Isomorphism means same shape, different costume
An isomorphism is a reversible map that preserves structure end to end. If one exists, the two objects share every property that depends only on that structure—so mathematicians often treat them as the same thing, "up to isomorphism."
The word blends Greek for equal and form. Interest lies in sameness of structural properties, ignoring labels and extra decoration. Different constructions of the real numbers—digit expansions, Dedekind cuts, Cauchy sequences—are identified as one continuum because they are isomorphic. Yet isomorphic substructures of a larger object need not be collapsed: all one-dimensional subspaces of a vector space are isomorphic, but identifying them would erase their distinct positions. A self-map that is an isomorphism—from a structure onto itself—is called an automorphism. A canonical isomorphism is the unique one, or the conspicuously natural one; all fields with a fixed prime number of elements are uniquely isomorphic.
For algebraic structures, the relevant maps are called homomorphisms, and a homomorphism is an isomorphism precisely when it is bijective. Other fields coin specialized names: diffeomorphisms for smooth manifolds, and various "transformations" in geometry—rigid, affine, projective. Category theory supplies a unifying language for structure-preserving maps across domains.
Familiar examples sharpen the idea. The exponential and logarithm give inverse isomorphisms between the additive reals and the positive reals under multiplication—turning products into sums, the principle behind slide rules. The Chinese remainder theorem yields a ring isomorphism between integers modulo a product of coprime moduli and a product of smaller modular rings. For sets with a binary relation, a bijective map that preserves and reflects the relation carries reflexivity, symmetry, transitivity, and order properties back and forth. Whenever two constructions feel interchangeable, an isomorphism is usually the warrant.
Source: Isomorphism
