Isogonal figure

In geometry, a polytope (a polygon, polyhedron or tiling, for example) is isogonal or vertex-transitive if, loosely speaking, all its vertices are the same. That implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces.

Technically, we say that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope is transitive on its vertices, or that the vertices lie within a single symmetry orbit.

All vertices of a finite n-dimensional isogonal figure exist on an (n-1)-sphere.

The term isogonal has long been used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry groups and graph theory.

The pseudorhombicuboctahedron — which is not isogonal — demonstrates that simply asserting that "all vertices look the same" is not as restrictive as the definition used here, which involves the group of isometries preserving the polyhedron or tiling.

Vertex

Vertex (Latin: peak; plural vertices or vertexes) may refer to:

Mathematics and computer science

Physics

Biology and anatomy

Companies

Vertex (graph theory)

In mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices). In a diagram of a graph, a vertex is usually represented by a circle with a label, and an edge is represented by a line or arrow extending from one vertex to another.

From the point of view of graph theory, vertices are treated as featureless and indivisible objects, although they may have additional structure depending on the application from which the graph arises; for instance, a semantic network is a graph in which the vertices represent concepts or classes of objects.

The two vertices forming an edge are said to be the endpoints of this edge, and the edge is said to be incident to the vertices. A vertex w is said to be adjacent to another vertex v if the graph contains an edge (v,w). The neighborhood of a vertex v is an induced subgraph of the graph, formed by all vertices adjacent to v.

Interaction point

In particle physics, an interaction point (IP) is the place where particles collide. One differentiates between the nominal IP, which is the design position of the IP, and the real or physics IP, which is the position where the particles actually collide. The real IP is the primary vertex of the particle collision.

For fixed target experiments, the IP is the point where beam and target interact. For colliders (and collider experiments), it is the place where the beams interact. Experiments (particle detectors) at particle accelerators are built around the nominal IPs of the accelerators. Therefore the whole region around the IP (the experimental hall) is called an interaction region. Particle colliders such as LEP, HERA, RHIC, Tevatron and LHC can host several interaction regions and therefore several experiments taking advantage of the same beam.