# Segmentotope (EntityClass, 11)

(Difference between revisions)
 Revision as of 17:35, 12 March 2016 (view source)Hayate (Talk | contribs) (→Full list of segmentochora: nomenclature)← Older edit Latest revision as of 11:04, 15 March 2016 (view source)Hayate (Talk | contribs) (→Classification: Assign mnemonics to each family) (12 intermediate revisions not shown) Line 127: Line 127: The current classification is as follows, where each number is a K 4.X index: The current classification is as follows, where each number is a K 4.X index: - ;3 regular polychora + ;Ur - 3 regular polychora :1, 2, 20 :1, 2, 20 - ;the 3-pyrotomochoron, a uniform polychoron + ;Ux - the pyrorectichoron, a uniform polychoron :5 :5 - ;21 members of infinite families + ;Up - all 17 prisms of the uniform polyhedra (not 18, because the cube prism is the K 4.20, already counted above) - :6, 10, 14, 18, 19, 22, 34, 39, 42, 46, 47, 53, 54, 58, 59, 65, 70, 93, 94, 96, 97 + - ;4 infinite families (not individual polychora) + - :174, 175, 176, 177 + - ;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 :9, 11, 36, 43, 57, 60, 66, 74, 89, 90, 99, 110, 111, 125, 127, 130, 150 - ;12 pyramids of CRF polyhedra + ;Inf - 4 infinite families (not individual polychora) and 21 members of those infinite families + {|class='wikitable' + |- + !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 :3, 4, 7, 17, 26, 80, 84, 85, 86, 87, 88, 141 - ;25 prisms of [[orbiform]] [[Johnson solids]] + ;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 :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 - ;30 cupolic forms + ;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 :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 - ;all 10 [[bicupolic ring]]s (excluding the prisms of the tetrahedron and square pyramid, already counted above) + ;Bicu - all 10 [[bicupolic ring]]s (excluding the prisms of the tetrahedron and square pyramid, already counted above) :8, 25, 27, 51, 64, 73, 105, 133, 154, 165 :8, 25, 27, 51, 64, 73, 105, 133, 154, 165 - ;3 fragments of cupolic forms (excluding the bicupolic rings) + ;Xcu - 3 fragments of cupolic forms (excluding the bicupolic rings) :109, 139, 146 :109, 139, 146 - ;the [[gyrated octahedral prism]] + ;Xg - the [[gyrated octahedral prism]] :13 :13 - ;the trigonal tridiminished-icosahedral wedge + ;Xt - the trigonal tridiminished-icosahedral wedge :33 :33 - ;49 gyrations and diminishes of the cupolic forms + ;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 :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 == == Full list of segmentochora == + *''See also: [[Segmentochoron construction table]]'' + {|class="wikitable" {|class="wikitable" !# !# Line 170: Line 181: |4||[[square dipyramid]]||xo4oo2ox||4x tetrahedron + 2x [[square pyramid]]||quarter of aerochoron |4||[[square dipyramid]]||xo4oo2ox||4x tetrahedron + 2x [[square pyramid]]||quarter of aerochoron |- |- - |5||[[3-pyrotomochoron]]||o3x3o3o||5x tetrahedron + 5x octahedron||tetrahedral semicupola + |5||[[pyrorectichoron]]||o3x3o3o||5x tetrahedron + 5x octahedron||tetrahedral semicupola |- |- - |6||[[K4.6]]||xox3oxo&#x||3x tetrahedron + 2x octahedron + 3x square pyramid + 1x [[trigonal prism]]|| + |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|| |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 3-pyrotomochoron + |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|| |9||[[tetrahedral prism]]||x3o3o2x||2x tetrahedron + 4x trigonal prism|| Line 188: Line 199: |13||[[gyrated octahedral prism]]||??||4x square pyramid + (4+4)x trigonal prism|| |13||[[gyrated octahedral prism]]||??||4x square pyramid + (4+4)x trigonal prism|| |- |- - |14||[[K4.14]]||xox4oxo&#x||4x tetrahedron + 4x square pyramid + 2x [[square antiprism]] + 1x cube|| + |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|| |15||[[staurohedral antiprism]]||xo4oo3ox||(8+12)x tetrahedron + 1x octahedron + 6x square pyramid + 1x cube|| Line 206: Line 217: |22||[[K4.22]]||xox5oxo&#x||5x tetrahedron + 5x square pyramid + 2x [[pentagonal antiprism]] + 1x [[pentagonal prism]]|| |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 [[cuboctahedron]]|| + |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]]|| |24||[[K4.24]]||??||2x tetrahedron + 6x trigonal prism + 2x [[trigonal cupola]]|| Line 216: Line 227: |27||[[triangular gyrobicupolic ring]]||xxo3xox&#x||1x octahedron + 6x square pyramid + 2x trigonal cupola|| |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 cuboctahedron|| + |28||[[K4.28]]||??||(4+8)x square pyramid + 2x cube + 1x stauromesohedron|| |- |- - |29||[[octahedral semicupola]]||xo3ox4oo||(1+8)x octahedron + 6x square pyramid + 1x cuboctahedron|| + |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|| |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 cuboctahedron|| + |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|| |32||[[K4.32]]||??||1x octahedron + (2+2+4)x square pyramid + 1x cube + 2x trigonal cupola|| Line 230: Line 241: |34||[[5,3-duoprism]]||x5o2x3o||5x trigonal prism + 3x pentagonal prism|| |34||[[5,3-duoprism]]||x5o2x3o||5x trigonal prism + 3x pentagonal prism|| |- |- - |35||[[cubic semicupola]]||xo4ox3oo||8x tetrahedron + 6x square antiprism + 1x cube + 1x cuboctahedron|| + |35||[[cubic semicupola]]||xo4ox3oo||8x tetrahedron + 6x square antiprism + 1x cube + 1x stauromesohedron|| |- |- |36||[[icosahedral prism]]||x3o5o2x||20x trigonal prism + 2x icosahedron|| |36||[[icosahedral prism]]||x3o5o2x||20x trigonal prism + 2x icosahedron|| Line 246: Line 257: |42||[[pentagonal diprism]]||x5o2x4o||5x cube + 4x pentagonal prism|| |42||[[pentagonal diprism]]||x5o2x4o||5x cube + 4x pentagonal prism|| |- |- - |43||[[cuboctahedron prism]]||o4x3o2x||8x trigonal prism + 6x cube + 2x cuboctahedron|| + |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]]|| |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 cuboctahedron prism + |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|| |46||[[K4.46]]||xox6oxo&#x||6x tetrahedron + 6x square pyramid + 2x [[hexagonal antiprism]] + 1x hexagonal prism|| Line 256: Line 267: |47||[[6,3-duoprism]]||x6o2x3o||6x trigonal prism + 3x hexagonal prism|| |47||[[6,3-duoprism]]||x6o2x3o||6x trigonal prism + 3x hexagonal prism|| |- |- - |48||[[tetrahedral semipericupola]]||xx3xo3ox||4x octahedron + 6x trigonal prism + 1x cuboctahedron + 4x trigonal cupola + 1x [[truncated tetrahedron]]|| + |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|| |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|| Line 282: Line 293: |60||[[snub cube prism]]||s4s3s2x||(8+24)x trigonal prism + 6x cube + 2x [[snub cube]]|| |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 cuboctahedron + 1x [[stauroperihedron]]|| + |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 cuboctahedron + 1x [[elongated square cupola]] + 1 [[square cupola]]|| + |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 cuboctahedron + 2x square cupola + 1x octagonal prism|| + |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|| |64||[[square gyrobicupolic ring]]||xxo4oxx&#x||8x square pyramid + 1x square antiprism + 2x square cupola|| Line 314: Line 325: |76||[[tetrahedral canticupola]]||xx3xx3xo||6x trigonal prism + 4x trigonal cupola + 4x hexagonal prism + 1 truncated tetrahedron + 1 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 [[icosidodecahedron]]|| + |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|| |78||[[rhodohedral antiprism]]||xo5oo3ox||(20+30)x tetrahedron + 1x icosahedron + 12x pentagonal pyramid + 1x dodecahedron|| Line 340: Line 351: |89||[[truncated octahedron prism]]||x3x4o2x||6x cube + 8x hexagonal prism + 2x truncated octahedron|| |89||[[truncated octahedron prism]]||x3x4o2x||6x cube + 8x hexagonal prism + 2x truncated octahedron|| |- |- - |90||[[icosidodecahedron prism]]||??||20x trigonal prism + 12x pentagonal prism + 2x icosidodecahedron|| + |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]]|| |91||[[orthobirotunda prism]]||??||(10+10)x trigonal prism + (2+10)x pentagonal prism + 2x [[orthobirotunda]]|| Line 350: Line 361: |94||[[10,3-duoprism]]||x10o2x3o||10x trigonal prism + 3x decagonal prism|| |94||[[10,3-duoprism]]||x10o2x3o||10x trigonal prism + 3x decagonal prism|| |- |- - |95||[[octahedral semimesocupola]]||xo3xx4oo||6x cube + 1x cuboctahedron + 8x trigonal cupola + 1x truncated octahedron|| + |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|| |96||[[decagonal antiprism prism]]||??||20x trigonal prism + 2x decagonal antiprism + 2x decagonal prism|| Line 410: Line 421: |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]]|| |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||[[truncated cuboctahedron prism]]||x4x3x2x||12x cube + 8x hexagonal prism + 6x octagonal prism + 2x [[truncated cuboctahedron]]|| + |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]]|| |126||[[icosahedral semipericupola]]||xx3xo5ox||30x trigonal prism + 12x pentagonal antiprism + 20x trigonal cupola + 1x rhodoperihedron + 1x [[truncated icosahedron]]|| Line 416: Line 427: |127||[[truncated icosahedron prism]]||x3x5o2x||12x pentagonal prism + 20x hexagonal prism + 2x 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 truncated cuboctahedron|| + |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 cuboctahedron + 6x square cupola + 1x truncated cube|| + |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]]|| |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 icosidodecahedron + 1x rhodoperihedron|| + |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 icosidodecahedron + 1x diminished rhodoperihedron + 1x pentagonal cupola|| + |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|| |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 icosidodecahedron + 2x pentagoal cupola + 1x parabidiminished rhodoperihedron|| + |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 icosidodecahedron + 2x pentagonal cupola + 1x metabidiminished 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 icosidodecahedron + 3x pentagonal cupola + 1x tridiminished 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 icosidodecahedron|| + |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 icosidodecahedron|| + |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|| |139||[[K4.139]]||??||5x octahedron + 5x square pyramid + (1+1+5)x pentagonal pyramid + 1x pentagonal rotnda +1x pentagonal cupola|| Line 444: Line 455: |141||[[pentagonal prism pyramid]]||xox5ooo&#x||5x square pyramid + 2x pentagonal pyramid + 1x pentagonal prism|| |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 icosidodecahedron|| + |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 icosidodecahedron|| + |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|| |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|| Line 454: Line 465: |146||[[K4.146]]||??||(5+5)x square pyramid + 5x pentagonal pyramid + 1x pentagonal prism + 1x 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 icosidodecahedron|| + |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|| |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]]||??||12x trigonal prism + 8x hexagonal prism + 6x square cupola + 1x truncated octahedron + 1x truncated cuboctahedron|| + |149||[[octahedral canticupola]]||xx3xx4ox||12x trigonal prism + 8x hexagonal prism + 6x square cupola + 1x truncated octahedron + 1x stauropantohedron|| |- |- - |150||[[rhodopantohedral prism]]||??||30x cube + 20x hexagonal prism + 12x decagonal prism + 2x [[rhodopantohedron]]|| + |150||[[rhodopantohedral prism]]||x5x3x2x||30x cube + 20x hexagonal prism + 12x decagonal prism + 2x [[rhodopantohedron]]|| |- |- - |151||[[rhodohedral semiantiprism]]||??||30x tetrahedron + 20x trigonal cupola + 12x pentagonal cupola + 1x truncated icosahedron + 1x truncated dodecahedron|| + |151||[[rhodohedral semiantiprism]]||xo5xx3ox||30x tetrahedron + 20x trigonal cupola + 12x pentagonal cupola + 1x truncated icosahedron + 1x truncated dodecahedron|| |- |- - |152||[[dodecahedral pericupola]]||??||20x tetrahedron + 30x trigonal prism + 12x pentagonal prism + 1x dodecahedron + 1x rhodoperihedron|| + |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|| |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]]||??||5x tetrahedron + 5x trigonal prism + 1x pentagonal prism + 2x 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|| |155||[[K4.155]]||??||10x tetrahedron + (10+10)x trigonal prism + 10x pentagonal prism + 1x dodecahedron + 2x pentagonal cupola + 1x parabidiminished rhodoperihedron|| Line 476: Line 487: |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|| |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]]||??||12x pentagonal prism + 20x trigonal cupola + 1x icosidodecahedron + 1x truncated icosahedron|| + |158||[[icosahedral semimesocupola]]||ox3xx5oo||12x pentagonal prism + 20x trigonal cupola + 1x rhodomesohedron + 1x truncated icosahedron|| |- |- - |159||[[dodecahedral semipericupola]]||??||20x octahedron + 30x trigonal prism + 1x rhodoperihedron + 12x pentagonal cupola + 1x truncated dodecahedron|| + |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|| |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|| Line 490: Line 501: |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|| |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]]||??||5x square pyramid + 5x trigonal prism + 1x decagonal prism + 2x pentagonal cupola|| + |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|| |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|| Line 506: Line 517: |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|| |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]]||??||30x trigonal prism + 20x trigonal cupola + 12x decagonal prism + 1x truncated dodecahedron + 1x rhodopantohedron|| + |173||[[dodecahedral canticupola]]||xx5xx3ox||30x trigonal prism + 20x trigonal cupola + 12x decagonal prism + 1x truncated dodecahedron + 1x rhodopantohedron|| |- |- - |174||[[K4.174]]||??||''n''x tetrahedron + ''n''x square pyramid + 2x ''n''-antiprism + 1x ''n''-prism|| + |174||[[biantiprismatic ring]]s||xox''n''oxo&#x||''n''x tetrahedron + ''n''x square pyramid + 2x ''n''-antiprism + 1x ''n''-prism|| |- |- - |175||[[K4.175]]||??||''n''x trigonal prism + 3x ''n''-gonal prism|| + |175||3,''n''-[[duoprism]]s||x''n''o2x3o||''n''x trigonal prism + 3x ''n''-gonal prism|| |- |- - |176||[[K4.176]]||??||2''n''x trigonal prism + 2x ''n''-gonal antiprism + 2x ''n''-gonal prism|| + |176||''n''-[[antiprism]] prisms||??||2''n''x trigonal prism + 2x ''n''-gonal antiprism + 2x ''n''-gonal prism|| |- |- - |177||[[K4.177]]||??||''n''x cube + 4x ''n''-gonal prism|| + |177||4,''n''-duoprisms||x''n''o2x4o||''n''x cube + 4x ''n''-gonal prism|| |} |}

## Latest revision as of 11:04, 15 March 2016

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 four-dimensional 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 lower-dimensional polytopes. For segmentochora, A and B are usually polyhedra, although one of them can be lower-dimensional, 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)CellsValues

K 4.7

• Height of triangular prism pyramid: E*sqrt(5/12)
• Dichoral angle between tetrahedron and triangular prism: atan(sqrt(5/3)) ≈ 52.24°
• Dichoral angle between square pyramid and triangular prism: atan(sqrt(5)) ≈ 65.91°

K 4.26

• Dichoral angle between square pyramid and cube: 45° (exact)
• Dichoral angle between adjacent square pyramids: 180° - acos(1/3) ≈ 109.47°

K 4.45

triangular_cupola || triangular_cupola

• Distance between hexagonal prism and antipodal triangular prism: E*sqrt(2/3). (Same as the height of a triangular cupola.)

K 4.51

• Distance from hexagonal prism to triangle: E*sqrt(5/12)
• Dichoral angle between trigonal cupola and hexagonal prism: atan(sqrt(5/3)) ≈ 52.23°
• Dichoral angle between triangular prism and hexagonal prism: acos(2/3) ≈ 48.19°
• Dichoral angle between square pyramid and hexagonal prism: atan(sqrt(5)) ≈ 65.91°

K 4.69

square_cupola || square_cupola

• Distance between octagonal prism and antipodal cube: E*sqrt(2)/2. (Same as height of square cupola.)

K 4.105

• Distance from octagonal prism to square: E*sqrt(2)/2
• Dichoral angle between square cupola and octagonal prism: 45° (exact)
• Dichoral angle between square pyramid and octagonal prism: 45° (exact)
• Dichoral angle between triangular prism and octagonal prism: acos(sqrt(2/3)) ≈ 35.26°

K 4.117

pentagonal_cupola || pentagonal_cupola

• Distance between pentagonal prism and decagonal prism: E*(sqrt(2*sqrt(2*(3*sqrt(5)+7)) - (12*sqrt(5)+20)/5)/2). (Same as height of pentagonal cupola.)

K 4.141

• Distance between pentagonal prism and antipodal point: (E/2)*sqrt((5-2*sqrt(5))/5)
• Dichoral angle between pentagonal pyramid and pentagonal prism: 18° (exact)
• Dichoral angle between square pyramid and pentagonal prism: atan(sqrt(5)-2) ≈ 13.28°

K 4.165

• Distance between decagonal prism and pentagon: (E/2)*sqrt((5-2*sqrt(5))/5)
• Dichoral angle between pentagonal cupola and decagonal prism: 18° (exact)
• Dichoral angle between square pyramid and decagonal prism: atan(sqrt(9-4*sqrt(5))) ≈ 13.28°
• Dichoral angle between triangular prism and decagonal prism: asin(sqrt((5-2*sqrt(5))/15)) ≈ 10.81°

## 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}
17461422465893
17510(18)34475994
176(11)1939536596
17718(20)42547097
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 tridiminished-icosahedral 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

1pyrochoronx3o3o3o5x tetrahedron
2aerochoronx3o3o4o16x tetrahedron
3octahedral pyramidxo3oo4oo8x tetrahedron + 1x octahedronhalf of aerochoron
4square dipyramidxo4oo2ox4x tetrahedron + 2x square pyramidquarter of aerochoron
5pyrorectichorono3x3o3o5x tetrahedron + 5x octahedrontetrahedral semicupola
6trigonal biantiprismatic ringxox3oxo&#x3x tetrahedron + 2x octahedron + 3x square pyramid + 1x trigonal prism(mono)diminished pyrorectichoron
7triangular prism pyramidxo3oo2xo2x tetrahedron + 3x square pyramid + 1x trigonal prism
8digonal gyrobicupolic ringxxo2xox&#x1x tetrahedron + 4x square pyramid + 2x trigonal prismbidiminished pyrorectichoron
9tetrahedral prismx3o3o2x2x tetrahedron + 4x trigonal prism
10duotrianglinderx3o2x3o6x trigonal prism
11octahedral prismx3o4o2x2x octahedron + 8x trigonal prism
12square pyramid prismxx2oo4xo2x square pyramid + 4x trigonal prism + 1x cube
13gyrated octahedral prism??4x square pyramid + (4+4)x trigonal prism
14square biantiprismatic ringxox4oxo&#x4x tetrahedron + 4x square pyramid + 2x square antiprism + 1x cube
15staurohedral antiprismxo4oo3ox(8+12)x tetrahedron + 1x octahedron + 6x square pyramid + 1x cube
16K4.16??(4+4+4)x tetrahedron + (1+1+4)x square pyramid + 1x square antiprism + 1x cube
17square antiprism pyramidoxo4oox&#x8x tetrahedron + 2x square pyramid + 1x square antiprism
18triangular diprismx3o2x4o4x trigonal prism + 3x cube
19square antiprism prism??8x trigonal prism + 2x square antiprism + 2x cube
20geochoronx4o3o3o8x cube
21snubdis antiprismxo4os3os8x tetrahedron + 12x square pyramid + 6x trigonal prism + 1x cube + 1x icosahedronsnubdis antiprism
22K4.22xox5oxo&#x5x tetrahedron + 5x square pyramid + 2x pentagonal antiprism + 1x pentagonal prism
23tetrahedral pericupolaxx3oo3ox(1+4)x tetrahedron + (4+6)x trigonal prism + 1x stauromesohedron
24K4.24??2x tetrahedron + 6x trigonal prism + 2x trigonal cupola
25triangular orthobicupolic ringxx3xo2ox3x tetrahedron + (1+3)x trigonal prism + 2x trigonal cupola
26cubic pyramidxo4oo3oo6x square pyramid + 1x cube
27triangular gyrobicupolic ringxxo3xox&#x1x octahedron + 6x square pyramid + 2x trigonal cupola
28K4.28??(4+8)x square pyramid + 2x cube + 1x stauromesohedron
29octahedral semicupolaxo3ox4oo(1+8)x octahedron + 6x square pyramid + 1x stauromesohedron
30K4.30??(2+3)x octahedron + 6x square pyramid + 2x trigonal cupola
31K4.31??4x octahedron + (1+1+4+4)x square pyramid + 1x cube + 1x stauromesohedron
32K4.32??1x octahedron + (2+2+4)x square pyramid + 1x cube + 2x trigonal cupola
33K4.33??3x tetrahedron + 1x octahedron + 3x square pyramid + 1x trigonal prism + 3x pentagonal pyramid + 1x tridiminished icosahedrontrigonal tridiminished-icosahedral wedge
345,3-duoprismx5o2x3o5x trigonal prism + 3x pentagonal prism
35cubic semicupolaxo4ox3oo8x tetrahedron + 6x square antiprism + 1x cube + 1x stauromesohedron
36icosahedral prismx3o5o2x20x trigonal prism + 2x icosahedron
37gyroelongated pentagonal pyramid prism??(5+5+5)x trigonal prism + 1x pentagonal prism + 2x gyroelongated pentagonal pyramid
38pentagonal pyramid prismxx2oo5xo5x trigonal prism + 2x pentagonal pyramid + 1x pentagonal prism
39pentagonal antiprism prism??10x trigonal prism + 2x pentagonal antiprism + 2x pentagonal prism
40metabidiminished icosahedron prism??(2+2+2+4)x trigonal prism + 2x pentagonal prism + 2x metabidiminished icosahedron
41tridiminished icosahedron prismxxx2xfo3oox(1+1+3)x trigonal prism + 3x pentagonal prism + 2x tridiminished icosahedron
42pentagonal diprismx5o2x4o5x cube + 4x pentagonal prism
43stauromesohedron prismo4x3o2x8x trigonal prism + 6x cube + 2x stauromesohedron
44trigonal orthobicupola prismxxx3oxo2xxx(2+6)x trigonal prism + 6x cube + 2x trigonal orthobicupola
45trigonal cupola prismxx3xo2xx(1+3)x trigonal prism + 3x cube + 2x trigonal cupola + 1x hexagonal prismhalf of stauromesohedron prism
46K4.46xox6oxo&#x6x tetrahedron + 6x square pyramid + 2x hexagonal antiprism + 1x hexagonal prism
476,3-duoprismx6o2x3o6x trigonal prism + 3x hexagonal prism
48tetrahedral semipericupolaxx3xo3ox4x octahedron + 6x trigonal prism + 1x stauromesohedron + 4x trigonal cupola + 1x truncated tetrahedron
49K4.49??1x octahedron + (3+3)x square pyramid + (3+3)x trigonal prism + 1x trigonal orthobicupola + (1+3)x trigonal cupola + 1x truncated tetrahedron
50K4.50??1x octahedron + 3x square pyramid + 3x trigonal prism + (1+3)x trigonal cupola + 1x hexagonal prism + 1x truncated tetrahedron
51triangular magnabicupolic ringxx3xo2xo3x square pyramid + 3x trigonal prism + 2x trigonal cupola + 1x hexagonal prism
52tetrahedral semimesocupolaxo3xx3oo1x octahedron + 4x trigonal prism + 4x trigonal cupola + 1x truncated tetrahedron
53hexagonal antiprism prism??12x trigonal prism + 2x hexagonal antiprism + 2x hexagonal prism
54hexagonal diprismx6o2x4o6x cube + 4x hexagonal prism
55tetrahedral semiantiprismxo3xx3ox6x tetrahedron + 8x trigonal cupola + 2x truncated tetrahedron
56tetrahedral quasicupolaxx3xo3oo(1+4)x tetrahedron + 4x trigonal cupola + 1x truncated tetrahedron
57truncated tetrahedron prismx3x3o2x4x trigonal prism + 4x hexagonal prism + 2x truncated tetrahedron
58K4.58xox8oxo&#x8x tetrahedron + 8x square pyramid + 2x octagonal antiprism + 1x octagonal prism
598,3-duoprismx8o2x3o8x trigonal prism + 3x octagonal prism
60snub cube prisms4s3s2x(8+24)x trigonal prism + 6x cube + 2x snub cube
61staurohedral mesocupolaxo4ox3xo8x octahedron + 12x square pyramid + 6x square antiprism + 1x stauromesohedron + 1x stauroperihedron
62K4.62??4x octahedron + (4+4+4)x square pyramid + (1+4)x square antiprism + 1x stauromesohedron + 1x elongated square cupola + 1 square cupola
63K4.63??(4+8)x square pyramid + 4x square antiprism + 1x stauromesohedron + 2x square cupola + 1x octagonal prism
64square gyrobicupolic ringxxo4oxx&#x8x square pyramid + 1x square antiprism + 2x square cupola
65octagonal antiprism prism??16x trigonal prism + 2x octagonal antiprism + 2x octagonal prism
66stauroperihedral prismx4o3x2x8x trigonal prism + (6+12)x cube + 2x stauroperihedron
67elongated square gyrobicupola prism??8x trigonal prism + (2+8+8)x cube + 2x elongated square gyrobicupola
68elongated square cupola prism??4x trigonal prism + (1+4+4+4)x cube + 2x elongated square cupola + 1x octagonal prism
69square cupola prismxx4xo2xx4x trigonal prism + (1+4)x cube + 2x square cupola + 1x octagonal prism
70octagonal diprismx8o2x4o8x cube + 4x octagonal prism
71cubic pericupolaxx4oo3ox8x tetrahedron + 12x trigonal prism + (1+6)x cube + 1x stauroperihedron
72K4.72??4x tetrahedron + (4+4)x trigonal prism + (1+1+4)x cube + 1x elongated square cupola + 1 square cupola
73square orthobicupolic ringxx4xo2ox4x tetrahedron + 2x square pyramid + 4x trigonal prism + 1x cube
74dodecahedron prismx5o3o2x12x pentagonal prism + 2x dodecahedron
75octahedral semipericupolaxx3xo4ox12x trigonal prism + 6x square antiprism + 8x trigonal cupola + 1x stauroperihedron + 1x truncated octahedron
76tetrahedral canticupolaxx3xx3xo6x trigonal prism + 4x trigonal cupola + 4x hexagonal prism + 1 truncated tetrahedron + 1 truncated octahedron
77dodecahedral semicupolaxo5ox3oo20x tetrahedron + 12x pentagonal antiprism + 1x dodecahedron + 1x rhodomesohedron
78rhodohedral antiprismxo5oo3ox(20+30)x tetrahedron + 1x icosahedron + 12x pentagonal pyramid + 1x dodecahedron
79K4.79??(5+5+5+5+5+5+10)x tetrahedron + 1x gyroelongated pentagonal pyramid + (1+5+5)x pentagonal pyramid + 1x pentagonal antiprism + 1x dodecahedron
80pentagonal antiprism pyramid??10x tetrahedron + 2x pentagonal pyramid + 1x pentagonal antiprism
81K4.81??(10+10+10)x tetrahedron + 10x pentagonal pyramid + (1+2)x pentagonal antiprism + 1x dodecahedron
82K4.82??(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
83K4.83??(1+1+3+3+3+3+6)x tetrahedron + (3+3+3)x pentagonal pyramid + 3x pentagonal antiprism + 1x tridiminished icosahedron + 1x dodecahedron
84icosahedron pyramidxo3oo5oo20x tetrahedron + 1x icosahedron
85gyroelongated pentagonal pyramid pyramid??(5+5+5)x tetrahedron + 1x gyroelongated pentagonal pyramid + 1x pentagonal pyramid
86pentagonal dipyramidxo5oo2ox5x tetrahedron + 2x pentagonal pyramid
87metabidiminished icosahedron pyramid??(2+2+2+4)x tetrahedron + 2x pentagonal pyramid + 1x metabidiminished icosahedron
88tridiminished icosahedron pyramid??(1+1+3)x tetrahedron + 3x pentagonal pyramid + 1x tridiminished icosahedron
89truncated octahedron prismx3x4o2x6x cube + 8x hexagonal prism + 2x truncated octahedron
90rhodomesohedron prism??20x trigonal prism + 12x pentagonal prism + 2x rhodomesohedron
91orthobirotunda prism??(10+10)x trigonal prism + (2+10)x pentagonal prism + 2x orthobirotunda
92rotunda prism??(5+5)x trigonal prism + (1+5)x pentagonal prism + 2x rotunda + 1x decagonal prism
93K4.93xox10oxo&#x10x tetrahedron + 10x square pyramid + 2x decagonal antiprism + 1x decagonal prism
9410,3-duoprismx10o2x3o10x trigonal prism + 3x decagonal prism
95octahedral semimesocupolaxo3xx4oo6x cube + 1x stauromesohedron + 8x trigonal cupola + 1x truncated octahedron
96decagonal antiprism prism??20x trigonal prism + 2x decagonal antiprism + 2x decagonal prism
97decagonal diprismx10o2x4o10x cube + 4x decagonal prism
98staurohedral semiantiprismxo4xx3ox12x tetrahedron + 8x trigonal cupola + 6x square cupola + 1x truncated octahedron + 1x truncated cube
99truncated cube prismx4x3o2x8x trigonal prism + 6x octagonal prism + 2x truncated cube
100cubic semipericupolaxx4xo3ox8x octahedron + 12x trigonal prism + 1x stauroperihedron + 6x square cupola + 1x truncated cube
101K4.101??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
102K4.102??(8+8)x square pyramid + (4+8)x trigonal prism + 1x stauroperihedron + (2+4)x square cupola + 1x truncated cube
103K4.103??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
104K4.104??(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
105square magnabicupolic ringxx4xo2xo4x square pyramid + 4x trigonal prism + 2x square cupola + 1x octagonal prism
106K4.106??8x square pyramid + 4x trigonal prism + 4x square cupola + (1+2)x octagonal prism + 1x truncated cube
107octahedral pericupolaxx3oo4ox1x octahedron + 6x square pyramid + (8+12)x trigonal prism + 1x stauroperihedron
108K4.108??(1+1+4)x square pyramid + (4+4+4)x trigonal prism + 1x elongated square cupola + 1x square cupola
109K4.109??8x square pyramid + 2x trigonal prism + 2x square cupola
110snub dodecahedron prisms5s3s2x(20+60)x trigonal prism + 12x pentagonal prism + 2x snub dodecahedron
111rhodoperihedron prismx5o3x2x20x trigonal prism + 30x cube + 12x pentagonal prism + 2x rhodoperihedron
112gyrated 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
113parabigyrated rhodoperihedron prism??(10+10)x trigonal prism + (10+10+10)x cube + (2+10)x pentagonal prism + 2x parabigyrated rhodoperihedron
114metabigyrated 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
115trigyrated 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
116diminished 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
117pentagonal cupola prismxx5xo2xx5x trigonal prism + 5x cube + 1x pentagonal prism + 1x decagonal prism + 2x pentagonal cupola
118gyrated 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
119gyrated 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
120bigyrated 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
121parabidiminished rhodoperihedron prism??10x trigonal prism + (10+10)x cube + 10x pentagonal prism + 2x decagonal prism + 2x parabidiminished rhodoperihedron
122metabidiminished 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
123gyrated 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
124tridiminished 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
125stauropantohedron prismx4x3x2x12x cube + 8x hexagonal prism + 6x octagonal prism + 2x stauropantohedron
126icosahedral semipericupolaxx3xo5ox30x trigonal prism + 12x pentagonal antiprism + 20x trigonal cupola + 1x rhodoperihedron + 1x truncated icosahedron
127truncated icosahedron prismx3x5o2x12x pentagonal prism + 20x hexagonal prism + 2x truncated icosahedron
128cubic canticupolaxx4xx3ox12x trigonal prism + 8x trigonal cupola + 6x octagonal prism + 1x truncated cube + 1x stauropantohedron
129cubic semimesocupolaxo4xx3oo8x trigonal prism + 1x stauromesohedron + 6x square cupola + 1x truncated cube
130truncated dodecahedron prismx5x3o2x20x trigonal prism + 12x decagonal prism + 2x truncated dodecahedron
131rhodohedral mesocupolaxo5ox3xo20x octahedron + 30x square pyramid + 12x pentagonal antiprism + 1x rhodomesohedron + 1x rhodoperihedron
132K4.132??(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
133pentagonal gyrobicupolic ringxxo5oxx&#x10x square pyramid + 1x pentagonal antiprism + 2x pentagonal cupola
134K4.134??10x octahedon + (10+10+10)x square pyramid + 10x pentagonal antiprism + 1x rhodomesohedron + 2x pentagoal cupola + 1x parabidiminished rhodoperihedron
135K4.135??(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
136K4.136??(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
137K4.137xo3ox5oo20x octahedron + 1x icosahedron + 12x pentagonal pyramid + 1x rhodomesohedron
138K4.138??(5+5+5)x octahedron + 5x square pyramid + 1x gyroelongated pentagonal pyramid + (1+5+5)x pentagonal pyramid + 1x pentagonal prism + 1x rhodomesohedron
139K4.139??5x octahedron + 5x square pyramid + (1+1+5)x pentagonal pyramid + 1x pentagonal rotnda +1x pentagonal cupola
140K4.140??(5+5)x octahedron + 5x square pyramid + 1x gyroelongated pentagonal pyramid + (1+5)x pentagonal pyramid + 1x pentagonal rotunda + 1x pentagonal cupola
141pentagonal prism pyramidxox5ooo&#x5x square pyramid + 2x pentagonal pyramid + 1x pentagonal prism
142K4.142??10x octahedron + 10x square pyramid + 10x pentagonal pyramid + 1x pentagonal antiprism + 2x pentagonal prism + 1x rhodomesohedron
143K4.143??(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
144K4.144??5x octahedron + (5+5)x square pyramid + 5x pentagonal pyramid +1x pentagonal antiprism + 1x pentagonal prism + 1x pentagonal rotunda + 1x pentagonal cupola
145K4.145??(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
146K4.146??(5+5)x square pyramid + 5x pentagonal pyramid + 1x pentagonal prism + 1x rotunda +1x pentagonal cupola
147K4.147??(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
148K4.148??(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
149octahedral canticupolaxx3xx4ox12x trigonal prism + 8x hexagonal prism + 6x square cupola + 1x truncated octahedron + 1x stauropantohedron
150rhodopantohedral prismx5x3x2x30x cube + 20x hexagonal prism + 12x decagonal prism + 2x rhodopantohedron
151rhodohedral semiantiprismxo5xx3ox30x tetrahedron + 20x trigonal cupola + 12x pentagonal cupola + 1x truncated icosahedron + 1x truncated dodecahedron
152dodecahedral pericupolaxx5oo3ox20x tetrahedron + 30x trigonal prism + 12x pentagonal prism + 1x dodecahedron + 1x rhodoperihedron
153K4.153??(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
154pentagonal orthobicupolic ringxx5xo2ox5x tetrahedron + 5x trigonal prism + 1x pentagonal prism + 2x pentagonal cupola
155K4.155??10x tetrahedron + (10+10)x trigonal prism + 10x pentagonal prism + 1x dodecahedron + 2x pentagonal cupola + 1x parabidiminished rhodoperihedron
156K4.156??(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
157K4.157??(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
158icosahedral semimesocupolaox3xx5oo12x pentagonal prism + 20x trigonal cupola + 1x rhodomesohedron + 1x truncated icosahedron
159dodecahedral semipericupolaxx5xo3ox20x octahedron + 30x trigonal prism + 1x rhodoperihedron + 12x pentagonal cupola + 1x truncated dodecahedron
160K4.160??(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
161K4.161??10x octahedron + (10+10)x square pyramid + (10+10+10)x trigonal prism + 1x parabigyrated rhodoperihedron + (2+10)x pentagonal cupola + 1x truncated dodecahedron
162K4.162??(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
163K4.163??(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
164K4.164??(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
165pentagonal magnabicupolic ringxx5xo2ox5x square pyramid + 5x trigonal prism + 1x decagonal prism + 2x pentagonal cupola
166K4.166??(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
167K4.167??(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
168K4.168??(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
169K4.169??10x octahedron + 10x square pyramid + (10+10)x trigonal prism + 2x decagonal prism + 10x pentagonal cupola + 1x parabidiminished rhodoperihedron + 1x truncated dodecahedron
170K4.170??(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
171K4.171??(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
172K4.172??(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
173dodecahedral canticupolaxx5xx3ox30x trigonal prism + 20x trigonal cupola + 12x decagonal prism + 1x truncated dodecahedron + 1x rhodopantohedron
174biantiprismatic ringsxoxnoxo&#xnx tetrahedron + nx square pyramid + 2x n-antiprism + 1x n-prism
1753,n-duoprismsxno2x3onx trigonal prism + 3x n-gonal prism
176n-antiprism prisms??2nx trigonal prism + 2x n-gonal antiprism + 2x n-gonal prism
1774,n-duoprismsxno2x4onx cube + 4x n-gonal prism

## Pages in this category (5)

 Square dipyramid