平方剩余函数及其图案美

Quadratic Residue Function and Its Pattern Beauty

在二次同余式的基础上，定义了一维、二维和多重平方剩余函数，得到了平方剩余函数的主要性质．平方剩余函数是一种非线性映射，具有优良的对称性和丰富的周期．利用计算机生成的图像显示了平方剩余函数“方中见圆，圆中见方”的古典图案美．与基于迭代的分形生成算法相比，这种基于取模的图像生成算法的时间复杂度很低．

Considering quadratic congruence formula, one dimension, two dimensions and multiple quadratic residue functions (QRF) are defined. QRF is a special nonlinear mapping which has excellent symmetrical property and abundant periods. The main properties of QRF is discussed in the paper. The images generated by computer based on QRF show “square in circle, and circle in square” characteristics in classic pattern art. Comparing with iteration algorithm for fractal image, the generation algorithm based on modulo has lower time complexity.