Symmetrysuch

Symmetrysuch is a term used to describe patterns, structures, or objects that exhibit multiple, interacting symmetries. It refers to configurations that remain invariant under more than one nontrivial transformation drawn from a symmetry group, with these invariances arranged across different scales or components of the system.

In mathematical discourse, symmetrysuch emphasizes hierarchical or multi-layer symmetry. If a shape S has a symmetry group G, and the space or the object contains several modules or scales each of which is invariant under a subgroup of G, S is said to be symmetrysuch with respect to that hierarchical structure.

Examples include nested tilings where a macro-pattern and its micro-pattern share the same rotational symmetry, or architectural façades that display the same dihedral symmetry both in the overall layout and in repeated decorative motifs. In crystallography and chemistry, symmetrysuch can describe molecules or crystal forms where local and global symmetries align, yielding higher-order invariance.

Origin and usage: The term is not part of a formal axiomatic theory and has appeared in interdisciplinary discussions since the early 21st century. It is used mainly as a descriptive heuristic to discuss complex symmetry, rather than as a precise mathematical category.

Related ideas include self-similarity, hierarchical symmetry, and dual symmetry. The concept overlaps with multi-scale symmetry, fractal patterns, and the study of symmetry groups in geometry and art.