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Vertex figure
 
 
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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 vertex figure is, broadly speaking, the figure exposed when a corner 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 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 sliced off.

Definitions - theme and variations
Take some vertex of a polyhedron.






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Triangular Prism Vertfig
Great Icosahedron Vertfig
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 vertex figure is, broadly speaking, the figure exposed when a corner 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 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 sliced off.

Definitions - theme and variations


Take some vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points. When done, these lines form a complete circuit, i.e. a polygon, around the vertex. This polygon is the vertex figure.

But compare the detailed descriptions across standard reference works, and you will soon find there is little agreement on a formal definition.

  • Coxeter (e.g. 1948, 1954) varies his definition as convenient for the current area of discussion.
  • Cromwell (1999) treats it as a spherical polygon marked on a sphere centered on the vertex.
  • Skilling (1975) and most abstract theorists use the definition discussed below.


General properties


One kind of vertex figure represents the arrangement of a connected set of points of all the neighboring vertices, in a 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 ....
 to a given vertex. This applies equally well to infinite tilings
Tessellation

A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces....
, or space-filling tessellation
Honeycomb

A honeycomb is a mass of hexagonal waxcells built by honey bees in their beehive to contain their larva and stores of honey and pollen.beekeeping may remove the entire honeycomb to harvest honey....
 with 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 ....
 cells
Cell (geometry)

In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object....
.

A vertex figure for an n-polytope is an (n-1)-polytope. For example, a vertex figure for 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....
 is a polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
 figure, and the vertex figure for 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"....
 is 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....
 figure.

By considering the connectivity of these neighboring vertices an (n-1)-polytope, the vertex figure, can be constructed for each vertex of a polytope:
  • Each vertex
    Vertex (geometry)

    In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
     of the vertex figure coincides with a vertex of the original polytope.
  • Each edge
    Graph theory

    In mathematics and computer science, graph theory is the study of graph : mathematical structures used to model pairwise relations between objects from a certain collection....
     of the vertex figure exists on or inside of a face of the original polytope connecting two alternate vertices from an original face.
  • Each 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 square s that bound a cube is a face of the cube....
     of the vertex figure exists on or inside a cell of the original n-polytope (for n>3).
  • ...and so on to higher order elements in higher order polytopes.


Vertex figures are the most useful for uniform polytope
Uniform polytope

A uniform polytope is a vertex-transitive polytope made from uniform polytope Facet . A uniform polytope must also have only regular polygon faces....
s because one vertex figure can imply the entire polytope.

For polyhedra, the vertex figure can be represented by a vertex configuration
Vertex configuration

In polyhedral geometry a vertex configuration is a short-hand notation for representing a polyhedron vertex figure as the sequence of faces around a vertex....
 notation, by listing the faces in sequence around a vertex. For example 3.4.4.4 is a vertex with one triangle and 3 squares, and it represents the rhombicuboctahedron
Rhombicuboctahedron

The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangle and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each....
.

If the polytope is vertex-transitive
Vertex-transitive

In geometry, a polytope is isogonal or vertex-transitive if all its vertex are the same. That is, each vertex is surrounded by the same kinds of face in the same order, and with the same angles between corresponding faces....
, the vertex figure will exist in a hyperplane
Hyperplane

A hyperplane is a concept in geometry. It is a higher-dimensional generalization of the concepts of a line in the plane and a plane in 3-dimensional space....
 surface of the n-space. In general the vertex figure need not be planar.

Also nonconvex polyhedra, the vertex figure may also be nonconvex. Uniform polytopes can have either star polygon
Star polygon

A star polygon is a non-convex polygon which looks in some way like a star. Only the regular ones have been studied in any depth; star polygons in general have never been formally defined....
 faces and vertex figures for instance.

Dorman Luke construction

For a uniform polyhedron, the face of the dual polyhedron
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
 may be found from the original polyhedron's vertex figure using the Dorman Luke
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
 construction.

Regular polytopes


If a polytope is regular, it can be represented by a Schläfli symbol
Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
 and both the cell
Cell (geometry)

In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object....
 and the vertex figure can be trivially extracted from this notation.

In general a regular polytope with Schläfli symbol has cells as , and vertex figures as .

  1. For a regular 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....
     , the vertex figure is , a q-gon.
    • Example, the vertex figure for a cube , is the triangle .
  2. For a regular 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"....
     or space-filling tessellation
    Honeycomb (geometry)

    In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions....
     , the vertex figure is .
    • Example, the vertex figure for a hypercube , the vertex figure is a regular tetrahedron .
    • Also the vertex figure for a cubic honeycomb
      Cubic honeycomb

      The cubic honeycomb is the only regular space-filling tessellation in Euclidean 3-space, made up of cubes. It is an analog of the square tiling of the plane, and part of a dimensional family called hypercube honeycombs....
       , the vertex figure is a regular octahedron .


Since the dual polytope of a regular polytope is also regular and represented by the Schläfli symbol indices reversed, it is easy to see the dual of the vertex figure is the cell of the dual polytope. For regular polyhedra, this is a special case of the Dorman Luke construction
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
.

an example vertex figure of a honeycomb

Truncated Cubic Honeycomb
The vertex figure of a truncated cubic honeycomb
Truncated cubic honeycomb

The truncated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is comprised of truncated cubes and octahedron in a ratio of 1:1....
 is a nonuniform square pyramid
Square pyramid

In geometry, a square pyramid is a Pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry....
. One octahedron and four truncated cube meet at each vertex for form a space-filling tessellation
Tessellation

A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces....
.
Vertex figure: A nonuniform square pyramid
Square pyramid

In geometry, a square pyramid is a Pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry....
Created as a square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
 base from an octahedron
Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
Octahedron Vertfig

(4.4.4.4)
And four isosceles triangle sides from truncated cube
Truncated cube

The truncated cube, or truncated hexahedron, is an Archimedean solid. It has 6 regular octagonal faces, 8 regular triangle faces, 24 vertices and 36 edges....
s
Truncated Cube Vertfig

(3.8.8)


See also

  • Vertex configuration
    Vertex configuration

    In polyhedral geometry a vertex configuration is a short-hand notation for representing a polyhedron vertex figure as the sequence of faces around a vertex....
  • List of regular polytopes
    List of regular polytopes

    This page lists the regular polytopes in Euclidean geometry, spherical geometry and hyperbolic geometry spaces.The Schl?fli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each....


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