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Symmetry and symmetric transformations in mathematical imaging
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Abstract
The article delves into the intricate relationship between symmetry and mathematical imaging, spanning various mathematical disciplines. Symmetry, a concept deeply ingrained in mathematics, manifests in art, nature, and physics, providing a powerful tool for understanding complex structures. The paper explores three types of symmetries—reflection, rotational, and translational—exemplified through concrete mathematical expressions. Evariste Galois’s Group Theory emerges as a pivotal tool, providing a formal framework to understand and classify symmetric operations, particularly in the roots of polynomial equations. Galois theory, a cornerstone of modern algebra, connects symmetries, permutations, and solvability of equations. Group theory finds practical applications in cryptography, physics, and coding theory. Sophus Lie extends group theory to continuous spaces with Lie Group Theory, offering a powerful framework for studying continuous symmetries. Lie groups find applications in robotics and control theory, streamlining the representation of transformations. Benoit Mandelbrot’s fractal geometry, introduced in the late 20th century, provides a mathematical framework for understanding complex, self-similar shapes. The applications of fractal geometry range from computer graphics to financial modeling. Symmetry’s practical applications extend to data visualization and cryptography. The article concludes by emphasizing symmetry’s foundational role in physics, chemistry, computer graphics, and beyond. A deeper understanding of symmetry not only enriches perspectives across scientific disciplines but also fosters interdisciplinary collaborations, unveiling hidden order and structure in the natural and designed world. The exploration of symmetry promises ongoing discoveries at the intersection of mathematics and diverse fields of study.
Keywords
Symmetry, mathematic, imaging
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Cite this article
Wang,Y. (2024). Symmetry and symmetric transformations in mathematical imaging. Theoretical and Natural Science,31,320-323.
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Volume title: Proceedings of the 3rd International Conference on Computing Innovation and Applied Physics
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