What is dual to line segment? From Wiki Dual polyhedron 
For example the smplices:
3-simplex (tetrahedron) is self-dual
2-simplex (triangle) is self-dual
1-simplex (line segment)  ??
hypercubes:
3-cube (cube) dual is octahedron
2-cube (square) dual is square, self-dual
1-cube (line segment) ??
In the above link you can see it says all regular simplices are self-dual, but I can't see how a line segment is self-dual. Shouldn't its dual have 1 vertex and 2 edges? How would you draw that? Is there a rigorous definition of duality in this geometric sense?
 A: To understand why a line segment (which is both the cube and the simplex in dimension 1) is "its own dual" you have to think a little more carefully about duality and dimensions.
The dual of an $n$-dimensional polytope $P$ has a face of dimension $n-d-1$ for each $d$-dimensional face of $P$. You can see that clearly for the cube and octahedron. The eight vertices of the cube (dimension $0$) give the eight faces of the octahedron (dimension $3-0-1=2$). The twelve edges of each naturally correspond.
The dual of a square shouldn't be thought of as a square, but rather as a diamond whose edges come from the vertices of the square. They are perpendicular to the edges of the original square and form a square, but that's a coincidence.
On the line, with dimension $1$, each of the two endpoints of a segment, with dimension $0$, gives a face of the dual of dimension $1-0-1 = 0$, which is just a point. The dual has two points, so is a line segment.
You can get all this from the algebra of polarities as described on the wikipedia page.
A: The duality of polytopes such as simplices arises in either of two ways:


*

*In the theory of abstract polytopes, the structure of the figure is described as a partially-ordered set of ranked elements, linked by an incidence relation. Vertex points are rank 0, edges rank 1 and so on. To obtain the dual polytope you simply reverse the order of ranking.

*Geometrically a dual may be obtained via a projective transformation known as reciprocation. Duality is a theorem of projective geometry, but its exact manifestation depends on the number of spatial dimensions. In 3-space a point is dual to a plane, a line to another line. In the plane a point is dual to a line. In 1-space a point is dual to a point. Reciprocating an n-polytope in n-space about an n-sphere yields the familiar dual figures. In the case of a 1-polytope bounded by a point pair on a line, the result is also a point pair.
