants, insects, or even life forms unknown on Earth), the spatio-temporal scale of a message could be inappropriate and may never be decoded due to incompatibilities at both ends. We devise a new method to encode messages that is agnostic vis-a-vis space and time scales. We propose the use of fractal functions as self-executable carriers for sending messages, given their properties of structural self-similarity and scale invariance. We call this ‘fractal messaging’. Starting from a spatial embedding, we introduce a framework for a space-time scale-free messaging approach. In creating a space and time agnostic framework for message transmission, encoding a message that could be decoded at several spatio-temporal scales is the objective. Our core idea is to encode a binary message as waves along infinitely many frequencies (in power-like distributions) and amplitudes, transmit such a message, and then decode and reproduce it. To do so, the components/cycles of the Weierstrass function, a known fractal, are used as carriers of the message. Each component will have its amplitude modulated to embed the binary stream, allowing for a space-time agnostic approach to messaging.