Striking patterns in natural magic squares’ associated electrostatic potentials: Matrices of the 4th and 5th order

Fahimi, Peyman; Toussi, Cyrus Ahmadi; Trump, Walter; Haddadnia, Javad; Matta, Chérif F.

doi:10.1016/j.disc.2020.112229

Cited by 4 publications

(6 citation statements)

References 23 publications

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“…Ahmed [14] described many features of Franklin squares, which is used in this paper in a cryptographic application of magic squares. Fahimi et al [15] classified twelve groups of 4 by 4 squares where he discussed these groups with the minimum values of electrostatic potential. The potential and its related properties are calculated on the grids numerically using MATLAB 2018a.…”

Section: Literature Reviewmentioning

confidence: 99%

An investigation of even ordered magic squares (4, 6, and 8): characteristic polynomials, eigenvalues, and encryption

Al-Ashhab

Al-qdah²

2022

IJEECS

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In this paper, we discuss and mathematically compute the eigenvalues and the characteristic polynomials of special even square matrices of orders 4x4 and 8x8. Also, we introduce two 8th order compound magic squares. The computed values are verified using Maple software. First, for the 4th order square matrix, the characteristic polynomial was derived to be: λ(λ-2s)( λ²+4Θ) with the eigenvalues: 0,2 s, and two other conjugates. In further analysis, we performed numerical classification of the squares for the matrices of order 4. Second, for the 8th order magic square, the characteristic polynomial was obtained in the form: λ3(λ-4s)(λ4+Ωλ2+θ) where Ω,Θ are constants; the eigenvalues are 0,4 s, ∓√λ1, ∓√λ2; where λ1, λ2 are the roots of the quadratic equation: λ2+Ωλ+Θ=0. Third, for the franklin square, we obtained the eigenvalues 0,4 s, and the roots of the equation: λ2+aλ+b. Finally, we suggested a hybrid image encryption technique based on Franklin magic square matrices and improved substitution technique. The proposed a grayscale image encryption/ decryption algorithm uses circular rotation of bits and Franklin magic squares’ properties in conjunction with substitution techniques to obtain a very secure algorithm against attacks.

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Section: Literature Reviewmentioning

confidence: 99%

An investigation of even ordered magic squares (4, 6, and 8): characteristic polynomials, eigenvalues, and encryption

Al-Ashhab

Al-qdah²

2022

IJEECS

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“…In recent years, there has been an increasing interest in using physics concepts to explore new patterns in magic squares [8][9][10][11][12][13][14][15][16][17]. A novel way to analyze magic squares is to represent the numbers of the square with point masses [12][13][14] or electric charges [8,10,11] distributed regularly on a 2D lattice (Fig. 2).…”

Section: Introductionmentioning

confidence: 99%

“…2). This approach provides insights into the center of mass [14], moment of inertia [12] (inertia tensor [13]), electric multipoles [11], and electrostatic potential [8,10] of magic squares. By applying the principles of physics to magic squares, researchers can gain a deeper understanding of the properties and characteristics of these intriguing mathematical objects that were previously undiscovered.…”

Section: Introductionmentioning

confidence: 99%

“…By applying the principles of physics to magic squares, researchers can gain a deeper understanding of the properties and characteristics of these intriguing mathematical objects that were previously undiscovered. For example, new research has shown that the maximum and minimum potential values within 4×4 and 5×5 magic squares are only possible at specific grid points, following a distinct pattern [8]. Additionally, the sums of potentials along horizontal and vertical lines within the grid contain certain constants [8].…”

Section: Introductionmentioning

confidence: 99%

“…For example, new research has shown that the maximum and minimum potential values within 4×4 and 5×5 magic squares are only possible at specific grid points, following a distinct pattern [8]. Additionally, the sums of potentials along horizontal and vertical lines within the grid contain certain constants [8]. These findings suggest that the distribution of electrostatic potential in magic squares is not random but rather follows a striking pattern.…”

Section: Introductionmentioning

confidence: 99%

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Quasi-Static Levitation of Magic Squares

Fahimi

2023

Preprint

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Magic squares are fascinating mathematical constructs that have captured the imagination of people for centuries. In this study, we investigate the behavior of electrically charged magic squares of order 4 under the influence of electrostatic forces. We treat the matrix elements of the magic squares as electric charges and numerically simulate their behavior using a dynamic controller. Our results demonstrate that the charges move towards a quasi-static equilibrium while preserving the original magic square arrangement and its magical properties. Additionally, the results reveal distinctive patterns in the displacement between the charges. Our study sheds light on the behavior of charged magic squares under electrostatic forces and provides insight into the physical properties of these intriguing mathematical constructs.

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Binary color-coded magic squares: A study of uniqueness under rotation/reflection, PCA, and LDA analysis

Fahimi

2024

Discrete Mathematics

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Striking patterns in natural magic squares’ associated electrostatic potentials: Matrices of the 4th and 5th order

Cited by 4 publications

References 23 publications

An investigation of even ordered magic squares (4, 6, and 8): characteristic polynomials, eigenvalues, and encryption

An investigation of even ordered magic squares (4, 6, and 8): characteristic polynomials, eigenvalues, and encryption

Quasi-Static Levitation of Magic Squares

Binary color-coded magic squares: A study of uniqueness under rotation/reflection, PCA, and LDA analysis

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