In
geometryGeometry 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 arrangement is a set of points in space described by their relative positions. They can be described by their use in
polytopeIn 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 .When referring to an...
s.
For example a
squareIn geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
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-skeletonIn mathematics, particularly in algebraic topology, the n-skeleton of a topological space X presented as a simplicial complex refers to the subspace X
n that is the union of the simplices of X of dimensions m ≤ n...
.
The same set of vertices can be connected by edges in different ways.
In
geometryGeometry 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 arrangement is a set of points in space described by their relative positions. They can be described by their use in
polytopeIn 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 .When referring to an...
s.
For example a
squareIn geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
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-skeletonIn mathematics, particularly in algebraic topology, the n-skeleton of a topological space X presented as a simplicial complex refers to the subspace X
n that is the union of the simplices of X of dimensions m ≤ n...
.
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.
Two polygons with same vertex arrangement.
pentagonIn geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540°.- Regular pentagons :...
|
pentagramA pentagram is the shape of a five-pointed star drawn with five straight strokes...
|
A
vertex arrangement is often described by the
convex hullIn mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....
polytope which contains it. For example, the regular
pentagram can be said to have a (regular)
pentagonal vertex arrangement.
 |
ABCD is a concave The word concave means curving in or hollowed inward. The term is most commonly used to refer to:* Concave lens, a lens with inward-curving surfaces.* Concave polygon, a polygon which is not convex....
quadrilateralIn geometry, a quadrilateral is a polygon with four 'sides' or edges and four vertices or corners. Sometimes, the term quadrangle is used, for analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on. The word quadrilateral is made of the words quad and...
(green). Its vertex arrangement is the set {A, B, C, D}. Its convex hull is the triangleA triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
ABC (blue). The vertex arrangement of the convex hull is the set {A, B, C}, which is not the same as that of the quadrilateral; so here, the convex hull is not a way to describe the vertex arrangement. |
Infinite tilings can also share common
vertex arrangements.
For example, this triangular lattice of points can be connected to form either isosceles triangles or
rhombicIn geometry, a rhombus or rhomb is a quadrilateral whose four sides all have the same length. The rhombus is often called a diamond, after the diamonds suit in playing cards, or a lozenge, though the latter sometimes refers specifically to a rhombus with a 45° angle.In general, a polygon whose...
faces.
Four tilingA 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. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in...
s with same vertex arrangement.
Lattice points |
Triangular tilingIn geometry, the triangular tiling is one of the three regular tilings of the Euclidean plane. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees...
|
rhombicIn geometry, a rhombus or rhomb is a quadrilateral whose four sides all have the same length. The rhombus is often called a diamond, after the diamonds suit in playing cards, or a lozenge, though the latter sometimes refers specifically to a rhombus with a 45° angle.In general, a polygon whose...
tiling |
Zig-zag rhombic tiling |
Rhombille tiling |
Edge arrangement
PolyhedraA polyhedron is often defined as a geometric solid with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is...
can also have the same
edge arrangement which means they have similar vertex and edge arrangements, but may differ in their faces.
For example the self-intersecting
great dodecahedron shares it edge arrangement with the convex
icosahedron.
Two polyhedra with same edge arrangement.
icosahedronIn geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of five Platonic solids....
(20 triangles) |
great dodecahedronIn geometry, the great dodecahedron is a Kepler-Poinsot polyhedron, with Schläfli symbol {5,5/2}. It is one of four nonconvex regular polyhedra...
(12 intersecting pentagons) |
Face arrangement
4-polytopesIn geometry, a four-dimensional polytope or 4-polytope is a connected and closed figure composed of lower dimensional polytopal elements: vertices, edges, faces , and cells...
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 polychoraIn four dimensional geometry, Schläfli-Hess polychora are the complete set of 10 regular self-intersecting star polychora . They are named in honor of their discoverers: Ludwig Schläfli and Edmund Hess. Each is represented by a Schläfli symbol {p,q,r} in which one of the numbers is 5/2...
, there are only 7 unique face arragements.
For example the
grand stellated 120-cellIn geometry, the grand stellated 120-cell is a star polychoron with Schläfli symbol {5/2,5,5/2}. It is one of 10 regular Schläfli-Hess polychora.- Related polytopes :...
and
great stellated 120-cellIn geometry, the great stellated 120-cell is a star polychoron with Schläfli symbol {5/2,3,5}. It is one of 10 regular Schläfli-Hess polychora.It is one of four regular star polychora discovered by Ludwig Schläfli...
, both with
pentagramA pentagram is the shape of a five-pointed star drawn with five straight strokes...
mic faces, appear visually indistinguishable without a representation of their
cellsIn 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...
:
Two (projected) polychora with same face arrangement
Grand stellated 120-cellIn geometry, the grand stellated 120-cell is a star polychoron with Schläfli symbol {5/2,5,5/2}. It is one of 10 regular Schläfli-Hess polychora.- Related polytopes :...
(120 small stellated dodecahedronIn geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, with Schläfli symbol {5/2,5}. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex....
s) |
Great stellated 120-cellIn geometry, the great stellated 120-cell is a star polychoron with Schläfli symbol {5/2,3,5}. It is one of 10 regular Schläfli-Hess polychora.It is one of four regular star polychora discovered by Ludwig Schläfli...
(120 great stellated dodecahedronIn geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedron, with Schläfli symbol {5/2,3}. It is one of four nonconvex regular polyhedra.It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each vertex....
s) |
Classes of similar polytopes
George OlshevskyGeorge Olshevsky is a freelance editor, writer, publisher, paleontologist, and mathematician living in San Diego, California.Olshevsky maintains the comprehensive online Dinosaur Genera List...
advocates the term
army for a class of polytopes that share an element arrangement. More generally he defines
n-regiments for polytopes that share elements up to dimension
n. So a regiment (
1-regiment) shares the same
edge and vertex arrangement. He called a set of polytopes with the same
2-regiment as a company.
See also
- n-skeleton
In mathematics, particularly in algebraic topology, the n-skeleton of a topological space X presented as a simplicial complex refers to the subspace X
n that is the union of the simplices of X of dimensions m ≤ n...
- a set of elements of dimension n and lower in a higher polytope.
- Vertex figure
In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...
- 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) (Same vertex, edge and face arrangement)