Vertex figure

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"Half-edge" vertex figure of the cube Cube-vertex-figure-middle.svg
"Half-edge" vertex figure of the cube

In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.

Contents

Definitions

"Whole-edge" vertex figure of the cube Cube-vertex-figure-large.svg
"Whole-edge" vertex figure of the cube
Spherical vertex figure of the cube Cube-vertex-figure-sphere.svg
Spherical vertex figure of the cube
Point-set vertex figure of the cube Cube-vertex-figure-points.svg
Point-set vertex figure of the cube

Take some corner or vertex of a polyhedron. 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 polytope cells and other higher-dimensional polytopes.

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. 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. [1] [2]

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 polytopes 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 polytopes, 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 Fn/V, where Fn 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 polyhedron is a 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 polygons for faces and/or for vertex figures.

Isogonal figures

Vertex figures are especially significant for uniforms 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 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 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:

Dorman Luke construction

For a uniform polyhedron, the face of the dual polyhedron may be found from the original polyhedron's vertex figure using the "Dorman Luke" construction.

Regular polytopes

The vertex figure of the great icosahedron is a regular pentagram or star polygon {5/2}. Great icosahedron vertfig.svg
The vertex figure of the great icosahedron is a regular pentagram or star polygon {5/2}.

If a polytope is regular, it can be represented by a Schläfli symbol and both the cell and the vertex figure can be trivially extracted from this notation.

In general a regular polytope with Schläfli symbol {a,b,c,...,y,z} has cells as {a,b,c,...,y}, and vertex figures as {b,c,...,y,z}.

  1. For a regular polyhedron {p,q}, the vertex figure is {q}, a q-gon.
    • Example, the vertex figure for a cube {4,3}, is the triangle {3}.
  2. For a regular 4-polytope or space-filling tessellation {p,q,r}, the vertex figure is {q,r}.
    • Example, the vertex figure for a hypercube {4,3,3}, the vertex figure is a regular tetrahedron {3,3}.
    • Also the vertex figure for a cubic honeycomb {4,3,4}, the vertex figure is a regular octahedron {3,4}.

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

truncated cubic honeycomb (partial). Truncated cubic honeycomb1.jpg
truncated cubic honeycomb (partial).

The vertex figure of a truncated cubic honeycomb is a nonuniform square pyramid. One octahedron and four truncated cubes meet at each vertex form a space-filling tessellation.

Vertex figure: A nonuniform square pyramid Truncated cubic honeycomb verf.png
Schlegel diagram
VF-truncated cubic.png
Perspective
Created as a square base from an octahedron Octahedron vertfig.png
(3.3.3.3)
And four isosceles triangle sides from truncated cubes Truncated cube vertfig.png
(3.8.8)

Edge figure

The truncated cubic honeycomb has two edge types, one with four truncated cubes, and the others with one octahedron, and two truncated cubes. These can be seen as two types of edge figures. These are seen as the vertices of the vertex figure. Truncated cubic honeycomb1.jpg
The truncated cubic honeycomb has two edge types, one with four truncated cubes, and the others with one octahedron, and two truncated cubes. These can be seen as two types of edge figures. These are seen as the vertices of the vertex figure.

Related to the vertex figure, an edge figure is the vertex figure of a vertex figure. [3] 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 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 {p,q,r,s,...,z}, the edge figure is {r,s,...,z}.

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 cubic honeycomb {4,3,4} is a square, and for a regular 4-polytope {p,q,r} is the polygon {r}.

Less trivially, the truncated cubic honeycomb t0,1{4,3,4}, has a square pyramid vertex figure, with truncated cube and octahedron 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.

See also

Related Research Articles

Cuboctahedron Polyhedron with 8 triangular faces and 6 square faces

A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral.

A polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.

4-polytope Four-dimensional geometric object with flat sides

In geometry, a 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853.

A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.

Schläfli symbol Notation that defines regular polytopes and tessellations

In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.

Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry group acts transitively 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 dimension ≤ n.

Uniform 4-polytope

In geometry, a uniform 4-polytope is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.

Rectification (geometry) Operation in Euclidean geometry

In Euclidean geometry, rectification, also known as critical truncation or complete-truncation is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the rectified facets of the original polytope.

Uniform polyhedron Class of mathematical solids

A uniform polyhedron has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.

Cubic honeycomb Only regular space-filling tessellation of the cube

The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a cubille.

Order-4 dodecahedral honeycomb

In the geometry of hyperbolic 3-space, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations. With Schläfli symbol {5,3,4}, it has four dodecahedra around each edge, and 8 dodecahedra around each vertex in an octahedral arrangement. Its vertices are constructed from 3 orthogonal axes. Its dual is the order-5 cubic honeycomb.

Truncation (geometry) Operation that cuts polytope vertices, creating a new facet in place of each vertex

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids.

Alternation (geometry) Operation on a polyhedron or tiling that removes alternate vertices

In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.

Expansion (geometry)

In geometry, expansion is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements. Equivalently this operation can be imagined by keeping facets in the same position but reducing their size.

5-polytope

In five-dimensional geometry, a five-dimensional polytope or 5-polytope is a 5-dimensional polytope, bounded by (4-polytope) facets. Each polyhedral cell being shared by exactly two 4-polytope facets.

Uniform polytope

A uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.

In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.

6-polytope

In six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets.

In geometry, a regular skew apeirohedron is an infinite regular skew polyhedron, with either skew regular faces or skew regular vertex figures.

References

Notes

  1. Coxeter, H. et al. (1954).
  2. Skilling, J. (1975).
  3. Klitzing: Vertex figures, etc.

Bibliography