page_title: Abstract polytope
section_title: Rank 1: the line segment
hierarchical_section_title: Abstract polytope / The simplest polytopes / Rank 1: the line segment
caption_attribution_description: English: An Edge (Line Segment) and its Hasse Diagram
context_page_description: In mathematics, an abstract polytope is an algebraic partially ordered set or poset which captures the combinatorial properties of a traditional polytope without specifying purely geometric properties such as angles or edge lengths. A polytope is a generalisation of polygons and polyhedra into any number of dimensions.
  An ordinary geometric polytope is said to be a realization in some real N-dimensional space, typically Euclidean, of the corresponding abstract polytope. The abstract definition allows some more general combinatorial structures than traditional definitions of a polytope, thus allowing many new objects that have no counterpart in traditional theory.
context_section_description: 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.
caption: The graph (left) and Hasse Diagram of a line segment
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