Symmetrioid

Symmetrioid is a term used in geometry to describe objects that exhibit a pronounced degree of symmetry. The name combines symmetria, from Greek for “congruence of parts,” with the suffix -oid, indicating likeness. In practice, a symmetrioid is an object whose shape is invariant under a nontrivial finite group of isometries, typically a dihedral, cyclic, or polyhedral symmetry group.

In two dimensions, a symmetrioid is a closed plane curve whose interior remains unchanged under a set of symmetry operations; typical examples include regular polygons, star polygons, and various rounded figures sharing the same symmetry as a regular n-gon. In three dimensions, a symmetrioid is a solid whose boundary and interior are invariant under the same type of symmetry group; common instances include cubes, octahedra, dodecahedra, and other solids with multiple mirror and rotational symmetries. The defining feature is the presence of a nontrivial symmetry group that constrains the object's geometry.

Mathematically, symmetrioids are often described by their invariance under a finite subgroup of the orthogonal group, linking them to group theory, tessellations, and polyhedral geometry. They find use in computer graphics, architectural design, and molecular modeling where symmetric forms are advantageous. Terminology varies: some authors reserve “symmetrioid” for objects with maximal symmetry in a given dimension, while others apply it more broadly to highly symmetric shapes. See also symmetry, dihedral group, polyhedron, and group theory.