Various visualizations of the icosahedron

In geometry, a **Schlegel diagram** is a projection of a polytope from into through a point just outside one of its facets. The resulting entity is a polytopal subdivision of the facet in 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.

The most elementary Schlegel diagram, that of a polyhedron, was described by Duncan Sommerville as follows:^{ [1] }

- A very useful method of representing a convex polyhedron is by plane projection. If it is projected from any external point, since each ray cuts it twice, it will be represented by a polygonal area divided twice over into polygons. It is always possible by suitable choice of the centre of projection to make the projection of one face completely contain the projections of all the other faces. This is called a
*Schlegel diagram*of the polyhedron. The Schlegel diagram completely represents the morphology of the polyhedron. It is sometimes convenient to project the polyhedron from a vertex; this vertex is projected to infinity and does not appear in the diagram, the edges through it are represented by lines drawn outwards.

Sommerville also considers the case of a simplex in four dimensions:^{ [2] } "The Schlegel diagram of simplex in S_{4} is a tetrahedron divided into four tetrahedra." More generally, a polytope in n-dimensions has a Schegel diagram constructed by a perspective projection viewed from a point outside of the polytope, above the center of a facet. All vertices and edges of the polytope are projected onto a hyperplane of that facet. If the polytope is convex, a point near the facet will exist which maps the facet outside, and all other facets inside, so no edges need to cross in the projection.

Dodecahedron | 120-cell |
---|---|

12 pentagon faces in the plane | 120 dodecahedral cells in 3-space |

- Net (polyhedron) – A different approach for visualization by lowering the dimension of a polytope is to build a net, disconnecting facets, and
*unfolding*until the facets can exist on a single hyperplane. This maintains the geometric scale and shape, but makes the topological connections harder to see.

In geometry, every polyhedron is associated with a second **dual** figure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron.

In elementary geometry, a **polytope** is a geometric object with flat sides (*faces*). It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or **n-polytope**. In this context, "flat sides" means that the sides of a (*k* + 1)-polytope consist of k-polytopes that may have (*k* – 1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope.

In geometry, a **4-polytope** is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853.

In geometry, a **hypercube** is an *n*-dimensional analogue of a square and a cube. It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in *n* dimensions is equal to .

In solid geometry, a **face** is a flat surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a *polyhedron*.

In geometry, the **Schläfli symbol** is a notation of the form that defines regular polytopes and tessellations.

A **zonohedron** is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric. Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as the three-dimensional projection of a hypercube. Zonohedra were originally defined and studied by E. S. Fedorov, a Russian crystallographer. More generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a **zonotope**.

In geometry, a **vertex figure**, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.

A **convex polytope** is a special case of a polytope, having the additional property that it is also a convex set contained in the -dimensional Euclidean space . Most texts use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary.

In geometry, a **pyramid** is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a *lateral face*. It is a conic solid with polygonal base. A pyramid with an *n*-sided base has *n* + 1 vertices, *n* + 1 faces, and 2*n* edges. All pyramids are self-dual.

In geometry, a **truncated 5-cell** is a uniform 4-polytope formed as the truncation of the regular 5-cell.

In five-dimensional geometry, a **five-dimensional polytope** or **5-polytope** is a 5-dimensional polytope, bounded by (4-polytope) facets. Each polyhedral cell being shared by exactly two 4-polytope facets.

A **uniform polytope** of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.

In geometry, an **edge** is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a line segment on the boundary, and is often called a **polygon side**. In a polyhedron or more generally a polytope, an edge is a line segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal.

In geometry, a **Petrie polygon** for a regular polytope of n dimensions is a skew polygon in which every *n* – 1 consecutive sides belongs to one of the facets. The **Petrie polygon** of a regular polygon is the regular polygon itself; that of a regular polyhedron is a skew polygon such that every two consecutive sides belongs to one of the faces. Petrie polygons are named for mathematician John Flinders Petrie.

In mathematics, a **regular 4-polytope** is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.

In 6-dimensional geometry, the **1 _{22}** polytope is a uniform polytope, constructed from the E

In 6-dimensional geometry, the **2 _{21}** polytope is a uniform 6-polytope, constructed within the symmetry of the E

In six-dimensional geometry, a **six-dimensional polytope** or **6-polytope** is a polytope, bounded by 5-polytope facets.

- ↑ Duncan Sommerville (1929).
*Introduction to the Geometry of N Dimensions*, p.100. E. P. Dutton. Reprint 1958 by Dover Books. - ↑ Sommerville (1929), p.101.

- Victor Schlegel (1883)
*Theorie der homogen zusammengesetzten Raumgebilde*, Nova Acta, Ksl. Leop.-Carol. Deutsche Akademie der Naturforscher, Band XLIV, Nr. 4, Druck von E. Blochmann & Sohn in Dresden. - Victor Schlegel (1886)
*Ueber Projectionsmodelle der regelmässigen vier-dimensionalen Körper*, Waren. - Coxeter, H.S.M.;
*Regular Polytopes*, (Methuen and Co., 1948). (p. 242)*Regular Polytopes*, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8

- Grünbaum, Branko (2003), Kaibel, Volker; Klee, Victor; Ziegler, Günter M. (eds.),
*Convex polytopes*(2nd ed.), New York & London: Springer-Verlag, ISBN 0-387-00424-6 .

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