Floating mandalas

This article considers geometric patterns arising from infinite series of concentric circles, whose radii converge to zero. As an archetype of such design, the initial focus is on level curves of the surface given by z=sin⁡(1/r) in cylindrical polar coordinates, before generalizing to other periodic functions. Since any horizontal section of this surface comprises an infinite number of circles, there is no physical medium on which a complete visualization can be achieved. As we explore different computer-based representations, the reader will be presented with a sample of surprisingly stunning patterns involving intricate arrangements of moiré and related patterns. Especially when considering rectangular grid approximations, these patterns are reminiscent of mandala figures. We discuss how they arise due to the finite nature of any computing or display device, epitomized by floating-point approximations of real numbers. Code is provided as a means to generate an endless supply of artworks.本文研究由半径收敛至零的无限同心圆弧序列所衍生的几何图案。作为此类设计的典型范本，本文首先聚焦圆柱极坐标系（cylindrical polar coordinates）下曲面z=sin(1/r)的等值线（level curves），随后将研究推广至其他周期函数。由于该曲面的任意水平截面均包含无穷多圆弧，故而不存在可完整呈现其全貌的物理介质。在探索各类基于计算机的可视化表达形式时，本文将为读者展示一系列惊艳绝伦的精美图案，其中涵盖莫尔条纹（moiré）及其相关图案的复杂排布结构。尤其当采用矩形网格近似时，此类图案与曼陀罗（mandala）纹样极为相似。本文将探讨这类图案的成因：其根源在于所有计算与显示设备的有限性，而实数的浮点近似正是这一特性的典型缩影。本文附带可生成海量艺术作品的代码实现。