Isomorphism: From the Greek:
"Having the same form". As spelled out with
some modern mathematics: A 11 mapping between
two sets
such that some (structural) relations 
the forms  are preserved.
The
mathematical statement just given has the
following import: Let A1..An and B1...Bn be
sets,; let
f be a 11 function
from Ai to Bi for 1<=i<=n; let S be a
relation of the elements
of A1..An and R a relation between the
elements of B1..Bn. Then using Cartesian
Products from settheory and
firstorder predicate logic:
f is an isomorphism
from A1*..*An to B1*..*Bn for S and R
IFF S(a1 ..
an) iff R(f(a1) .. f(an))
In words: IFF a1 .. an stand in
relation S precisely if f(a1) .. f(am), that
is the values of a1 .. an of f, in that order, stand in
relation R.
Often the relations S and R are taken to be
the same: Then the same relation  such as a
shape, form, curve  obtains in B and in A
for the elements f maps from A to B.
Then again, this may not hold, or may hold only in fairly fanciful ways, such as the curves and bends in a line on a map that is supposed to represent the curves and bends in a river.
There are many variants, applications and
precifisications in various fields of
mathematics of the ntionn of isomorphism and also outside it.
