In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class. Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave unchanged. 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.
Examples A simple example of invariance is expressed in our ability to count. For a finite set of objects of any kind, there is a number to which we always arrive, regardless of the order in which we count the objects in the set. The quantity—a cardinal number—is associated with the set, and is invariant under the process of counting. An identity is an equation that remains true for all values of its variables. There are also inequalities that remain true when the values of their variables change. The distance between two points on a number line is not changed by adding the same quantity to both numbers. On the other hand, multiplication does not have this same property, as distance is not invariant under multiplication. Angles and ratios of distances are invariant under scalings, rotations, translations and reflections. These transformations produce similar shapes, which are the basis of trigonometry. In contrast, angles and ratios are not invariant under non-uniform scaling (such as stretching). The sum of a triangle's interior angles (180°) is invariant under all the above operations. 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)). Some more complicated examples:
The real part and the absolute value of a complex number are invariant under complex conjugation. The tricolorability of knots. The degree of a polynomial is invariant under a linear change of variables. The dimension and homology groups of a topological object are invariant under homeomorphism. The number of fixed points of a dynamical system is invariant under many mathematical operations. Euclidean distance is invariant under orthogonal transformations. Area is invariant under linear maps which have determinant ±1 (see Equiareal map § Linear transformations). Some invariants of projective transformations include collinearity of three or more points, concurrency of three or more lines, conic sections, and the cross-ratio. The determinant, trace, eigenvectors, and eigenvalues of a linear endomorphism are invariant under a change of basis. In other words, the spectrum of a matrix is invariant under a change of basis. The principal invariants of tensors do not change with rotation of the coordinate system (see Invariants of tensors). The singular values of a matrix are invariant under orthogonal transformations. Lebesgue measure is invariant under translations. The variance of a probability distribution is invariant under translations of the real line. Hence the variance of a random variable is unchanged after the addition of a constant. The fixed points of a transformation are the elements in the domain that are invariant under the transformation. They may, depending on the application, be called symmetric with respect to that transformation. For example, objects with translational symmetry are invariant under certain translations. The integral ∫ M K d μ {\textstyle \int _{M}K\,d\mu } of the Gaussian curvature K {\displaystyle K} of a two-dimensional Riemannian manifold ( M , g ) {\displaystyle (M,g)} is invariant under changes of the Riemannian metric g {\displaystyle g} . This is the Gauss–Bonnet theorem.
MU puzzle The MU puzzle is a good example of a logical problem where determining an invariant is of use for an impossibility proof. The puzzle asks one to start with the word MI and transform it into the word MU, using in each step one of the following transformation rules:
If a string ends with an I, a U may be appended (xI → xIU) The string after the M may be completely duplicated (Mx → Mxx) Any three consecutive I's (III) may be replaced with a single U (xIIIy → xUy) Any two consecutive U's may be removed (xUUy → xy) An example derivation (with superscripts indicating the applied rules) is
MI →2 MII →2 MIIII →3 MUI →2 MUIUI →1 MUIUIU →2 MUIUIUUIUIU →4 MUIUIIUIU → ... In light of this, one might wonder whether it is possible to convert MI into MU, using only these four transformation rules. One could spend many hours applying these transformation rules to strings. 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. This makes the following invariant interesting to consider:
The number of I's in the string is not a multiple of 3. This is an invariant to the problem, if for each of the transformation rules the following holds: if the invariant held before applying the rule, it will also hold after applying it. Looking at the net effect of applying the rules on the number of I's and U's, one can see this actually is the case for all rules:
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