Isomorphism

In mathematics, two objects are said to be isomorphic when one is just a "relabeling" of the other. Formally, an isomorphism is defined as a bijective homomorphism, and two objects are isomorphic when there exists an isomorphism between them. Most relevant to polytopes are isomorphisms between abstract polytopes or between groups.

For most practical purposes, two isomorphic objects may be considered the exact same, and mathematicians will often treat them as such. This rule is however not set in stone, and sometimes different representations of the same structure can be interesting in their own right, as is the case with symmetry groups.

Isomorphisms between an object and itself are called automorphisms.

Between polytopes
Recall that an abstract polytope is a partially ordered set satisfying certain properties. An isomorphism between two abstract polytopes $$\mathcal P=(P,\le_P)$$ and $$\mathcal Q=(Q,\le_Q)$$ is a bijective function $$\varphi:P\to Q$$ such that for all $$p_1,p_2\in P,$$
 * $$p_1\le_P p_2\Rightarrow\varphi(p_1)\le_Q\varphi(p_2).$$

This condition alone guarantees that the improper elements of $$\mathcal P$$ will be mapped to the corresponding improper elements of $$\mathcal Q$$, and that the rank of an element in $$\mathcal P$$ will equal the rank of the corresponding element in $$\mathcal Q$$.

The realizations of two abstract polytopes may be called isomorphic when the underlying abstract polytopes are.

Between groups
An isomorphism between two groups $$(G,\cdot)$$ and $$(H,*)$$ is a bijective function $$\varphi:G\to H$$ such that for all $$g_1,g_2\in G$$,
 * $$\varphi(g_1\cdot g_2)=\varphi(g_1)*\varphi(g_2).$$

Most of the mathematical study of groups does not really distinguish between isomorphic groups. However, this is sometimes necessary when dealing with symmetry groups. For instance, chiral icosahedral symmetry and chiral pentachoric symmetry are isomorphic, yet describe very different polytopes, as the first is three-dimensional and the second is four-dimensional. Furthermore, there are isomorphic symmetry groups in the same dimension that still describe different sets of polytopes, such as the symmetry groups of the pentagonal prism and pentagonal antiprism. Formally, two symmetry groups describe the same set of polytopes if they are s in the broader group of.