Abstract A magic square is a square matrix whereby the sum of any row, column, or any one of the two principal diagonals is equal. A surrogate of this abstract mathematical construct, introduced in 2012 by Fahimi and Jaleh, is the “electrostatic potential (ESP)” that results from treating the matrix elements of the magic square as electric charges. The overarching idea is to characterize patterns associated with these matrices that can possibly be used, in the future, in reverse to generate these squares. This study focuses on squares of order 4 and 5 with 880 and 275,305,224 distinct (irreducible/unique) realizations, respectively. It is shown that characteristic patterns emerge from plots of the ESPs of the matrices representing the studied squares. The electrostatic potentials for natural magic squares exhibit a striking pattern of maxima and minima in all distinct 880 of the 4th order and all distinct 275,305,224 of the 5th order matrices. The minimum values of ESP of Dudeney groups are discussed. Equipotential points and certain constants are found among the ESP sums along horizontal and vertical lines on the square lattice. These findings may help to open a new perspective regarding magic squares unsolved problems. While mathematics often leads discovery in physics, the latter (physics) is used here to detect otherwise invisible patterns in a mathematical object such as magic squares.

中文翻译：

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自然幻方相关静电势中的显着模式：四阶和五阶矩阵

摘要 幻方是一个方阵，其中任何行、列或两条主对角线中的任何一条的和都相等。Fahimi 和 Jaleh 于 2012 年引入的这种抽象数学结构的替代物是“静电势 (ESP)”，它是将幻方的矩阵元素视为电荷而产生的。总体思路是表征与这些矩阵相关的模式，将来可能会反向使用这些矩阵来生成这些方块。本研究侧重于分别具有 880 和 275,305,224 个不同（不可约/唯一）实现的 4 阶和 5 阶平方。结果表明，特征模式从代表所研究方格的矩阵的 ESP 图中出现。自然幻方的静电势在所有 4 阶矩阵的所有不同 880 和所有 5 阶矩阵的所有不同 275,305,224 中表现出极大和极小值的惊人模式。讨论了杜德尼群的 ESP 最小值。等势点和某些常数在 ESP 和沿着方形点阵上的水平线和垂直线被发现。这些发现可能有助于开启关于幻方未解决问题的新视角。虽然数学常常引领物理学的发现，但后者（物理学）在此用于检测数学对象（如幻方）中原本不可见的模式。等势点和某些常数在 ESP 和沿着方形点阵上的水平线和垂直线被发现。这些发现可能有助于开启关于幻方未解决问题的新视角。虽然数学常常引领物理学的发现，但后者（物理学）在此用于检测数学对象（如幻方）中原本不可见的模式。等势点和某些常数在 ESP 和沿着方形点阵上的水平线和垂直线被发现。这些发现可能有助于开启关于幻方未解决问题的新视角。虽然数学常常引领物理学的发现，但后者（物理学）在此用于检测数学对象（如幻方）中原本不可见的模式。