In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a peak is an (n-3)-face of an n-dimensional polytope

Polytope

In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...

. A peak attaches at least three facets (and, accordingly, at least three ridges).

A regular

Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...

 n-polytope with Schläfli symbol {p₁,p₂,p₃,...,p_n-2,p_n-1} has a sequence of p_n-1 {p₁,p₂,p₃,...,p_n-2} facets around every peak. For example, the 600-cell, with Schläfli symbol {3,3,5} has 5 {3,3} (tetrahedra) around each peak (edge).

By dimension, this corresponds to:

• a vertex
  Vertex (geometry)
  In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
  
   of a polyhedron
  Polyhedron
  In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
  
  ;
• an edge
  Edge (geometry)
  In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....
  
   of a polychoron
  Polychoron
  In geometry, a polychoron or 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower dimensional polytopal elements: vertices, edges, faces , and cells...
  
   (4-polytope);
• a face
  Face (geometry)
  In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...
  
   of a 5-polytope
  5-polytope
  In five-dimensional geometry, a 5-polytope is a 5-dimensional polytope, bounded by facets. Each polyhedral cell being shared by exactly two polychoron facets. A proposed name for 5-polytopes is polyteron.-Definition:...
  
  ;
• a cell
  Cell (geometry)
  In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object.- In polytopes :A cell is a three-dimensional polyhedron element that is part of the boundary of a higher-dimensional polytope, such as a polychoron or honeycomb For example, a cubic honeycomb is made...
  
   of a 6-polytope
  6-polytope
  In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform polytera....
  
  ;
• a 4-face of a 7-polytope
  7-polytope
  In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets....
  
  ;
• and so forth.