# Face **A flat surface forming part of the boundary of a solid object or an element of any dimension of any polytope.** ![[CubeFaces.svg|500]] *A cube and its faces* ## Solid geometry In solid geometry, a *face* most commonly refers to a polygon that forms *part of the boundary* of a polyhedron. An example is the 6 squares that form the boundary of a cube or the sphere that forms the boundary of a ball.[^1] This definition is used in [[Euler Characteristic#Euler's polyhedron formula|Euler's polyhedron formula]] which states that all *convex polyhedra* has an Euler characteristic of $2$. ## $k$-face In higher-dimensional geometry, a *face* of a polytope refers more generally to elements of any dimensions. A face of dimension $k$ is referred to as a $k$-face and for some $k$ there exists special names. - *0-face* or *vertex* - a point, especially the point of intersecting edges - *1-face* or *edge* - a line connecting two vertices of a polytope - *2-face* or *face* - a polygon; typically the term "face" without context refers to a 2-face - *3-face* or *cell* - a polyhedron - *(n-1)-face* or *facet* - also known as a *hyperface*, an element of an $n$-polytope with $n-1$ dimensions ^bb0034 - *(n-2)-face* or *ridge* - also known as a *subfacet*, an element of an $n$-polytope with $n-2$ dimensions - *(n-3)-face* or *peak* - an element of an $n$-polytope with $n-3$ dimensions [^1]: In mathematics, a *ball* refers strictly to a solid object that is bounded by a sphere, also known as a solid sphere.