7simplex (Octaexon) 
7orthoplex, 4_{11} (Heptacross) 

7cube (Hepteract) 
7demicube 1_{41} (Demihepteract) 

3_{21} polytope 
2_{31} polytope 
1_{31} polytope 
In geometry, a sevendimensional polytope, or 7polytope, is a polytope in 7dimensional space. Each 5polytope ridge being shared by exactly two 6polytope facets.
A proposed name for 7polytope is polyexon (plural: polyexa), created from poly, exa (short for hexa, meaning six) and on.
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Regular 7polytopes are represented by the Schläfli symbol {p,q,r,s,t,u} with u {p,q,r,s,t} 6polytopes facets around each 4face.
There are exactly three such convex regular 7polytopes:
There are no nonconvex regular 7polytopes.
The Euler characteristic for 7polytopes that are topological 6spheres (including all convex 7polytopes) is two. χ=VE+FC+f_{4}f_{5}+f_{6}=2.
For the 3 convex regular 7polytopes, their elements are:
Name  Schläfli symbol 
Vertices  Edges  Faces  Cells  4faces  5faces  6faces  χ 

7simplex  {3,3,3,3,3,3}  8  28  56  70  56  28  8  2 
7cube  {4,3,3,3,3,3}  128  448  672  560  280  84  14  2 
7orthoplex  {3,3,3,3,3,4}  14  84  280  560  672  448  128  2 
For 4 convex uniform 7polytopes, their elements are:
Name  Schläfli symbol 
Vertices  Edges  Faces  Cells  4faces  5faces  6faces  χ 

7demicube  {3^{1,4,1}}  64  672  2240  2800  1624  532  78  2 
3_{21} polytope  {3^{3,2,1}}  56  756  4032  10080  12096  6048  702  2 
2_{31} polytope  {3^{2,3,1}}  126  2016  10080  20160  16128  4788  632  2 
1_{32} polytope  {3^{1,3,2}}  576  10080  40320  50400  23688  4284  182  2 
Uniform 7polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the CoxeterDynkin diagrams:
#  Coxeter group  CoxeterDynkin diagram  

1  A_{7}  [3^{6}]  
2  B_{7}  [4,3^{5}]  
3  D_{7}  [3^{4,1,1}]  
4  E_{7}  [3^{3,2,1}] 
Selected regular and uniform 7polytopes from each family include:
There are 16 uniform prismatic families based on the uniform 6polytopes.
#  Coxeter group  CoxeterDynkin diagram  

1  A_{6}×A_{1}  [3^{5}] × [ ]  
2  B_{6}×A_{1}  [4,3^{4}] × [ ]  
3  D_{6}×A_{1}  [3^{3,1,1}] × [ ]  
4  E_{6}×A_{1}  [3^{2,2,1}] × [ ]  
5  A_{4}×I_{2}(p)×A_{1}  [3,3,3] × [p] × [ ]  
6  B_{4}×I_{2}(p)×A_{1}  [4,3,3] × [p] × [ ]  
7  F_{4}×I_{2}(p)×A_{1}  [3,4,3] × [p] × [ ]  
8  H_{4}×I_{2}(p)×A_{1}  [5,3,3] × [p] × [ ]  
9  D_{4}×I_{2}(p)×A_{1}  [3^{1,1,1}] × [p] × [ ]  
10  A_{3}×A_{3}×A_{1}  [3,3] × [3,3] × [ ]  
11  A_{3}×B_{3}×A_{1}  [3,3] × [4,3] × [ ]  
12  A_{3}×H_{3}×A_{1}  [3,3] × [5,3] × [ ]  
13  B_{3}×B_{3}×A_{1}  [4,3] × [4,3] × [ ]  
14  B_{3}×H_{3}×A_{1}  [4,3] × [5,3] × [ ]  
15  H_{3}×A_{3}×A_{1}  [5,3] × [5,3] × [ ]  
16  I_{2}(p)×I_{2}(q)×I_{2}(r)×A_{1}  [p] × [q] × [r] × [ ] 
There are 18 uniform duoprismatic forms based on Cartesian products of lower dimensional uniform polytopes.
#  Coxeter group  CoxeterDynkin diagram  

1  A_{5}×I_{2}(p)  [3,3,3] × [p]  
2  B_{5}×I_{2}(p)  [4,3,3] × [p]  
3  D_{5}×I_{2}(p)  [3^{2,1,1}] × [p]  
4  A_{4}×A_{3}  [3,3,3] × [3,3]  
5  A_{4}×B_{3}  [3,3,3] × [4,3]  
6  A_{4}×H_{3}  [3,3,3] × [5,3]  
7  B_{4}×A_{3}  [4,3,3] × [3,3]  
8  B_{4}×B_{3}  [4,3,3] × [4,3]  
9  B_{4}×H_{3}  [4,3,3] × [5,3]  
10  H_{4}×A_{3}  [5,3,3] × [3,3]  
11  H_{4}×B_{3}  [5,3,3] × [4,3]  
12  H_{4}×H_{3}  [5,3,3] × [5,3]  
13  F_{4}×A_{3}  [3,4,3] × [3,3]  
14  F_{4}×B_{3}  [3,4,3] × [4,3]  
15  F_{4}×H_{3}  [3,4,3] × [5,3]  
16  D_{4}×A_{3}  [3^{1,1,1}] × [3,3]  
17  D_{4}×B_{3}  [3^{1,1,1}] × [4,3]  
18  D_{4}×H_{3}  [3^{1,1,1}] × [5,3] 
There are 3 uniform triprismatic families based on Cartesian products of uniform polyhedrons and two regular polygons.
#  Coxeter group  CoxeterDynkin diagram  

1  A_{3}×I_{2}(p)×I_{2}(q)  [3,3] × [p] × [q]  
2  B_{3}×I_{2}(p)×I_{2}(q)  [4,3] × [p] × [q]  
3  H_{3}×I_{2}(p)×I_{2}(q)  [5,3] × [p] × [q] 
There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in 6space:
#  Coxeter group  CoxeterDynkin diagram  

1  A^{~}_{6}  [3^{[7]}]  
2  B^{~}_{6}  [4,3^{4},4]  
3  C^{~}_{6}  h[4,3^{4},4] [4,3^{3},3^{1,1}] 

4  D^{~}_{6}  q[4,3^{4},4] [3^{1,1},3^{2},3^{1,1}] 

5  E^{~}_{6}  [3^{2,2,2}] 
Regular and uniform tessellations include:
