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{{Infobox Polytope | name = Polytope

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A polytope

A polytope

| caption = A polytope

Definition

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A polytope is a higher-dimensional analogue of a polygon, with a finite number of vertices, edges, and faces.

Properties

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1. Number of Faces

• A polytope has at least one face, which is a lower-dimensional polytope.
• The number of faces of a polytope is limited to a finite number.

1. Number of Edges

• A polytope has at least one edge, which is a line segment connecting two vertices.
• The number of edges of a polytope is limited to a finite number.

1. Number of Vertices

• A polytope has at least one vertex, which is a point where two or more edges meet.
• The number of vertices of a polytope is limited to a finite number.

1. Dimensionality

• A polytope can be classified by its dimensionality, which is the number of dimensions required to describe its geometry.

1. Regularity

• A regular polytope is a polytope with all sides and angles equal.
• An irregular polytope is a polytope that does not have all sides and angles equal.

Types

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1. Polyhedra

• A polyhedron is a three-dimensional polytope.
• Examples include the tetrahedron, cube, and octahedron.

1. Polychora

• A polychoron is a four-dimensional polytope.
• Examples include the tesseract and the 16-cell.

1. Higher-Dimensional Polytopes

• Higher-dimensional polytopes are generalizations of the above types.
• Examples include the 5-simplex and the 24-cell.

History

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1. Ancient Greece

• The study of polytopes dates back to ancient Greece, where mathematicians such as Euclid and Archimedes studied the properties of polygons and polyhedra.

1. Modern Developments

• The study of higher-dimensional polytopes has continued to evolve, with advances in geometry and topology.

References

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