Invariant (mathematics)
For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane.
For example, conformal maps are defined as transformations of the plane that preserve angles.
The discovery of invariants is an important step in the process of classifying mathematical objects.
Angles and ratios of distances are invariant under scalings, rotations, translations and reflections.
In contrast, angles and ratios are not invariant under non-uniform scaling (such as stretching).
As another example, all circles are similar: they can be transformed into each other and the ratio of the circumference to the diameter is invariant (denoted by the Greek letter π (pi)).
However, it might be quicker to find a property that is invariant to all rules (that is, not changed by any of them), and that demonstrates that getting to MU is impossible.
By looking at the puzzle from a logical standpoint, one might realize that the only way to get rid of any I's is to have three consecutive I's in the string.
An invariant set of an operation T is also said to be stable under T. For example, the normal subgroups that are so important in group theory are those subgroups that are stable under the inner automorphisms of the ambient group.
The notion of invariance is formalized in three different ways in mathematics: via group actions, presentations, and deformation.
However, if one allows scaling in addition to rigid motions, then the AAA similarity criterion shows that this is a complete set of invariants.
Secondly, a function may be defined in terms of some presentation or decomposition of a mathematical object; for instance, the Euler characteristic of a cell complex is defined as the alternating sum of the number of cells in each dimension.
One may forget the cell complex structure and look only at the underlying topological space (the manifold) – as different cell complexes give the same underlying manifold, one may ask if the function is independent of choice of presentation, in which case it is an intrinsically defined invariant.
The most common examples are: Thirdly, if one is studying an object which varies in a family, as is common in algebraic geometry and differential geometry, one may ask if the property is unchanged under perturbation (for example, if an object is constant on families or invariant under change of metric).
The theory of optimizing compilers, the methodology of design by contract, and formal methods for determining program correctness, all rely heavily on invariants.
Some object oriented programming languages have a special syntax for specifying class invariants.
