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

TL;DR

The paper addresses the challenge of communicating across unknown and varying space-time scales by proposing fractal messaging that uses the Weierstrass function as a self-similar, scale-free carrier. Binary data is embedded across infinitely many frequencies and amplitudes via amplitude modulation of the Weierstrass components, formalized as $\mathcal{F}_{a,b}(msg,t)=\sum_{m=0}^{\infty} a^m A_m(t,msg) \cos(b^m \pi t)$ with $A_m(t,msg)$ defined by per-bit pulses; the construction converges since $\mathcal{F}_{a,b}(msg,t) \le W_{a,b}(t)$. The framework supports decoding through inverse Fourier techniques and suggests hardware implementations (e.g., trapped-ion systems) and energy-tuning strategies, while also extending the idea to fractal-based computation via IFSs and Turing-machine encodings. The work positions fractal, scale-free signaling as a foundation for robust inter-domain communication with potential generalizations to higher-dimensional fractals and integrated sender–receiver designs.

Abstract

In many communication contexts, the capabilities of the involved actors cannot be known beforehand, whether it is a cell, a plant, an insect, or even a life form unknown to Earth. Regardless of the recipient, the message space and time scale could be too fast, too slow, too large, or too small and may never be decoded. Therefore, it pays to devise a way to encode messages agnostic of space and time scales. We propose the use of fractal functions as self-executable infinite-frequency carriers for sending messages, given their properties of structural self-similarity and scale invariance. We call it `fractal messaging'. Starting from a spatial embedding, we introduce a framework for a space-time scale-free messaging approach to this challenge. When considering a space and time-agnostic framework for message transmission, it would be interesting to encode a message such that it could be decoded at several spatio-temporal scales. Hence, the core idea of the framework proposed herein 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 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.

Figures (13)

• Figure 1: General behaviour of $W_{a,b}(t)$ as more terms (components) are considered in the summation when $a=1/2$ and $b=3$.
• Figure 2: Simple illustration of amplitude modulation (or masking) of a signal $f(t)$.
• Figure 3: Simple illustration of embedding $Re(z_e) \geq 0$ (left) and $Im(z_e) \geq 0$ (right) for $R_e = 10$, that is, the encoding of the message '01' in the Mandelbrot set. Other fractals can be used for the same purposes, including, for example, the Koch snowflake, which has a mathematical description between the Mandelbrot 2D nature and a discrete version of the signal structure of the Weiertrass function.
• Figure 4: Encoded message "hi!" using the components of $W_{1/2,3}$ as carriers. Ten components were considered, for computational simplicity.
• Figure 5: First three components of the fractal message embedding the ASCII representation of "hi!"