Editing 5-cell (section)

{{Short description|Four-dimensional analogue of the tetrahedron}}
{{For|the sequence of fifth element numbers of Pascal's triangle|Pentatope number}}
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{{Infobox polychoron
 | Name=5-cell<BR>(4-simplex)
 | Image_File=5-cell.gif
 | Image_Caption=A 3D orthogonal projection of a 5-cell performing a [[SO(4)#Geometry of 4D rotations|simple rotation]]
 | Type=[[Convex regular 4-polytope]]
 | Family=[[Simplex]]
 | Last= 
 | Index=1
 | Next=[[Rectified 5-cell|2]]
 | Schläfli={3,3,3}
 | CD={{CDD|node_1|3|node|3|node|3|node}}
 | Cell_List=5 [[tetrahedron|{3,3}]] [[Image:3-simplex t0.svg|20px]] 
 | Face_List= 10 {3} [[Image:2-simplex t0.svg|20px]]
 | Edge_Count= 10
 | Vertex_Count= 5
 | Petrie_Polygon=[[pentagon]]
 | Coxeter_Group= A<sub>4</sub>, [3,3,3]
 | Vertex_Figure=[[Image:5-cell verf.svg|80px]]<BR>([[tetrahedron]])
 | Dual=[[Self-dual polytope|Self-dual]]
 | Property_List=[[Convex polytope|convex]], [[isogonal figure|isogonal]], [[isotoxal figure|isotoxal]], [[isohedral figure|isohedral]]
}}

In [[geometry]], the '''5-cell''' is the convex [[4-polytope]] with [[Schläfli symbol]] {3,3,3}. It is a 5-vertex [[four-dimensional space|four-dimensional]] object bounded by five tetrahedral cells. It is also known as a '''C<sub>5</sub>''', '''hypertetrahedron''', '''pentachoron''',{{sfn|Johnson|2018|p=249}} '''pentatope''', '''pentahedroid''',{{sfn|Ghyka|1977|p=[https://books.google.com/books?id=h1Wm6tHAKHcC&pg=PA68 68]}} '''tetrahedral pyramid''', or '''4-[[simplex]]''' (Coxeter's <math>\alpha_4</math> polytope),{{sfn|Coxeter|1973|p=120|loc=§7.2. see illustration Fig 7.2<small>A</small>}} the simplest possible convex 4-polytope, and is analogous to the [[tetrahedron]] in three dimensions and the [[triangle]] in two dimensions. The 5-cell is a [[Hyperpyramid|4-dimensional pyramid]] with a tetrahedral base and four tetrahedral sides.

The '''regular 5-cell''' is bounded by five [[regular tetrahedron|regular tetrahedra]], and is one of the six [[regular convex 4-polytope]]s (the four-dimensional analogues of the [[Platonic solids]]). A regular 5-cell can be constructed from a regular tetrahedron by adding a fifth vertex one edge length distant from all the vertices of the tetrahedron. This cannot be done in 3-dimensional space. The regular 5-cell is a solution to the problem: ''Make 10 equilateral triangles, all of the same size, using 10 matchsticks, where each side of every triangle is exactly one matchstick, and none of the triangles and matchsticks intersect one another.'' No solution exists in three dimensions.