Segmentotope (EntityClass, 11)
From Hi.gher. Space
A segmentotope is an orbiform polytope whose vertices lie on two parallel hyperplanes and whose edge lengths are all equal. The set of all convex segmentotopes up to and including 4D has been enumerated by Dr. Richard Klitzing. There are 177 fourdimensional segmentotopes, including some polychora from other categories (such as cube  cube, which is the same as the tesseract). The full list can be obtained from Klitzing's paper (PDF format).
Nomenclature
A segmentotope is denoted by the notation A  B, where A and B are lowerdimensional polytopes. For segmentochora, A and B are usually polyhedra, although one of them can be lowerdimensional, as is the case with the wedges and pyramids.
Some segmentotopes may have multiple designations, for example, (triangular_prism  hexagonal_prism) is the same as (triangular_cupola  triangular_cupola). Where multiple names are possible, the name listed by Klitzing takes precedence.
Properties
Below are some useful properties of selected segmentochora. Klitzing's numbering is written as "K 4.n", as given in his PhD dissertation. Measurements are given in terms of E, the edge length.
#  Name(s)  Cells  Values 

K 4.7 

 
K 4.26 



K 4.45 
triangular_cupola  triangular_cupola 
 
K 4.51 
 
K 4.69 
square_cupola  square_cupola 
 
K 4.105 
 
K 4.117 
pentagonal_cupola  pentagonal_cupola 
 
K 4.141 
 
K 4.165 

Classification
Keiji has begun a project to classify all of Klitzing's segmentochora, so that more interesting ones stand out to help future research.
The current classification is as follows, where each number is a K 4.X index:
 Ur  3 regular polychora
 1, 2, 20
 Ux  the pyrorectichoron, a uniform polychoron
 5
 Up  all 17 prisms of the uniform polyhedra (not 18, because the cube prism is the K 4.20, already counted above)
 9, 11, 36, 43, 57, 60, 66, 74, 89, 90, 99, 110, 111, 125, 127, 130, 150
 Inf  4 infinite families (not individual polychora) and 21 members of those infinite families
family  {3}  {4}  {5}  {6}  {8}  {10} 

174  6  14  22  46  58  93 
175  10  (18)  34  47  59  94 
176  (11)  19  39  53  65  96 
177  18  (20)  42  54  70  97 
 Py  12 pyramids of CRF polyhedra
 3, 4, 7, 17, 26, 80, 84, 85, 86, 87, 88, 141
 Po  25 prisms of orbiform Johnson solids
 12, 37, 38, 40, 41, 44, 45, 67, 68, 69, 91, 92, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124
 Cu  30 monostratic cupolic forms
 15, 21, 23, 29, 35, 48, 52, 55, 56, 61, 71, 75, 76, 77, 78, 95, 98, 100, 107, 126, 128, 129, 131, 137, 149, 151, 152, 158, 159, 173
 Bicu  all 10 bicupolic rings (excluding the prisms of the tetrahedron and square pyramid, already counted above)
 8, 25, 27, 51, 64, 73, 105, 133, 154, 165
 Xcu  3 fragments of cupolic forms (excluding the bicupolic rings)
 109, 139, 146
 Xg  the gyrated octahedral prism
 13
 Xt  the trigonal tridiminishedicosahedral wedge
 33
 Gy  49 gyrations and diminishes of the cupolic forms
 16, 24, 28, 30, 31, 32, 49, 50, 62, 63, 72, 79, 81, 82, 83, 101, 102, 103, 104, 106, 108, 132, 134, 135, 136, 138, 140, 142, 143, 144, 145, 147, 148, 153, 155, 156, 157, 160, 161, 162, 163, 164, 166, 167, 168, 169, 170, 171, 172
Full list of segmentochora
 See also: Segmentochoron construction table
#  Name  Symbol  Cells  Comments 

1  pyrochoron  x3o3o3o  5x tetrahedron  
2  aerochoron  x3o3o4o  16x tetrahedron  
3  octahedral pyramid  xo3oo4oo  8x tetrahedron + 1x octahedron  half of aerochoron 
4  square dipyramid  xo4oo2ox  4x tetrahedron + 2x square pyramid  quarter of aerochoron 
5  pyrorectichoron  o3x3o3o  5x tetrahedron + 5x octahedron  tetrahedral semicupola 
6  trigonal biantiprismatic ring  xox3oxo&#x  3x tetrahedron + 2x octahedron + 3x square pyramid + 1x trigonal prism  (mono)diminished pyrorectichoron 
7  triangular prism pyramid  xo3oo2xo  2x tetrahedron + 3x square pyramid + 1x trigonal prism  
8  digonal gyrobicupolic ring  xxo2xox&#x  1x tetrahedron + 4x square pyramid + 2x trigonal prism  bidiminished pyrorectichoron 
9  tetrahedral prism  x3o3o2x  2x tetrahedron + 4x trigonal prism  
10  duotrianglinder  x3o2x3o  6x trigonal prism  
11  octahedral prism  x3o4o2x  2x octahedron + 8x trigonal prism  
12  square pyramid prism  xx2oo4xo  2x square pyramid + 4x trigonal prism + 1x cube  
13  gyrated octahedral prism  ??  4x square pyramid + (4+4)x trigonal prism  
14  square biantiprismatic ring  xox4oxo&#x  4x tetrahedron + 4x square pyramid + 2x square antiprism + 1x cube  
15  staurohedral antiprism  xo4oo3ox  (8+12)x tetrahedron + 1x octahedron + 6x square pyramid + 1x cube  
16  K4.16  ??  (4+4+4)x tetrahedron + (1+1+4)x square pyramid + 1x square antiprism + 1x cube  
17  square antiprism pyramid  oxo4oox&#x  8x tetrahedron + 2x square pyramid + 1x square antiprism  
18  triangular diprism  x3o2x4o  4x trigonal prism + 3x cube  
19  square antiprism prism  ??  8x trigonal prism + 2x square antiprism + 2x cube  
20  geochoron  x4o3o3o  8x cube  
21  snubdis antiprism  xo4os3os  8x tetrahedron + 12x square pyramid + 6x trigonal prism + 1x cube + 1x icosahedron  snubdis antiprism 
22  K4.22  xox5oxo&#x  5x tetrahedron + 5x square pyramid + 2x pentagonal antiprism + 1x pentagonal prism  
23  tetrahedral pericupola  xx3oo3ox  (1+4)x tetrahedron + (4+6)x trigonal prism + 1x stauromesohedron  
24  K4.24  ??  2x tetrahedron + 6x trigonal prism + 2x trigonal cupola  
25  triangular orthobicupolic ring  xx3xo2ox  3x tetrahedron + (1+3)x trigonal prism + 2x trigonal cupola  
26  cubic pyramid  xo4oo3oo  6x square pyramid + 1x cube  
27  triangular gyrobicupolic ring  xxo3xox&#x  1x octahedron + 6x square pyramid + 2x trigonal cupola  
28  K4.28  ??  (4+8)x square pyramid + 2x cube + 1x stauromesohedron  
29  octahedral semicupola  xo3ox4oo  (1+8)x octahedron + 6x square pyramid + 1x stauromesohedron  
30  K4.30  ??  (2+3)x octahedron + 6x square pyramid + 2x trigonal cupola  
31  K4.31  ??  4x octahedron + (1+1+4+4)x square pyramid + 1x cube + 1x stauromesohedron  
32  K4.32  ??  1x octahedron + (2+2+4)x square pyramid + 1x cube + 2x trigonal cupola  
33  K4.33  ??  3x tetrahedron + 1x octahedron + 3x square pyramid + 1x trigonal prism + 3x pentagonal pyramid + 1x tridiminished icosahedron  trigonal tridiminishedicosahedral wedge 
34  5,3duoprism  x5o2x3o  5x trigonal prism + 3x pentagonal prism  
35  cubic semicupola  xo4ox3oo  8x tetrahedron + 6x square antiprism + 1x cube + 1x stauromesohedron  
36  icosahedral prism  x3o5o2x  20x trigonal prism + 2x icosahedron  
37  gyroelongated pentagonal pyramid prism  ??  (5+5+5)x trigonal prism + 1x pentagonal prism + 2x gyroelongated pentagonal pyramid  
38  pentagonal pyramid prism  xx2oo5xo  5x trigonal prism + 2x pentagonal pyramid + 1x pentagonal prism  
39  pentagonal antiprism prism  ??  10x trigonal prism + 2x pentagonal antiprism + 2x pentagonal prism  
40  metabidiminished icosahedron prism  ??  (2+2+2+4)x trigonal prism + 2x pentagonal prism + 2x metabidiminished icosahedron  
41  tridiminished icosahedron prism  xxx2xfo3oox  (1+1+3)x trigonal prism + 3x pentagonal prism + 2x tridiminished icosahedron  
42  pentagonal diprism  x5o2x4o  5x cube + 4x pentagonal prism  
43  stauromesohedron prism  o4x3o2x  8x trigonal prism + 6x cube + 2x stauromesohedron  
44  trigonal orthobicupola prism  xxx3oxo2xxx  (2+6)x trigonal prism + 6x cube + 2x trigonal orthobicupola  
45  trigonal cupola prism  xx3xo2xx  (1+3)x trigonal prism + 3x cube + 2x trigonal cupola + 1x hexagonal prism  half of stauromesohedron prism 
46  K4.46  xox6oxo&#x  6x tetrahedron + 6x square pyramid + 2x hexagonal antiprism + 1x hexagonal prism  
47  6,3duoprism  x6o2x3o  6x trigonal prism + 3x hexagonal prism  
48  tetrahedral semipericupola  xx3xo3ox  4x octahedron + 6x trigonal prism + 1x stauromesohedron + 4x trigonal cupola + 1x truncated tetrahedron  
49  K4.49  ??  1x octahedron + (3+3)x square pyramid + (3+3)x trigonal prism + 1x trigonal orthobicupola + (1+3)x trigonal cupola + 1x truncated tetrahedron  
50  K4.50  ??  1x octahedron + 3x square pyramid + 3x trigonal prism + (1+3)x trigonal cupola + 1x hexagonal prism + 1x truncated tetrahedron  
51  triangular magnabicupolic ring  xx3xo2xo  3x square pyramid + 3x trigonal prism + 2x trigonal cupola + 1x hexagonal prism  
52  tetrahedral semimesocupola  xo3xx3oo  1x octahedron + 4x trigonal prism + 4x trigonal cupola + 1x truncated tetrahedron  
53  hexagonal antiprism prism  ??  12x trigonal prism + 2x hexagonal antiprism + 2x hexagonal prism  
54  hexagonal diprism  x6o2x4o  6x cube + 4x hexagonal prism  
55  tetrahedral semiantiprism  xo3xx3ox  6x tetrahedron + 8x trigonal cupola + 2x truncated tetrahedron  
56  tetrahedral quasicupola  xx3xo3oo  (1+4)x tetrahedron + 4x trigonal cupola + 1x truncated tetrahedron  
57  truncated tetrahedron prism  x3x3o2x  4x trigonal prism + 4x hexagonal prism + 2x truncated tetrahedron  
58  K4.58  xox8oxo&#x  8x tetrahedron + 8x square pyramid + 2x octagonal antiprism + 1x octagonal prism  
59  8,3duoprism  x8o2x3o  8x trigonal prism + 3x octagonal prism  
60  snub cube prism  s4s3s2x  (8+24)x trigonal prism + 6x cube + 2x snub cube  
61  staurohedral mesocupola  xo4ox3xo  8x octahedron + 12x square pyramid + 6x square antiprism + 1x stauromesohedron + 1x stauroperihedron  
62  K4.62  ??  4x octahedron + (4+4+4)x square pyramid + (1+4)x square antiprism + 1x stauromesohedron + 1x elongated square cupola + 1 square cupola  
63  K4.63  ??  (4+8)x square pyramid + 4x square antiprism + 1x stauromesohedron + 2x square cupola + 1x octagonal prism  
64  square gyrobicupolic ring  xxo4oxx&#x  8x square pyramid + 1x square antiprism + 2x square cupola  
65  octagonal antiprism prism  ??  16x trigonal prism + 2x octagonal antiprism + 2x octagonal prism  
66  stauroperihedral prism  x4o3x2x  8x trigonal prism + (6+12)x cube + 2x stauroperihedron  
67  elongated square gyrobicupola prism  ??  8x trigonal prism + (2+8+8)x cube + 2x elongated square gyrobicupola  
68  elongated square cupola prism  ??  4x trigonal prism + (1+4+4+4)x cube + 2x elongated square cupola + 1x octagonal prism  
69  square cupola prism  xx4xo2xx  4x trigonal prism + (1+4)x cube + 2x square cupola + 1x octagonal prism  
70  octagonal diprism  x8o2x4o  8x cube + 4x octagonal prism  
71  cubic pericupola  xx4oo3ox  8x tetrahedron + 12x trigonal prism + (1+6)x cube + 1x stauroperihedron  
72  K4.72  ??  4x tetrahedron + (4+4)x trigonal prism + (1+1+4)x cube + 1x elongated square cupola + 1 square cupola  
73  square orthobicupolic ring  xx4xo2ox  4x tetrahedron + 2x square pyramid + 4x trigonal prism + 1x cube  
74  dodecahedron prism  x5o3o2x  12x pentagonal prism + 2x dodecahedron  
75  octahedral semipericupola  xx3xo4ox  12x trigonal prism + 6x square antiprism + 8x trigonal cupola + 1x stauroperihedron + 1x truncated octahedron  
76  tetrahedral canticupola  xx3xx3xo  6x trigonal prism + 4x trigonal cupola + 4x hexagonal prism + 1 truncated tetrahedron + 1 truncated octahedron  
77  dodecahedral semicupola  xo5ox3oo  20x tetrahedron + 12x pentagonal antiprism + 1x dodecahedron + 1x rhodomesohedron  
78  rhodohedral antiprism  xo5oo3ox  (20+30)x tetrahedron + 1x icosahedron + 12x pentagonal pyramid + 1x dodecahedron  
79  K4.79  ??  (5+5+5+5+5+5+10)x tetrahedron + 1x gyroelongated pentagonal pyramid + (1+5+5)x pentagonal pyramid + 1x pentagonal antiprism + 1x dodecahedron  
80  pentagonal antiprism pyramid  ??  10x tetrahedron + 2x pentagonal pyramid + 1x pentagonal antiprism  
81  K4.81  ??  (10+10+10)x tetrahedron + 10x pentagonal pyramid + (1+2)x pentagonal antiprism + 1x dodecahedron  
82  K4.82  ??  (1+1+2+2+2+2+4+4+4+4+4)x tetrahedron + (2+2+2+4)x pentagonal pyramid + 2x pentagonal antiprism + 1x metabidiminished icosahedron + 1x dodecahedron  
83  K4.83  ??  (1+1+3+3+3+3+6)x tetrahedron + (3+3+3)x pentagonal pyramid + 3x pentagonal antiprism + 1x tridiminished icosahedron + 1x dodecahedron  
84  icosahedron pyramid  xo3oo5oo  20x tetrahedron + 1x icosahedron  
85  gyroelongated pentagonal pyramid pyramid  ??  (5+5+5)x tetrahedron + 1x gyroelongated pentagonal pyramid + 1x pentagonal pyramid  
86  pentagonal dipyramid  xo5oo2ox  5x tetrahedron + 2x pentagonal pyramid  
87  metabidiminished icosahedron pyramid  ??  (2+2+2+4)x tetrahedron + 2x pentagonal pyramid + 1x metabidiminished icosahedron  
88  tridiminished icosahedron pyramid  ??  (1+1+3)x tetrahedron + 3x pentagonal pyramid + 1x tridiminished icosahedron  
89  truncated octahedron prism  x3x4o2x  6x cube + 8x hexagonal prism + 2x truncated octahedron  
90  rhodomesohedron prism  ??  20x trigonal prism + 12x pentagonal prism + 2x rhodomesohedron  
91  orthobirotunda prism  ??  (10+10)x trigonal prism + (2+10)x pentagonal prism + 2x orthobirotunda  
92  rotunda prism  ??  (5+5)x trigonal prism + (1+5)x pentagonal prism + 2x rotunda + 1x decagonal prism  
93  K4.93  xox10oxo&#x  10x tetrahedron + 10x square pyramid + 2x decagonal antiprism + 1x decagonal prism  
94  10,3duoprism  x10o2x3o  10x trigonal prism + 3x decagonal prism  
95  octahedral semimesocupola  xo3xx4oo  6x cube + 1x stauromesohedron + 8x trigonal cupola + 1x truncated octahedron  
96  decagonal antiprism prism  ??  20x trigonal prism + 2x decagonal antiprism + 2x decagonal prism  
97  decagonal diprism  x10o2x4o  10x cube + 4x decagonal prism  
98  staurohedral semiantiprism  xo4xx3ox  12x tetrahedron + 8x trigonal cupola + 6x square cupola + 1x truncated octahedron + 1x truncated cube  
99  truncated cube prism  x4x3o2x  8x trigonal prism + 6x octagonal prism + 2x truncated cube  
100  cubic semipericupola  xx4xo3ox  8x octahedron + 12x trigonal prism + 1x stauroperihedron + 6x square cupola + 1x truncated cube  
101  K4.101  ??  4x octahedron + (4+4)x square pyramid + (4+4+4)x trigonal prism + 1x elongated square gyrobicupola + (1+1+4)x square cupola + 1x truncated cube  
102  K4.102  ??  (8+8)x square pyramid + (4+8)x trigonal prism + 1x stauroperihedron + (2+4)x square cupola + 1x truncated cube  
103  K4.103  ??  4x octahedron + 4x square pyramid + (4+4)x triangular prism + 1x elongated square cupola + (1+4)x square cupola + 1x octagonal prism + 1x truncated cube  
104  K4.104  ??  (4+4+4)x square pyramid + (4+4)x trigonal prism + 1x elongated square cupola + (1+4)x square cupola + 1x octagonal prism + 1x truncated cube  
105  square magnabicupolic ring  xx4xo2xo  4x square pyramid + 4x trigonal prism + 2x square cupola + 1x octagonal prism  
106  K4.106  ??  8x square pyramid + 4x trigonal prism + 4x square cupola + (1+2)x octagonal prism + 1x truncated cube  
107  octahedral pericupola  xx3oo4ox  1x octahedron + 6x square pyramid + (8+12)x trigonal prism + 1x stauroperihedron  
108  K4.108  ??  (1+1+4)x square pyramid + (4+4+4)x trigonal prism + 1x elongated square cupola + 1x square cupola  
109  K4.109  ??  8x square pyramid + 2x trigonal prism + 2x square cupola  
110  snub dodecahedron prism  s5s3s2x  (20+60)x trigonal prism + 12x pentagonal prism + 2x snub dodecahedron  
111  rhodoperihedron prism  x5o3x2x  20x trigonal prism + 30x cube + 12x pentagonal prism + 2x rhodoperihedron  
112  gyrated rhodoperihedron prism  ??  (5+5+5+5)x trigonal prism + (5+5+5+5+10)x cube + (1+1+5+5)x pentagonal prism + 2x gyrated rhodoperihedron  
113  parabigyrated rhodoperihedron prism  ??  (10+10)x trigonal prism + (10+10+10)x cube + (2+10)x pentagonal prism + 2x parabigyrated rhodoperihedron  
114  metabigyrated rhodoperihedron  ??  (2+2+2+2+4+4+4)x trigonal prism + (1+1+2+2+4+4+4+4+4+4)x cube + (2+2+2+2+4)x pentagonal prism + 2x metabigyrated rhodoperihedron  
115  trigyrated rhodoperihedron  ??  (1+1+3+3+6+6)x trigonal prism + (3+3+3+3+6+6+6)x cube + (3+3+3+3)x pentagonal prism + 2x trigyrated rhodoperihedron  
116  diminished rhodoperihedron prism  ??  (5+5+5)x trigonal prism + (5+5+5+10)x cube + (1+5+5)x pentagonal prism + 1x decagonal prism + 2x diminished rhodoperihedron  
117  pentagonal cupola prism  xx5xo2xx  5x trigonal prism + 5x cube + 1x pentagonal prism + 1x decagonal prism + 2x pentagonal cupola  
118  gyrated paradiminished rhodoperihedron prism  ??  (5+5+5)x trigonal prism + (5+5+5+10)x cube + (1+5+5)x pentagonal prism + 1x decagonal prism + 2x gyrated paradiminished rhodoperihedron  
119  gyrated metadiminished rhodoperihedron prism  ??  (1+1+1+2+2+2+2+2+2)x trigonal prism + (1+1+1+2+2+2+2+2+2+2+2+2+2+2)x cube + (1+1+1+2+2+2+2)x pentagonal prism + 1x decagonal prism + 2x gyrated metadiminished rhodoperihedron  
120  bigyrated diminished rhodoperihedron prism  ??  (1+1+1+2+2+2+2+2+2)x trigonal prism + (1+1+1+2+2+2+2+2+2+2+2+2+2+2)x cube + (1+1+1+2+2+2+2)x pentagonal prism + 1x decagonal prism + 2x bigyrated diminished rhodoperihedron  
121  parabidiminished rhodoperihedron prism  ??  10x trigonal prism + (10+10)x cube + 10x pentagonal prism + 2x decagonal prism + 2x parabidiminished rhodoperihedron  
122  metabidiminished rhodoperihedron prism  ??  (2+2+2+4)x trigonal prism + (1+1+2+4+4+4+4)x cube + (2+2+2+4)x pentagonal prism + 2x decagonal prism + 2x metabidiminished rhodoperihedron  
123  gyrated bidiminished rhodoperihedron prism  ??  (1+1+1+1+2+2+2)x trigonal prism + (1+1+1+1+2+2+2+2+2+2+2+2+2)x cube + (1+1+1+1+2+2+2)x pentagonal prism + 2x decagonal prism + 2x gyrated bidiminished rhodoperihedron  
124  tridiminished rhodoperihedron prism  ??  (1+1+3)x trigonal prism + (3+3+3+6)x cube + (3+3+3)x pentagonal prism + 3x decagonal prism + 2x tridiminished rhodoperihedron  
125  stauropantohedron prism  x4x3x2x  12x cube + 8x hexagonal prism + 6x octagonal prism + 2x stauropantohedron  
126  icosahedral semipericupola  xx3xo5ox  30x trigonal prism + 12x pentagonal antiprism + 20x trigonal cupola + 1x rhodoperihedron + 1x truncated icosahedron  
127  truncated icosahedron prism  x3x5o2x  12x pentagonal prism + 20x hexagonal prism + 2x truncated icosahedron  
128  cubic canticupola  xx4xx3ox  12x trigonal prism + 8x trigonal cupola + 6x octagonal prism + 1x truncated cube + 1x stauropantohedron  
129  cubic semimesocupola  xo4xx3oo  8x trigonal prism + 1x stauromesohedron + 6x square cupola + 1x truncated cube  
130  truncated dodecahedron prism  x5x3o2x  20x trigonal prism + 12x decagonal prism + 2x truncated dodecahedron  
131  rhodohedral mesocupola  xo5ox3xo  20x octahedron + 30x square pyramid + 12x pentagonal antiprism + 1x rhodomesohedron + 1x rhodoperihedron  
132  K4.132  ??  (5+5+5)x octahedron + (5+5+5+5+10)x square pyramid + (1+5+5)x pentagonal antiprism + 1x rhodomesohedron + 1x diminished rhodoperihedron + 1x pentagonal cupola  
133  pentagonal gyrobicupolic ring  xxo5oxx&#x  10x square pyramid + 1x pentagonal antiprism + 2x pentagonal cupola  
134  K4.134  ??  10x octahedon + (10+10+10)x square pyramid + 10x pentagonal antiprism + 1x rhodomesohedron + 2x pentagoal cupola + 1x parabidiminished rhodoperihedron  
135  K4.135  ??  (2+2+2+4)x octahedron + (1+1+2+2+4+4+4+4+4+4)x square pyramid + (2+2+2+4)x pentagonal antiprism + 1x rhodomesohedron + 2x pentagonal cupola + 1x metabidiminished rhodoperihedron  
136  K4.136  ??  (1+1+3)x octahedron + (3+3+3+3+6+6+6)x square pyramid + (3+3+3)x pentagonal antiprism + 1x rhodomesohedron + 3x pentagonal cupola + 1x tridiminished rhodoperihedron  
137  K4.137  xo3ox5oo  20x octahedron + 1x icosahedron + 12x pentagonal pyramid + 1x rhodomesohedron  
138  K4.138  ??  (5+5+5)x octahedron + 5x square pyramid + 1x gyroelongated pentagonal pyramid + (1+5+5)x pentagonal pyramid + 1x pentagonal prism + 1x rhodomesohedron  
139  K4.139  ??  5x octahedron + 5x square pyramid + (1+1+5)x pentagonal pyramid + 1x pentagonal rotnda +1x pentagonal cupola  
140  K4.140  ??  (5+5)x octahedron + 5x square pyramid + 1x gyroelongated pentagonal pyramid + (1+5)x pentagonal pyramid + 1x pentagonal rotunda + 1x pentagonal cupola  
141  pentagonal prism pyramid  xox5ooo&#x  5x square pyramid + 2x pentagonal pyramid + 1x pentagonal prism  
142  K4.142  ??  10x octahedron + 10x square pyramid + 10x pentagonal pyramid + 1x pentagonal antiprism + 2x pentagonal prism + 1x rhodomesohedron  
143  K4.143  ??  (2+2+2+4)x octahedron +(2+4+4)x square pyramid + (2+2+2+4)x pentagonal pyramid + 1x metabidiminished icosahedron + 1x pentagonal prism + 1x rhodomesohedron  
144  K4.144  ??  5x octahedron + (5+5)x square pyramid + 5x pentagonal pyramid +1x pentagonal antiprism + 1x pentagonal prism + 1x pentagonal rotunda + 1x pentagonal cupola  
145  K4.145  ??  (1+1+2+2)x octahedron + (2+2+2+2)x square pyramid + (1+2+2)x pentagonal pyramid + 1x metabidiminished icosahedron + 1x pentagonal prism + 1x pentagonal rotunda + 1x pentagonal cupola  
146  K4.146  ??  (5+5)x square pyramid + 5x pentagonal pyramid + 1x pentagonal prism + 1x rotunda +1x pentagonal cupola  
147  K4.147  ??  (1+1+3)x octahedron + (3+6+6)x square pyramid + (3+3+3)x pentagonal pyramid + 1x tridiminished icosahedron + 3x pentagonal prism + 1x rhodomesohedron  
148  K4.148  ??  (1+1)x octahedron + (1+2+2+2+2+2)x square pyramid + (1+1+2)x pentagonal pyramid + 1x tridiminished icosahedron + 2x pentagonal prism + 1x pentagonal rotunda + 1x pentagonal cupola  
149  octahedral canticupola  xx3xx4ox  12x trigonal prism + 8x hexagonal prism + 6x square cupola + 1x truncated octahedron + 1x stauropantohedron  
150  rhodopantohedral prism  x5x3x2x  30x cube + 20x hexagonal prism + 12x decagonal prism + 2x rhodopantohedron  
151  rhodohedral semiantiprism  xo5xx3ox  30x tetrahedron + 20x trigonal cupola + 12x pentagonal cupola + 1x truncated icosahedron + 1x truncated dodecahedron  
152  dodecahedral pericupola  xx5oo3ox  20x tetrahedron + 30x trigonal prism + 12x pentagonal prism + 1x dodecahedron + 1x rhodoperihedron  
153  K4.153  ??  (5+5+5)x tetrahedron + (5+5+5+10)x trigonal prism + (1+5+5+5)x pentagonal prism + 1x dodecahedron + 1x diminished rhodoperihedron + 1x pentagonal cupola  
154  pentagonal orthobicupolic ring  xx5xo2ox  5x tetrahedron + 5x trigonal prism + 1x pentagonal prism + 2x pentagonal cupola  
155  K4.155  ??  10x tetrahedron + (10+10)x trigonal prism + 10x pentagonal prism + 1x dodecahedron + 2x pentagonal cupola + 1x parabidiminished rhodoperihedron  
156  K4.156  ??  (2+2+2+4)x tetrahedron + (1+1+2+4+4+4+4)x trigonal prism + (2+2+2+4)x pentagonal prism + 1x dodecahedron + 2x pentagonal cupola + 1x metabidiminished rhodoperihedron  
157  K4.157  ??  (1+1+3)x tetrahedron + (3+3+3+6)x trigonal prism + (3+3+3)x pentagonal prism + 1x dodecahedron + 3x pentagonal cupola + 1x tridiminished rhodoperihedron  
158  icosahedral semimesocupola  ox3xx5oo  12x pentagonal prism + 20x trigonal cupola + 1x rhodomesohedron + 1x truncated icosahedron  
159  dodecahedral semipericupola  xx5xo3ox  20x octahedron + 30x trigonal prism + 1x rhodoperihedron + 12x pentagonal cupola + 1x truncated dodecahedron  
160  K4.160  ??  (5+5+5)x octahedron + (5+5)x square pyramid + (5+5+5+5+10)x trigonal prism + 1x gyrated rhodoperihedron + (1+1+5+5)x pentagonal cupola + 1x truncated dodecahedron  
161  K4.161  ??  10x octahedron + (10+10)x square pyramid + (10+10+10)x trigonal prism + 1x parabigyrated rhodoperihedron + (2+10)x pentagonal cupola + 1x truncated dodecahedron  
162  K4.162  ??  (2+2+2+4)x octahedron + (2+2+4+4+4+4)x square pyramid + (1+1+2+2+4+4+4+4+4+4)x trigonal prism + 1x metabigyrated rhodoperihedron + (2+2+2+2+4)x pentagonal cupola + 1x truncated dodecahedron  
163  K4.163  ??  (1+1+3)x octahedron + (3+3+6+6+6+6)x square pyramid + (3+3+3+3+6+6+6)x trigonal prism + 1x trigyrated rhodoperihedron + (3+3+3+3)x pentagonal cupola + 1x truncated dodecahedron  
164  K4.164  ??  (5+5+5)x octahedron + 5x square pyramid + (5+5+5+10)x trigonal prism + 1x decagonal prism + 1x diminished rhodoperihedron + (1+5+5)x pentagonal cupola + 1x truncated dodecahedron  
165  pentagonal magnabicupolic ring  xx5xo2ox  5x square pyramid + 5x trigonal prism + 1x decagonal prism + 2x pentagonal cupola  
166  K4.166  ??  (5+5)x octahedron + (5+5+5)x square pyramid + (5+5+5+10)x trigonal prism + (1+1+5)x pentagonal cupola + 1x gyrated paradiminished rhodoperihedron + 1x truncated dodecahedron  
167  K4.167  ??  (1+1+2+2+2+2)x octahedron + (1+1+1+2+2+2+2+2+2)x square pyramid + (1+1+1+2+2+2+2+2+2+2+2+2+2+2)x trigonal prism + (1+1+1+2+2+2+2)x pentagonal cupola + 1x gyrated metadiminished rhodoperihedron + 1x truncated dodecahedron  
168  K4.168  ??  (1+1+1+2)x octahedron + (1+2+2+2+2+2+2+2+2+2+2+2+2)x square pyramid + (1+1+1+2+2+2+2+2+2+2+2+2+2+2)x trigonal prism + (1+1+1+2+2+2+2)x pentagonal cupola + 1x bigyrated diminished rhodoperihedron + 1x truncated dodecahedron  
169  K4.169  ??  10x octahedron + 10x square pyramid + (10+10)x trigonal prism + 2x decagonal prism + 10x pentagonal cupola + 1x parabidiminished rhodoperihedron + 1x truncated dodecahedron  
170  K4.170  ??  (2+2+2+4)x octahedron + (2+4+4)x square pyramid + (1+1+2+4+4+4+4)x trigonal prism + 2x decagonal prism + (2+2+2+4)x pentagonal cupola + 1x metabidiminished rhodoperihedron + 1x truncated dodecahedron  
171  K4.171  ??  (1+1+1+2)x octahedron + (1+1+2+2+2+2+2+2+2+2+2)x square pyramid + (1+1+1+2+2+2+2+2+2+2+2+2+2+2)x trigonal prism + 2x decagonal prism + (1+1+1+1+2+2+2)x pentagonal cupola + 1x gyrated bidiminished rhodoperihedron + 1x truncated dodecahedron  
172  K4.172  ??  (1+1+3)x octahedron + (3+3+6)x square pyramid + (3+3+3+6)x trigonal prism + 3x decagonal prism + (3+3+3)x pentagonal cupola + 1x tridiminished rhodoperihedron + 1x truncated dodecahedron  
173  dodecahedral canticupola  xx5xx3ox  30x trigonal prism + 20x trigonal cupola + 12x decagonal prism + 1x truncated dodecahedron + 1x rhodopantohedron  
174  biantiprismatic rings  xoxnoxo&#x  nx tetrahedron + nx square pyramid + 2x nantiprism + 1x nprism  
175  3,nduoprisms  xno2x3o  nx trigonal prism + 3x ngonal prism  
176  nantiprism prisms  ??  2nx trigonal prism + 2x ngonal antiprism + 2x ngonal prism  
177  4,nduoprisms  xno2x4o  nx cube + 4x ngonal prism 