# Invariant We can define an invariant as follows: > An **invariant** is a feature of a mathematical object that cannot be changed with respect to a [Symmetry](Symmetry.md). For example, *area* is invariant under the rigid motion symmetries; no matter how we rotate, reflect, or translate a triangle, it's area will remain unchanged. A function $f$ is **invariant** with respect to a transformation $T$ if $f(T(x)) = f(x)$. In other words, the result of $f$ on $x$ does not change if $T$ is applied to $x$ first. A picture is often worth a thousand words. Below we have a transformation $S$ (in this case a [Translation](Translations.md)). We have a function, $f$, and an input (image), $x$. We see that $f$ is invariant to $S$ because we can apply $f$ to both $x$ and to $T(x)$ and either way we get the output $cat$.  Note that in effectively all cases our mathematical objects can be thought of as functions. --- Date: 20211117 Links to: [Symmetry](Symmetry.md) [Geometry](Geometry.md) [Equivariance](Equivariance.md) Tags: References: * []()