石头-剪刀-布曼德博集合
A rock-paper-scissors Mandelbrot set
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中文总结 AI 辅助
本文构造了基于石头-剪刀-布运算的三维实代数上的曼德博集合类似物，引入有限维实代数函数全纯性概念，证明其等价于解推广的柯西-黎曼方程组，并提供分形动画生成与游戏探索代码。
中文摘要 AI 辅助
本文所研究的对象是三维实代数上的曼德博集合类似物，该代数的乘法是石头-剪刀-布运算的双线性扩展。为研究该环境下映射$x\mapsto x^2+c$的动力学，本文引入了有限维实代数上函数的全纯性概念。在一个温和假设下，证明此类代数的全纯性等价于求解推广柯西-黎曼方程的有限一阶线性偏微分方程组。本文提供了生成这些分形动画及以电子游戏形式探索它们的代码。
英文摘要
The titular object of this paper is an analogue of the Mandelbrot set over the 3-dimensional real algebra whose multiplication is the bilinear extension of the rock-paper-scissors operation. In order to study the dynamics of the mappings $x\mapsto x^2+c$ in this setting, a notion of holomorphy for functions on general finite-dimensional real algebras is introduced. Under a mild assumption it is shown that for such algebras holomorphy is always equivalent to solving a finite system of linear first-order PDEs generalizing the Cauchy-Riemann equations. Code is provided for generating animations of these fractals and for exploring them in a video game format.
