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. Euclidean area is invariant under linear maps which have determinant ±1 (see ). Some invariants of projective transformations include collinearity of three or more points, concurrency of three or more lines, conic sections, the cross-ratio. The determinant, trace, and eigenvectors and eigenvalues of a square matrix are invariant under changes of basis. In other words, the spectrum of a matrix is invariant to the 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 of the Gaussian curvature of a 2-dimensional Riemannian manifold is invariant under changes of the Riemannian metric . This is the Gauss–Bonnet theorem. Differential invariants for differential equations 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