Facet (mathematics)

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• A facet
  Facetting
  
  |}In geometry, facetting is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices.Facetting is the reciprocal or dual process to stellation....
   of a geometric polyhedron
  Polyhedron
  
  |}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
   is traditionally any polygon whose corners are vertices of the polyhedron. By extension to higher dimensions, it is any j-tope (j-dimensional polytope
  Polytope
  
  In geometry, polytope is a generic term that can refer to a two-dimensional polygon, a three-dimensional polyhedron, or any of the various generalizations thereof, including generalizations to higher dimensions and other abstractions ....
  ) whose vertices are shared by some n-tope (n-dimensional polytope where 0<j<n). To facet a polytope is to find and join such facets to form a new polytope - this process is called facetting
  Facetting
  
  |}In geometry, facetting is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices.Facetting is the reciprocal or dual process to stellation....
   or faceting and is the reciprocal process to stellation
  Stellation
  
  Stellation is a process of constructing new polygons , new polyhedron in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again....
  .
• A facet of an n-polytope
  Polytope
  
  In geometry, polytope is a generic term that can refer to a two-dimensional polygon, a three-dimensional polyhedron, or any of the various generalizations thereof, including generalizations to higher dimensions and other abstractions ....
   is, more recently, an (n-1)-dimensional face or (n-1)-face. The informal term side can mean the same thing, edges of a polygon and faces of a polyhedron.

  For example:

  1. The facets of a polygon are edge
     Edge (geometry)
     
     In geometry, an edge is a one-dimensional line segment joining two zero-dimensional vertex in a polytope. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....
     s. (1-faces)
  2. The facets of a polyhedron
     Polyhedron
     
     |}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
      or tiling
     Uniform tiling
     
     In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-uniform.Uniform tilings can exist in both the Euclidean plane and hyperbolic plane....
      are faces
     Face (geometry)
     
     In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the square s that bound a cube is a face of the cube....
     . (2-faces)
  3. The facets of a polychoron
     Polychoron
     
     In geometry, a four-dimensional polytope is sometimes called a polychoron , from the Greek language root poly, meaning "many", and choros meaning "room" or "space"....
      (4-polytope) or honeycomb
     Convex uniform honeycomb
     
     In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedron cells....
      are cells
     Cell (geometry)
     
     In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object....
     . (3-faces)
  4. The facets of a polyteron (5-polytope) or 4-honeycomb are hypercell
     Hypercell
     
     In geometry, a hypercell is a descriptive term for an element of a polytope or tessellation, usually representing an element one dimension higher than a Cell ....
     s. (4-faces)

  Exactly two facets meet at any ridge

  Ridge (geometry)

  
  In geometry, a ridge is an -dimensional element of an n-dimensional polytope. It is also sometimes called a subfacet for having one lower dimension than a Facet ....
   in a polytope. By extension, facet or j-facet is sometimes used to mean any j-dimensional element of a polytope.

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