Schlegel diagram

Various visualizations of the icosahedron

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perspective

Net

Orthogonal

Petrie

Schlegel

Vertex figure

In geometry, a Schlegel diagram is a projection of a polytope from ${\textstyle \mathbb {R} ^{d}}$ into ${\textstyle \mathbb {R} ^{d-1}}$ through a point just outside one of its facets. The resulting entity is a polytopal subdivision of the facet in ${\textstyle \mathbb {R} ^{d-1}}$ that, together with the original facet, is combinatorially equivalent to the original polytope. The diagram is named for Victor Schlegel, who in 1886 introduced this tool for studying combinatorial and topological properties of polytopes. In dimension 3, a Schlegel diagram is a projection of a polyhedron into a plane figure; in dimension 4, it is a projection of a 4-polytope to 3-space. As such, Schlegel diagrams are commonly used as a means of visualizing four-dimensional polytopes.