Editing 4-polytope (section)

===Criteria===
Like all polytopes, 4-polytopes may be classified based on properties like "[[convex set|convexity]]" and "[[symmetry]]".

*A 4-polytope is ''[[Convex polytope|convex]]'' if its boundary (including its cells, faces and edges) does not intersect itself and the line segment joining any two points of the 4-polytope is contained in the 4-polytope or its interior; otherwise, it is ''non-convex''.  Self-intersecting 4-polytopes are also known as [[star 4-polytope]]s, from analogy with the star-like shapes of the non-convex [[star polygon]]s and [[Kepler–Poinsot polyhedra]].
* A 4-polytope is ''[[regular polytope|regular]]'' if it is [[transitive group action|transitive]] on its [[Flag (geometry)|flags]]. This means that its cells are all [[congruence (geometry)|congruent]] [[regular polyhedra]], and similarly its [[vertex figures]] are congruent and of another kind of regular polyhedron.
*A convex 4-polytope is ''[[semiregular polytope|semi-regular]]'' if it has a [[symmetry group]] under which all vertices are equivalent ([[vertex-transitive]]) and its cells are [[regular polyhedron|regular polyhedra]]. The cells may be of two or more kinds, provided that they have the same kind of face. There are only 3 cases identified by [[Thorold Gosset]] in 1900: the [[rectified 5-cell]], [[rectified 600-cell]], and [[snub 24-cell]].
*A 4-polytope is ''[[uniform polytope|uniform]]'' if it has a [[symmetry group]] under which all vertices are equivalent, and its cells are [[uniform polyhedron|uniform polyhedra]]. The faces of a uniform 4-polytope must be [[regular polygon|regular]].
* A 4-polytope is ''[[scaliform polytope|scaliform]]'' if it is vertex-transitive, and has all equal length edges. This allows cells which are not uniform, such as the regular-faced convex [[Johnson solid]]s.
*A regular 4-polytope which is also [[convex polytope|convex]] is said to be a [[convex regular 4-polytope]].
*A 4-polytope is ''prismatic'' if it is the [[Cartesian product]] of two or more lower-dimensional polytopes. A prismatic 4-polytope is uniform if its factors are uniform. The [[tesseract|hypercube]] is prismatic (product of two [[square (geometry)|square]]s, or of a [[cube]] and [[line segment]]), but is considered separately because it has symmetries other than those inherited from its factors.
*A ''[[tessellation|tiling]] or [[honeycomb (geometry)|honeycomb]] of 3-space'' is the division of three-dimensional [[Euclidean space]] into a repetitive [[Grid (spatial index)|grid]] of polyhedral cells. Such tilings or tessellations are infinite and do not bound a "4D" volume, and are examples of infinite 4-polytopes. A ''uniform tiling of 3-space'' is one whose vertices are congruent and related by a [[space group]] and whose cells are [[uniform polyhedron|uniform polyhedra]].