Editing Abstract polytope (section)

==The simplest polytopes==

===Rank < 1===
There is just one poset for each rank −1 and 0. These are, respectively, the null face and the point. These are not always considered to be valid abstract polytopes.

===Rank 1: the line segment===
[[Image:An Edge (Line Segment) and its Hasse Diagram.PNG|thumb|240px|The graph (left) and Hasse Diagram of a line segment]]
There is only one polytope of rank 1, which is the line segment. It has a least face, just two 0-faces and a greatest face, for example {ø, '''a, b, ab'''}. It follows that the vertices '''a''' and '''b''' have rank 0, and that the greatest face '''ab''', and therefore the poset, both have rank 1.

===Rank 2: polygons===
For each ''p'', 3 ≤ ''p'' < <math>\infty</math>, we have (the abstract equivalent of) the traditional polygon with ''p'' vertices and ''p'' edges, or a ''p''-gon. For p = 3, 4, 5, ... we have the triangle, square, pentagon, ....

For ''p'' = 2, we have the [[digon]], and ''p'' = <math>\infty</math> we get the [[apeirogon]].

====The digon====
[[Image:Digon and Hasse Diagram.PNG|thumb|220px|The graph (left) and Hasse Diagram of a digon]]
A [[digon]] is a polygon with just 2 edges. Unlike any other polygon, both edges have the same two vertices. For this reason, it is ''degenerate'' in the [[Euclidean plane]].

Faces are sometimes described using "vertex notation" - e.g. {'''ø''', '''a''', '''b''', '''c''', '''ab''', '''ac''', '''bc''', '''abc'''} for the triangle '''abc'''. This method has the advantage of ''implying'' the '''<''' relation.

With the digon this vertex notation ''cannot be used''. It is necessary to give the faces individual symbols and specify the subface pairs F < G.

Thus, a digon is defined as a set {'''ø''', '''a''', '''b''', E', E", G} with the relation '''<''' given by

:::{'''ø'''<'''a''', '''ø'''<'''b''', '''a'''<E', '''a'''<E",  '''b'''<E', '''b'''<E", E'<G, E"<G}

where E' and E" are the two edges, and G the greatest face.

This need to identify each element of the polytope with a unique symbol applies to many other abstract polytopes and is therefore common practice.

A polytope can only be fully described using vertex notation if ''every face is incident with a unique set of vertices''. A polytope having this property is said to be [[Atom (order theory)|atomistic]].