One special kind of polytope is the convex hull of a finite set of points. Roughly speaking this is the set of all possible weighted averages, with weights going from zero to one, of the points that lie at the vertices of the hull. When the points are in general position (are affinelinearly independent, no splane contains more than s of them), this defines an rsimplex (where r is the number of points).
For instance a 1simplex is a line segment, a 2simplex is a triangle, and a 3simplex is a tetrahedron (in each case here with interior, although the word simplex is also used to mean other variations). Note an rsimplex will not fit into an (r1)plane ((r1)dimensional space, if you prefer). Note also that any subset containing s of the r points defines a subsimplex, called an sface. The 0faces are just the vertices and the unique rface is the simplex itself.
Now given any convex hull in rdimensional space (but not in any (r1)plane, say) we can take linearly independent subsets of the vertices, and define rsimplexes with them. In fact you can choose several simplexes in this way such that their union as sets is the original hull, and the intersection of any two is either empty or an ssimplex (for some s < r).
For example, in the plane a square (convex hull of its corners) is the union of the two triangles (2simplexes), defined by a diagonal 1simplex which is their intersection?
In general, the definition (attributed to Alexandrov)is that an rpolytope is defined as a set with an rsimplicial decomposition. It is a union of ssimplices for values of s with s at most r, that is closed under intersection, and such that the only time that one of simplices is contained in another is as a face.
What does this let us build? Let's start with 1polytopes. Then we have the line segment, of course, and anything that you can get by joining line segments endtoend:
** ** ** ** ***    X  * ** ** *
If two segments meet at each vertex (so not the case for the final one), we get a topological curve, called a polygonal curve. You can categorize these as open or closed, depending on whether the ends match up, and as simple or complex, depending on whether they intersect themselves. Closed polygonal curves are called polygons.
Simple polygons in the plane are Jordan curves[?]: they have an interior that is a topological disk. And also a 2polytope (as you can see in the third example above), and these are often treated interchangeably with their boundary, the word polygon referring to either.
Now we can rinse and repeat! Joining polygons along edges (1faces) gives you a polyhedral surface, called a skew polygon when open and a polyhedron when closed. Simple polyhedra are interchangeable with their interiors, which are 3polytopes that can be used to build 4dimensional forms (sometimes called polychora), and so on to higher polytopes.
For a more abstract treatment, see simplicial complex.
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