Vertices, Edges and Faces
Euler emphasized five major components of a polyhedron in an attempt to find the Each side of each polygon meets exactly one other polygon along an edge. More vocabulary. The flat polygons and the regions they enclose are called the polyhedron's faces. The line segment where two faces meet is called an edge of . Polyhedron Edge. PolyhedronEdge. A line segment where two faces of a polyhedron meet, also called a side. SEE ALSO: Polygon Edge, Polyhedron Vertex.
These can be defined as partially ordered sets whose elements are the vertices, edges, and faces of a polyhedron. A vertex or edge element is less than an edge or face element in this partial order when the vertex or edge is part of the edge or face.
Additionally, one may include a special bottom element of this partial order representing the empty set and a top element representing the whole polyhedron. If the sections of the partial order between elements three levels apart that is, between each face and the bottom element, and between the top element and each vertex have the same structure as the abstract representation of a polygon, then these partially ordered sets carry exactly the same information as a topological polyhedron.
However, these requirements are often relaxed, to instead require only that sections between elements two levels apart have the same structure as the abstract representation of a line segment. Geometric polyhedra, defined in other ways, can be described abstractly in this way, but it is also possible to use abstract polyhedra as the basis of a definition of geometric polyhedra.
- Edge (geometry)
- Euler's polyhedron formula
- Vertices, Edges and Faces
A realization of an abstract polyhedron is generally taken to be a mapping from the vertices of the abstract polyhedron to geometric points, such that the points of each face are coplanar. A geometric polyhedron can then be defined as a realization of an abstract polyhedron. However, without additional restrictions, this definition allows degenerate or unfaithful polyhedra for instance, by mapping all vertices to a single point and the question of how to constrain realizations to avoid these degeneracies has not been settled.
In all of these definitions, a polyhedron is typically understood as a three-dimensional example of the more general polytope in any number of dimensions. For example, a polygon has a two-dimensional body and no faces, while a 4-polytope has a four-dimensional body and an additional set of three-dimensional "cells". However, some of the literature on higher-dimensional geometry uses the term "polyhedron" to mean something else: For instance, some sources define a convex polyhedron to be the intersection of finitely many half-spacesand a polytope to be a bounded polyhedron.
Characteristics[ edit ] Number of faces[ edit ] Polyhedra may be classified and are often named according to the number of faces. It turns out that it is quite a difficult task to identify whether or not a solid has a Hamiltonian circuit or not, and it may require a proof by exhaustion to determine whether or not a path is possible.
However, it has been proven that all Platonic solids, Archimedean solids, and planarconnected graphs have Hamiltonian circuits. Therefore, by counting the number of edges that we shade red we can determine the number of vertices.
To prove this, assume this is not the case.
Euler's polyhedron formula | promovare-site.info
Then there would be a vertex that is not coloured, which means we were not done colouring our edges red. Next we will begin by examining faces. Pick a face and shade it green, as well as any edges that have not yet been coloured red.
Proceed by finding another face who has only one green side, and shade green as before. Continue until there are either no more faces to colour that satisfy the condition.
Since it is impossible that a face exists with all four sides coloured, all faces must be shaded green. The relationship between green edges and faces can be described as: