In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a vertex arrangement is a set of
points in space described by their relative positions. They can be described by their use in
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 -d ...
s.
For example, a ''
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
vertex arrangement'' is understood to mean four points in a plane, equal distance and angles from a center point.
Two polytopes share the same ''vertex arrangement'' if they share the same
0-skeleton
In mathematics, particularly in algebraic topology, the of a topological space presented as a simplicial complex (resp. CW complex) refers to the subspace that is the union of the simplices of (resp. cells of ) of dimensions In other wo ...
.
A group of polytopes that shares a vertex arrangement is called an ''army''.
Vertex arrangement
The same set of vertices can be connected by edges in different ways. For example, the ''pentagon'' and ''pentagram'' have the same ''vertex arrangement'', while the second connects alternate vertices.
A ''vertex arrangement'' is often described by the
convex hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
polytope which contains it. For example, the regular ''pentagram'' can be said to have a (regular) ''pentagonal vertex arrangement''.
Infinite tilings can also share common ''vertex arrangements''.
For example, this
triangular lattice
The hexagonal lattice or triangular lattice is one of the five two-dimensional Bravais lattice types. The symmetry category of the lattice is wallpaper group p6m. The primitive translation vectors of the hexagonal lattice form an angle of 120° ...
of points can be connected to form either
isosceles triangle
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
s or
rhombic faces.
Edge arrangement
Polyhedra
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on t ...
can also share an ''edge arrangement'' while differing in their faces.
For example, the self-intersecting ''great dodecahedron'' shares its edge arrangement with the convex ''icosahedron'':
A group polytopes that share both a ''vertex arrangement'' and an ''edge arrangement'' are called a ''regiment''.
Face arrangement
4-polytopes
In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), an ...
can also have the same ''face arrangement'' which means they have similar vertex, edge, and face arrangements, but may differ in their cells.
For example, of the ten nonconvex regular
Schläfli-Hess polychora, there are only 7 unique face arrangements.
For example, the
grand stellated 120-cell
In geometry, the grand stellated 120-cell or grand stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes.
It is also one of two such polytopes that is self-dual.
Rela ...
and
great stellated 120-cell
In geometry, the great stellated 120-cell or great stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes.
It is one of four ''regular star 4-polytopes'' discovered by Lud ...
, both with
pentagram
A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle aroun ...
mic faces, appear visually indistinguishable without a representation of their
cells
Cell most often refers to:
* Cell (biology), the functional basic unit of life
Cell may also refer to:
Locations
* Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...
:
Classes of similar polytopes
George Olshevsky advocates the term ''regiment'' for a set of polytopes that share an edge arrangement, and more generally ''n-regiment'' for a set of polytopes that share elements up to dimension ''n''. Synonyms for special cases include ''company'' for a 2-regiment (sharing faces) and ''army'' for a 0-regiment (sharing vertices).
See also
*
n-skeleton
In mathematics, particularly in algebraic topology, the of a topological space presented as a simplicial complex (resp. CW complex) refers to the subspace that is the union of the simplices of (resp. cells of ) of dimensions In other word ...
- a set of elements of dimension ''n'' and lower in a higher polytope.
*
Vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
- A local arrangement of faces in a polyhedron (or arrangement of cells in a polychoron) around a single vertex.
External links
* (Same vertex arrangement)
* (Same vertex and edge arrangement)
* {{GlossaryForHyperspace , anchor=Company , title=Company (Same vertex, edge and face arrangement)
Polytopes