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R. Ranjitham,
S. Ravichandran,
A. Ajitha,
- Assistant Professor, Department of Physics, Tagore Institute of Engineering and Technology, Salem, Tamil Nadu, India
- Assistant Professor, Department of Physics, Tagore Institute of Engineering and Technology, Salem, Tamil Nadu, India
- Assistant Professor, Department of Physics, Tagore Institute of Engineering and Technology, Salem, Tamil Nadu, India
Abstract
Mathematical symmetry is a fundamental concept describing invariance under transformations and providing a common framework for understanding structure, pattern, equivalence, and organization across mathematics and the natural sciences. From elementary reflections and rotations to abstract groups, group actions, representations, and invariant structures, symmetry supplies a language for identifying what remains unchanged when a system is transformed. This review examines mathematical symmetry from an interdisciplinary perspective, emphasizing concepts that connect pure mathematics with chemistry, physics, crystallography, engineering, computer science, and computational modeling. The discussion covers geometric, algebraic, translational, rotational, reflectional, glide, and scaling symmetries; transformation groups and group actions; discrete and continuous symmetry; invariants and automorphisms; and the role of symmetry in mathematical structures. Applications are considered in crystallography and molecular chemistry, where point groups and space groups organize structural information; in mathematical physics, where continuous symmetries are connected with conservation laws; in computational science, where symmetry can reduce redundant computation and support invariant or equivariant representations; and in pattern recognition, engineering, architecture, and biological systems. Particular attention is given to the distinction between exact, approximate, and broken symmetry and to emerging computational approaches for symmetry detection and symmetry-aware modeling. The review highlights symmetry not only as a descriptive property but also as a practical strategy for classification, reduction, design, and interpretation of complex systems.
Keywords: Mathematical symmetry; symmetry groups; group theory; transformations; geometric symmetry; invariance; rotational symmetry; reflection symmetry; crystallographic symmetry; molecular symmetry
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Emerging Trends in Symmetry
| Volume | 02 | |
| 02 | ||
| Received | 22/09/2026 | |
| Accepted | 23/09/2026 | |
| Published | 08/10/2026 | |
| Publication Time | 16 Days |