Uniform 10polytope
In tendimensional geometry, a 10polytope is a 10dimensional polytope whose boundary consists of 9polytope facets, exactly two such facets meeting at each 8polytope ridge.
10simplex 
Truncated 10simplex 
Rectified 10simplex  
Cantellated 10simplex 
Runcinated 10simplex  
Stericated 10simplex 
Pentellated 10simplex 
Hexicated 10simplex  
Heptellated 10simplex 
Octellated 10simplex 
Ennecated 10simplex  
10orthoplex 
Truncated 10orthoplex 
Rectified 10orthoplex  
10cube 
Truncated 10cube 
Rectified 10cube  
10demicube 
Truncated 10demicube 
A uniform 10polytope is one which is vertextransitive, and constructed from uniform facets.
Regular 10polytopes
Regular 10polytopes can be represented by the Schläfli symbol {p,q,r,s,t,u,v,w,x}, with x {p,q,r,s,t,u,v,w} 9polytope facets around each peak.
There are exactly three such convex regular 10polytopes:
 {3,3,3,3,3,3,3,3,3}  10simplex
 {4,3,3,3,3,3,3,3,3}  10cube
 {3,3,3,3,3,3,3,3,4}  10orthoplex
There are no nonconvex regular 10polytopes.
Euler characteristic
The topology of any given 10polytope is defined by its Betti numbers and torsion coefficients.[1]
The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 10polytopes, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.[1]
Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.[1]
Uniform 10polytopes by fundamental Coxeter groups
Uniform 10polytopes with reflective symmetry can be generated by these three Coxeter groups, represented by permutations of rings of the CoxeterDynkin diagrams:
#  Coxeter group  CoxeterDynkin diagram  

1  A_{10}  [3^{9}]  
2  B_{10}  [4,3^{8}]  
3  D_{10}  [3^{7,1,1}] 
Selected regular and uniform 10polytopes from each family include:
 Simplex family: A_{10} [3^{9}] 
 527 uniform 10polytopes as permutations of rings in the group diagram, including one regular:
 {3^{9}}  10simplex 
 {3^{9}}  10simplex 
 527 uniform 10polytopes as permutations of rings in the group diagram, including one regular:
 Hypercube/orthoplex family: B_{10} [4,3^{8}] 
 1023 uniform 10polytopes as permutations of rings in the group diagram, including two regular ones:
 {4,3^{8}}  10cube or dekeract 
 {3^{8},4}  10orthoplex or decacross 
 h{4,3^{8}}  10demicube
.
 {4,3^{8}}  10cube or dekeract 
 1023 uniform 10polytopes as permutations of rings in the group diagram, including two regular ones:
 Demihypercube D_{10} family: [3^{7,1,1}] 
 767 uniform 10polytopes as permutations of rings in the group diagram, including:
 1_{7,1}  10demicube or demidekeract 
 7_{1,1}  10orthoplex 
 1_{7,1}  10demicube or demidekeract 
 767 uniform 10polytopes as permutations of rings in the group diagram, including:
The A_{10} family
The A_{10} family has symmetry of order 39,916,800 (11 factorial).
There are 512+161=527 forms based on all permutations of the CoxeterDynkin diagrams with one or more rings. 31 are shown below: all one and two ringed forms, and the final omnitruncated form. Bowersstyle acronym names are given in parentheses for crossreferencing.
#  Graph  CoxeterDynkin diagram Schläfli symbol Name 
Element counts  

9faces  8faces  7faces  6faces  5faces  4faces  Cells  Faces  Edges  Vertices  
1 

11  55  165  330  462  462  330  165  55  11  
2 

495  55  
3 

1980  165  
4 

4620  330  
5 

6930  462  
6 

550  110  
7 

4455  495  
8 

2475  495  
9 

15840  1320  
10 

17820  1980  
11 

6600  1320  
12 

32340  2310  
13 

55440  4620  
14 

41580  4620  
15 

11550  2310  
16 

41580  2772  
17 

97020  6930  
18 

110880  9240  
19 

62370  6930  
20 

13860  2772  
21 

34650  2310  
22 

103950  6930  
23 

161700  11550  
24 

138600  11550  
25 

18480  1320  
26 

69300  4620  
27 

138600  9240  
28 

5940  495  
29 

27720  1980  
30 

990  110  
31  t_{0,1,2,3,4,5,6,7,8,9}{3,3,3,3,3,3,3,3,3} Omnitruncated 10simplex 
199584000  39916800 
The B_{10} family
There are 1023 forms based on all permutations of the CoxeterDynkin diagrams with one or more rings.
Twelve cases are shown below: ten singlering (rectified) forms, and two truncations. Bowersstyle acronym names are given in parentheses for crossreferencing.
#  Graph  CoxeterDynkin diagram Schläfli symbol Name 
Element counts  

9faces  8faces  7faces  6faces  5faces  4faces  Cells  Faces  Edges  Vertices  
1  t_{0}{4,3,3,3,3,3,3,3,3} 10cube (deker) 
20  180  960  3360  8064  13440  15360  11520  5120  1024  
2  t_{0,1}{4,3,3,3,3,3,3,3,3} Truncated 10cube (tade) 
51200  10240  
3  t_{1}{4,3,3,3,3,3,3,3,3} Rectified 10cube (rade) 
46080  5120  
4  t_{2}{4,3,3,3,3,3,3,3,3} Birectified 10cube (brade) 
184320  11520  
5  t_{3}{4,3,3,3,3,3,3,3,3} Trirectified 10cube (trade) 
322560  15360  
6  t_{4}{4,3,3,3,3,3,3,3,3} Quadrirectified 10cube (terade) 
322560  13440  
7  t_{4}{3,3,3,3,3,3,3,3,4} Quadrirectified 10orthoplex (terake) 
201600  8064  
8  t_{3}{3,3,3,3,3,3,3,4} Trirectified 10orthoplex (trake) 
80640  3360  
9  t_{2}{3,3,3,3,3,3,3,3,4} Birectified 10orthoplex (brake) 
20160  960  
10  t_{1}{3,3,3,3,3,3,3,3,4} Rectified 10orthoplex (rake) 
2880  180  
11  t_{0,1}{3,3,3,3,3,3,3,3,4} Truncated 10orthoplex (take) 
3060  360  
12  t_{0}{3,3,3,3,3,3,3,3,4} 10orthoplex (ka) 
1024  5120  11520  15360  13440  8064  3360  960  180  20 
The D_{10} family
The D_{10} family has symmetry of order 1,857,945,600 (10 factorial × 2^{9}).
This family has 3×256−1=767 Wythoffian uniform polytopes, generated by marking one or more nodes of the D_{10} CoxeterDynkin diagram. Of these, 511 (2×256−1) are repeated from the B_{10} family and 256 are unique to this family, with 2 listed below. Bowersstyle acronym names are given in parentheses for crossreferencing.
#  Graph  CoxeterDynkin diagram Schläfli symbol Name 
Element counts  

9faces  8faces  7faces  6faces  5faces  4faces  Cells  Faces  Edges  Vertices  
1  10demicube (hede) 
532  5300  24000  64800  115584  142464  122880  61440  11520  512  
2  Truncated 10demicube (thede) 
195840  23040 
Regular and uniform honeycombs
There are four fundamental affine Coxeter groups that generate regular and uniform tessellations in 9space:
#  Coxeter group  CoxeterDynkin diagram  

1  [3^{[10]}]  
2  [4,3^{7},4]  
3  h[4,3^{7},4] [4,3^{6},3^{1,1}]  
4  q[4,3^{7},4] [3^{1,1},3^{5},3^{1,1}] 
Regular and uniform tessellations include:
 Regular 9hypercubic honeycomb, with symbols {4,3^{7},4},
 Uniform alternated 9hypercubic honeycomb with symbols h{4,3^{7},4},
Regular and uniform hyperbolic honeycombs
There are no compact hyperbolic Coxeter groups of rank 10, groups that can generate honeycombs with all finite facets, and a finite vertex figure. However, there are 3 noncompact hyperbolic Coxeter groups of rank 9, each generating uniform honeycombs in 9space as permutations of rings of the Coxeter diagrams.
= [3^{1,1},3^{4},3^{2,1}]: 
= [4,3^{5},3^{2,1}]: 
or = [3^{6,2,1}]: 
Three honeycombs from the family, generated by endringed Coxeter diagrams are:
References
 Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.
 T. Gosset: On the Regular and SemiRegular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
 A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
 H.S.M. Coxeter:
 H.S.M. Coxeter, M.S. LonguetHiggins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954
 H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10]
 (Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591]
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
 N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
 Klitzing, Richard. "10D uniform polytopes (polyxenna)".
External links
 Polytope names
 Polytopes of Various Dimensions, Jonathan Bowers
 Multidimensional Glossary
 Glossary for hyperspace, George Olshevsky.