# Schlegel Diagram **A projection of a polytope into the next lower dimension through one of its [[Face#^bb0034|facets]].** ![[SchlegelDiagramCube.svg|500]] *A cube and its Schlegel diagram* The *Schlegel diagram* of a polytope in $\mathbb{R}^n$ is a projection in $\mathbb{R}^{n-1}$ and is commonly used to visualise four-dimensional polytopes as three-dimensional polyhedra. The projection appears as though the polytope is being *viewed through one of its facets*; for example, the Schlegel diagram above is the view through the one of the square faces of the cube. ## Convex polyhedra The Schlegel diagrams of all *convex* polyhedra are also [[Connected Graph|connected]] [[Planar Graph|planar graphs]] and exhibit a [[Euler characteristic]] of $2$. The face through which the polyhedron is projected forms the outside of the graph thus satisfying Euler's polyhedron formula. ![[SchlegelDiagramCubeEulerCharacteristic.svg|600]]