Assertion: polyhedron's edge | /r/CapableOf | meet at vertex
Facets:
value: to another
type: TRANSITIVE-OBJECT
value: in the plane
type: LOCATION
value: together
type: MANNER
value: on a sphere
type: LOCATION
value: (intersect)
type: LOCATION
Source sentences:
text: A plane graph is one which has been drawn in the plane in such a way that the edges meet (intersect) only at a vertex.
text: 1 dimension: An edge joins one vertex to another and one face to another, and is usually a line segment.
text: All other combinatorial information about these solids, such as total number of vertices (V), edges (E), and faces (F), can be determined from p and q. Since any edge joins two vertices and has two adjacent faces we must have:
text: The angle α, is the angle between the two edges connecting the vertex d to the vertices b and c.
text: where a, b, and c represent three edges that meet at one vertex, and a · (b × c) is a scalar triple product.
text: At each vertex at least three edges come together, so the total number of ends of edges is at least 3 times the number of vertices.
text: The 4 Archimedean polyhedra illustrated so far are(i.e., only 3 edges meet at each vertex).
text: That is, each edge should be shared by exactly two polygons, and the link of each vertex should be a single cycle that alternates between the edges and polygons that meet at that vertex.
text: Archimedean solids for which 4 edges meet at every vertex.
text: A midplane is defined as a plane that is orthogonal to an edge joining any two vertices that also contains the centroid of an opposite edge formed by joining the other two vertices.
text: Denote the circumradius of a tetrahedron as R. Let a, b, c be the lengths of the three edges that meet at a vertex, and A, B, C the length of the opposite edges.
text: Although there are other ways of seeing that the graphs of convex 3-dimensional polyhedra or graphs on the sphere have planar graphs (i.e. at some stage use stereographic projection), this argument hopefully makes clear that the graphs that can be drawn on a sphere (polyhedra whose graphs are homeomorphic to a sphere) where edges meet only at vertices are the same as the graphs that can be drawn in a plane where edges cross only at vertices.
text: It holds for graphs embedded so that edges meet only at vertices on a sphere (or in the plane), but not for graphs embedded on the torus, a one-holed donut.
text: This concept involves the notion of the genus of a graph: the smallest number of handles g that must be added to the surface of a sphere so that a graph can be embedded on the augmented surface in such a way that edges meet only at vertices.
text: Adjacent faces meet at edges, and edges meet at vertices.
text: * 1 dimension: An edge joins one vertex to another and one face to another, and is usually a line of some kind.
text: For a polygon an edge is a line segment on the boundary joining one vertex (corner point) to another.