Editing 4-polytope (section)

==Classification==

===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]].

=== Classes ===

The following lists the various categories of 4-polytopes classified according to the criteria above:
[[File:Schlegel half-solid truncated 120-cell.png|150px|thumb|The [[truncated 120-cell]] is one of 47 convex non-prismatic uniform 4-polytopes]]
'''[[Uniform 4-polytope]]''' ([[vertex-transitive]]):
* '''Convex uniform 4-polytopes''' (64, plus two infinite families)
** 47 non-prismatic [[uniform 4-polytope#Convex uniform 4-polytope|convex uniform 4-polytope]] including:
*** 6 [[Convex regular 4-polytope]]
** [[uniform 4-polytope#Prismatic uniform 4-polytope|Prismatic uniform 4-polytopes]]:
*** {} × {p,q} : 18 [[Uniform 4-polytope#Polyhedral hyperprisms|polyhedral hyperprisms]] (including cubic hyperprism, the regular [[hypercube]])
*** Prisms built on antiprisms (infinite family)
*** {p} × {q} : [[duoprism]]s (infinite family)
* '''Non-convex uniform 4-polytopes''' (10 + unknown)[[File:Ortho solid 016-uniform polychoron p33-t0.png|150px|thumb|The [[great grand stellated 120-cell]] is the largest of 10 regular star 4-polytopes, having 600 vertices.]]
** 10 (regular) [[Schläfli-Hess polytope]]s
** 57 hyperprisms built on [[Nonconvex uniform polyhedron|nonconvex uniform polyhedra]]
** Unknown total number of nonconvex uniform 4-polytopes: [[Norman Johnson (mathematician)|Norman Johnson]] and other collaborators have identified 2191 forms (convex and star, excluding the infinite families), all constructed by [[vertex figures]] by [[Stella (software)|Stella4D software]].<ref>[https://www.mit.edu/~hlb/Associahedron/program.pdf Uniform Polychora], Norman W. Johnson (Wheaton College), 1845 cases in 2005</ref>

'''Other convex 4-polytopes''':
* [[Polyhedral pyramid]]
* [[Polyhedral bipyramid]]
* [[Polyhedral prism]]
<!--* [[Polyhedral antiprism]]-->
[[File:Cubic honeycomb.png|150px|thumb|The regular [[cubic honeycomb]] is the only infinite regular 4-polytope in Euclidean 3-dimensional space.]]
'''Infinite uniform 4-polytopes of [[Euclidean space|Euclidean 3-space]]''' (uniform tessellations of convex uniform cells)
* 28 [[convex uniform honeycomb]]s: uniform convex polyhedral tessellations, including:
** 1 regular tessellation, [[cubic honeycomb]]: {4,3,4}

'''Infinite uniform 4-polytopes of [[Hyperbolic space|hyperbolic 3-space]]''' (uniform tessellations of convex uniform cells)
* 76 Wythoffian [[convex uniform honeycombs in hyperbolic space]], including:
** [[List of regular polytopes#Tessellations of hyperbolic 3-space|4 regular tessellation of compact hyperbolic 3-space]]: {3,5,3}, {4,3,5}, {5,3,4}, {5,3,5}

'''Dual [[uniform 4-polytope]]''' ([[cell-transitive]]):
* 41 unique dual convex uniform 4-polytopes
* 17 unique dual convex uniform polyhedral prisms
* infinite family of dual convex uniform duoprisms (irregular tetrahedral cells)
* 27 unique convex dual uniform honeycombs, including:
** [[Rhombic dodecahedral honeycomb]]
** [[Disphenoid tetrahedral honeycomb]]

'''Others:'''
* [[Weaire–Phelan structure]] periodic space-filling honeycomb with irregular cells
[[File:Hemi-icosahedron coloured.svg|150px|thumb|The [[11-cell]] is an abstract regular 4-polytope, existing in the [[real projective plane]], it can be seen by presenting its 11 hemi-icosahedral vertices and cells by index and color.]]
'''[[Abstract polytope|Abstract regular 4-polytopes]]''':
* [[11-cell]]
* [[57-cell]]

These categories include only the 4-polytopes that exhibit a high degree of symmetry. Many other 4-polytopes are possible, but they have not been studied as extensively as the ones included in these categories.