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Invariant (mathematics)

Property that is not changed by mathematical transformations

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Example of wallpaper group type p2.
Owen Jones , uploaded to the English Wikipedia by w:en:User:Dmharvey · Public domain

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Lab · Class 10Coordinate GeometryLab · Class 9–10Coordinate Geometry

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:

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:

  1. If a string ends with an I, a U may be appended (xI → xIU)
  2. The string after the M may be completely duplicated (Mx → Mxx)
  3. Any three consecutive I's (III) may be replaced with a single U (xIIIy → xUy)
  4. 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:

The table above shows clearly that the invariant holds for each of the possible transformation rules, which means that whichever rule one picks, at whatever state, if the number of I's was not a multiple of three before applying the rule, then it will not be afterwards either.

Given that there is a single I in the starting string MI, and one is not a multiple of three, one can then conclude that it is impossible to go from MI to MU (as the number of I's will never be a multiple of three).

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