Compatible Euler's Identity Based on Golden Ratio Self-Similar Scaling and Its Geometric Modeling Research

The classical Euler's identity e^{i\pi}+1=0 describes phase symmetry of a unit circle with fixed radius yet lacks degrees of freedom for scale transformation. By constructing a solid of revolution cone constrained by golden section, this paper endows the coefficient \varphi^{-n} in the algebraic deformation \varphi^{-n}(e^{i\pi}+1)=0 with rigorous geometric meaning of self-similar radius sequence, realizing the generalization from single-circle symmetry to global symmetry of an infinitely nested self-similar annular family. Following the recursive rule that upper-layer golden section points impose geometric constraints on lower-layer counterparts, explicit parametric equations for golden logarithmic spirals and 3D self-similar spatial helices are derived rigorously. Preliminary theoretical frameworks are established for electromagnetic fields of tapered conical horn antennas and one-dimensional fractal potential fields. The proposed symbolic system unifies six fundamental mathematical constants: e,i,\pi,\varphi,1,0, constructing a geometric bridge linking classical complex analysis and naturally occurring self-similar growth morphologies.

经典欧拉恒等式 e^{i\pi}+1=0 描述固定半径单位圆的相位对称,缺乏尺度变换自由度。本文通过构造黄金分割回转圆锥,赋予代数变形 \varphi^{-n}(e^{i\pi}+1)=0 中的系数 \varphi^{-n} 以客观的自相似半径序列几何意义,实现了从“单圆对称”到“无穷嵌套自相似圆环族全域对称”的推广。基于“上层黄金分割点约束下层分割点”的递推关系,严格导出了黄金对数螺线与三维自相似空间螺旋的显式参数方程,并在渐变圆锥喇叭天线电磁场与一维分形势场中建立了初步理论模型。该符号体系统一了 e,i,\pi,\varphi,1,0 六大常数,为连接经典复分析与自相似生长结构提供了几何桥梁。