geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

is sliced off.

Definitions

Take some corner or vertex of a

polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...

. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points around the face. When done, these lines form a complete circuit, i.e. a polygon, around the vertex. This polygon is the vertex figure. More precise formal definitions can vary quite widely, according to circumstance. For example Coxeter (e.g. 1948, 1954) varies his definition as convenient for the current area of discussion. Most of the following definitions of a vertex figure apply equally well to infinite tilings or, by extension, to space-filling tessellation with

cells and other higher-dimensional

As a flat slice

Make a slice through the corner of the polyhedron, cutting through all the edges connected to the vertex. The cut surface is the vertex figure (a plane figure). This is perhaps the most common approach, and the most easily understood. Different authors make the slice in different places. Wenninger (2003) cuts each edge a unit distance from the vertex, as does Coxeter (1948). For uniform polyhedra the Dorman Luke construction cuts each connected edge at its midpoint. Other authors make the cut through the vertex at the other end of each edge. For an irregular polyhedron, cutting all edges incident to a given vertex at equal distances from the vertex may produce a figure that does not lie in a plane. A more general approach, valid for arbitrary convex polyhedra, is to make the cut along any plane which separates the given vertex from all the other vertices, but is otherwise arbitrary. This construction determines the combinatorial structure of the vertex figure, similar to a set of connected vertices (see below), but not its precise geometry; it may be generalized to

convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

s in any dimension. However, for non-convex polyhedra, there may not exist a plane near the vertex that cuts all of the faces incident to the vertex.

As a spherical polygon

Cromwell (1999) forms the vertex figure by intersecting the polyhedron with a sphere centered at the vertex, small enough that it intersects only edges and faces incident to the vertex. This can be visualized as making a spherical cut or scoop, centered on the vertex. The cut surface or vertex figure is thus a spherical polygon marked on this sphere. One advantage of this method is that the shape of the vertex figure is fixed (up to the scale of the sphere), whereas the method of intersecting with a plane can produce different shapes depending on the angle of the plane. Additionally, this method works for non-convex polyhedra.

As the set of connected vertices

Many combinatorial and computational approaches (e.g. Skilling, 1975) treat a vertex figure as the ordered (or partially ordered) set of points of all the neighboring (connected via an edge) vertices to the given vertex.

Abstract definition

In the theory of

abstract polytope In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines. A geometric polytope is said to be ...

s, the vertex figure at a given vertex ''V'' comprises all the elements which are incident on the vertex; edges, faces, etc. More formally it is the (''n''−1)-section ''F_n''/''V'', where ''F_n'' is the greatest face. This set of elements is elsewhere known as a ''vertex star''. The geometrical vertex figure and the vertex star may be understood as distinct ''realizations'' of the same abstract section.

General properties

A vertex figure of an ''n''-polytope is an (''n''−1)-polytope. For example, a vertex figure of a

is a

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

, and the vertex figure for a 4-polytope is a polyhedron. In general a vertex figure need not be planar. For nonconvex polyhedra, the vertex figure may also be nonconvex. Uniform polytopes, for instance, can have

star polygon In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, Decagram (geometry)#Related figures, certain notable ones can ...

s for faces and/or for vertex figures.

Isogonal figures

Vertex figures are especially significant for

uniform A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...

s and other isogonal (vertex-transitive) polytopes because one vertex figure can define the entire polytope. For polyhedra with regular faces, a vertex figure can be represented in

vertex configuration In geometry, a vertex configuration is a shorthand notation for representing a polyhedron or Tessellation, tiling as the sequence of Face (geometry), faces around a Vertex (geometry), vertex. It has variously been called a vertex description, vert ...

notation, by listing the faces in sequence around the vertex. For example 3.4.4.4 is a vertex with one triangle and three squares, and it defines the uniform rhombicuboctahedron. If the polytope is isogonal, the vertex figure will exist in a

hyperplane In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...

surface of the ''n''-space.

Constructions

From the adjacent vertices

By considering the connectivity of these neighboring vertices, a vertex figure can be constructed for each vertex of a polytope: *Each vertex of the ''vertex figure'' coincides with a vertex of the original polytope. *Each edge 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 The face is the front of the head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affect th ...

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.

Dorman Luke construction

For a uniform polyhedron, the face of the

dual polyhedron In geometry, every polyhedron is associated with a second dual structure, 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 ...

may be found from the original polyhedron's vertex figure using the " Dorman Luke" construction.

Regular polytopes

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

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

and both the cell 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 . #For a

regular polyhedron A regular polyhedron is a polyhedron whose symmetry group acts transitive group action, transitively on its Flag (geometry), flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In ...

, the vertex figure is , a ''q''-gon. #*Example, the vertex figure for a cube , is the triangle . #For a

regular 4-polytope In mathematics, a regular 4-polytope or regular polychoron is a regular polytope, regular 4-polytope, four-dimensional polytope. They are the four-dimensional analogues of the Regular polyhedron, regular polyhedra in three dimensions and the regul ...

or space-filling tessellation , 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 The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb (geometry), honeycomb) in Euclidean 3-space made up of cube, cubic cells. It has 4 cubes around every edge, and 8 cubes around each verte ...

, 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.

An example vertex figure of a honeycomb

The vertex figure of a truncated cubic honeycomb is a nonuniform

square pyramid In geometry, a square pyramid is a Pyramid (geometry), pyramid with a square base and four triangles, having a total of five faces. If the Apex (geometry), apex of the pyramid is directly above the center of the square, it is a ''right square p ...

. One octahedron and four truncated cubes meet at each vertex form a space-filling

tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...

Edge figure

Related to the ''vertex figure'', an ''edge figure'' is the ''vertex figure'' of a ''vertex figure''.Klitzing: Vertex figures, etc.
/ref> Edge figures are useful for expressing relations between the elements within regular and uniform polytopes. An ''edge figure'' will be a (''n''−2)-polytope, representing the arrangement of facets around a given edge. Regular and single-ringed

coxeter diagram Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated ...

uniform polytopes will have a single edge type. In general, a uniform polytope can have as many edge types as active mirrors in the construction, since each active mirror produces one edge in the fundamental domain. Regular polytopes (and honeycombs) have a single ''edge figure'' which is also regular. For a regular polytope , the ''edge figure'' is . In four dimensions, the edge figure of a 4-polytope or 3-honeycomb is a polygon representing the arrangement of a set of facets around an edge. For example, the ''edge figure'' for a regular

is a

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

, and for a regular 4-polytope is the polygon . Less trivially, the truncated cubic honeycomb t_0,1, has a

vertex figure, with truncated cube and

octahedron In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of i ...

cells. Here there are two types of ''edge figures''. One is a square edge figure at the apex of the pyramid. This represents the four ''truncated cubes'' around an edge. The other four edge figures are isosceles triangles on the base vertices of the pyramid. These represent the arrangement of two truncated cubes and one octahedron around the other edges.

References

Notes

Bibliography

* H. S. M. Coxeter, ''Regular Polytopes'', Hbk (1948), ppbk (1973). *H.S.M. Coxeter (et al.), Uniform Polyhedra, ''Phil. Trans''. 246 A (1954) pp. 401–450. *P. Cromwell, ''Polyhedra'', CUP pbk. (1999). *H.M. Cundy and A.P. Rollett, '' Mathematical Models'', Oxford Univ. Press (1961). *J. Skilling, The Complete Set of Uniform Polyhedra, ''Phil. Trans''. 278 A (1975) pp. 111–135. *M. Wenninger, ''Dual Models'', CUP hbk (1983) ppbk (2003). *''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, , (p. 289 Vertex figures).

External links

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Vertex Figures
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