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.

翻译：在许多通信场景中，无论通信主体是细胞、植物、昆虫，甚至是地球未知的生命形式，其能力范围都无法预先知晓。无论接收者是谁，消息的时空尺度可能过快、过慢、过大或过小，因而可能永远无法被解码。因此，设计一种不依赖时空尺度的消息编码方法具有重要意义。基于分形函数的结构自相似性和尺度不变性特征，我们提出将其用作自执行无限频率载波来发送消息，并称之为"分形消息传递"。从空间嵌入出发，我们建立了一种面向这一挑战的时空无标度消息传递框架。在构建时空无关的消息传输框架时，若能实现消息编码后可在多个时空尺度上解码会颇具价值。因此，本文提出框架的核心思想是：将二进制消息编码为包含无穷多频率（服从幂律分布）和振幅的波，传输后再解码还原。为此，我们采用已知分形函数——魏尔斯特拉斯函数的各分量作为消息载体，通过对每个分量进行振幅调制来嵌入二进制数据流，从而实现与时空无关的消息传递方法。