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Uniform polyhedron compound
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<p>In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, a uniform polyhedron compound is a <a href="/facts/Polyhedral_compound/L13pItNN">polyhedral compound</a> whose constituents are identical (although possibly <a href="/facts/Chirality/hjx0eGrB">enantiomorphous</a>) <a href="/facts/Uniform_polyhedron/PHnprqRC">uniform polyhedra</a>, in an arrangement that is also uniform, i.e. the <a href="/facts/Symmetry_group/CNqeseMp">symmetry group</a> of the compound acts <a href="/facts/Orbit_(group_theory)/Vh9VtUUw">transitively</a> on the compound's <a href="/facts/Vertex_(geometry)/ID3cea9z">vertices</a>. </p><p>The uniform polyhedron compounds were first enumerated by John Skilling in 1976, with a proof that the enumeration is complete. The following table lists them according to his numbering. </p><p>The prismatic compounds of {<i>p</i>/<i>q</i>}-gonal <a href="/facts/Prism_(geometry)/baIRRkCy">prisms</a> (<a href="/facts/Prismatic_compound_of_prisms_with_rotational_freedom/UrHF9cPz">UC20</a> and <a href="/facts/Prismatic_compound_of_prisms/049MhzQA">UC21</a>) exist only when <i>p</i>/<i>q</i> > 2, and when p and q are <a href="/facts/Coprime_integers/bnCxCXxs">coprime</a>. The uniform prismatic compounds of {<i>p</i>/<i>q</i>}-gonal <a href="/facts/Antiprism/FgP5jIVd">antiprisms</a> (<a href="/facts/Prismatic_compound_of_antiprisms_with_rotational_freedom/3BkR9f8R">UC22</a>, <a href="/facts/Prismatic_compound_of_antiprisms/dVrCqK0n">UC23</a>, <a href="/facts/Prismatic_compound_of_antiprisms_with_rotational_freedom/3BkR9f8R">UC24</a> and <a href="/facts/Prismatic_compound_of_antiprisms/dVrCqK0n">UC25</a>) exist only when <i>p</i>/<i>q</i> > 3/2, and when p and q are coprime. Furthermore, when <i>p</i>/<i>q</i> = 2, the antiprisms <a href="/facts/Degeneracy_(mathematics)/BqhqlXPH">degenerate</a> into <a href="/facts/Tetrahedra/7mkSrl6j">tetrahedra</a> with <a href="/facts/Digon/vur5IQ5d">digonal</a> bases. </p> <table><tbody><tr><th>Compound</th><th>Bowersacronym</th><th>Picture</th><th>Polyhedralcount</th><th><a href="/facts/Polyhedron/UC0p9FTF">Polyhedral type</a></th><th>Faces</th><th>Edges</th><th>Vertices</th><th>Notes</th><th><a href="/facts/Symmetry_group/CNqeseMp">Symmetry group</a></th><th><a href="/facts/Subgroup/r91mBgpa">Subgroup</a>restrictingto oneconstituent</th></tr><tr><td><a href="/facts/Compound_of_six_tetrahedra_with_rotational_freedom/S8em3umU">UC01</a></td><td>sis</td><td></td><td>6</td><td><a href="/facts/Tetrahedron/7mkSrl6j">tetrahedra</a></td><td>24{3}</td><td>36</td><td>24</td><td>Rotational freedom</td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>d</a></td><td><a href="/facts/Cyclic_symmetries/wQUCezbQ"><i>S</i>4</a></td></tr><tr><td><a href="/facts/Compound_of_twelve_tetrahedra_with_rotational_freedom/5H1H6R4C">UC02</a></td><td>dis</td><td></td><td>12</td><td><a href="/facts/Tetrahedron/7mkSrl6j">tetrahedra</a></td><td>48{3}</td><td>72</td><td>48</td><td>Rotational freedom</td><td><a href="/facts/Octahedral_symmetry/qLqhYxgm"><i>O</i>h</a></td><td><a href="/facts/Cyclic_symmetries/wQUCezbQ"><i>S</i>4</a></td></tr><tr><td><a href="/facts/Compound_of_six_tetrahedra/U6oRV7De">UC03</a></td><td>snu</td><td></td><td>6</td><td><a href="/facts/Tetrahedron/7mkSrl6j">tetrahedra</a></td><td>24{3}</td><td>36</td><td>24</td><td></td><td><a href="/facts/Octahedral_symmetry/qLqhYxgm"><i>O</i>h</a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i>2d</a></td></tr><tr><td><a href="/facts/Compound_of_two_tetrahedra/0j6l294C">UC04</a></td><td>so</td><td></td><td>2</td><td><a href="/facts/Tetrahedron/7mkSrl6j">tetrahedra</a></td><td>8{3}</td><td>12</td><td>8</td><td>Regular</td><td><a href="/facts/Octahedral_symmetry/qLqhYxgm"><i>O</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>d</a></td></tr><tr><td><a href="/facts/Compound_of_five_tetrahedra/KKFAnLxh">UC05</a></td><td>ki</td><td></td><td>5</td><td><a href="/facts/Tetrahedron/7mkSrl6j">tetrahedra</a></td><td>20{3}</td><td>30</td><td>20</td><td>Regular</td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i></a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i></a></td></tr><tr><td><a href="/facts/Compound_of_ten_tetrahedra/z9LgGAPL">UC06</a></td><td>e</td><td></td><td>10</td><td><a href="/facts/Tetrahedron/7mkSrl6j">tetrahedra</a></td><td>40{3}</td><td>60</td><td>20</td><td>Regular<p>2 polyhedra per vertex</p></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i></a></td></tr><tr><td><a href="/facts/Compound_of_six_cubes_with_rotational_freedom/zu6HeqxT">UC07</a></td><td>risdoh</td><td></td><td>6</td><td><a href="/facts/Cube/65lUaAqN">cubes</a></td><td>(12+24){4}</td><td>72</td><td>48</td><td>Rotational freedom</td><td><a href="/facts/Octahedral_symmetry/qLqhYxgm"><i>O</i>h</a></td><td><a href="/facts/Cyclic_symmetries/wQUCezbQ"><i>C</i>4h</a></td></tr><tr><td><a href="/facts/Compound_of_three_cubes/zpUzDzBV">UC08</a></td><td>rah</td><td></td><td>3</td><td><a href="/facts/Cube/65lUaAqN">cubes</a></td><td>(6+12){4}</td><td>36</td><td>24</td><td></td><td><a href="/facts/Octahedral_symmetry/qLqhYxgm"><i>O</i>h</a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i>4h</a></td></tr><tr><td><a href="/facts/Compound_of_five_cubes/PNIAfKTN">UC09</a></td><td>rhom</td><td></td><td>5</td><td><a href="/facts/Cube/65lUaAqN">cubes</a></td><td>30{4}</td><td>60</td><td>20</td><td>Regular<p>2 polyhedra per vertex</p></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>h</a></td></tr><tr><td>UC10</td><td>dissit</td><td></td><td>4</td><td><a href="/facts/Octahedron/G38q92xW">octahedra</a></td><td>(8+24){3}</td><td>48</td><td>24</td><td>Rotational freedom</td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>h</a></td><td><a href="/facts/Cyclic_symmetries/wQUCezbQ"><i>S</i>6</a></td></tr><tr><td>UC11</td><td>daso</td><td></td><td>8</td><td><a href="/facts/Octahedron/G38q92xW">octahedra</a></td><td>(16+48){3}</td><td>96</td><td>48</td><td>Rotational freedom</td><td><a href="/facts/Octahedral_symmetry/qLqhYxgm"><i>O</i>h</a></td><td><a href="/facts/Cyclic_symmetries/wQUCezbQ"><i>S</i>6</a></td></tr><tr><td><a href="/facts/Compound_of_four_octahedra/VsMRBAKo">UC12</a></td><td>sno</td><td></td><td>4</td><td><a href="/facts/Octahedron/G38q92xW">octahedra</a></td><td>(8+24){3}</td><td>48</td><td>24</td><td></td><td><a href="/facts/Octahedral_symmetry/qLqhYxgm"><i>O</i>h</a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i>3d</a></td></tr><tr><td><a href="/facts/Compound_of_twenty_octahedra_with_rotational_freedom/rPenzSfK">UC13</a></td><td>addasi</td><td></td><td>20</td><td><a href="/facts/Octahedron/G38q92xW">octahedra</a></td><td>(40+120){3}</td><td>240</td><td>120</td><td>Rotational freedom</td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Cyclic_symmetries/wQUCezbQ"><i>S</i>6</a></td></tr><tr><td><a href="/facts/Compound_of_twenty_octahedra/dNCua5vk">UC14</a></td><td>dasi</td><td></td><td>20</td><td><a href="/facts/Octahedron/G38q92xW">octahedra</a></td><td>(40+120){3}</td><td>240</td><td>60</td><td>2 polyhedra per vertex</td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Cyclic_symmetries/wQUCezbQ"><i>S</i>6</a></td></tr><tr><td><a href="/facts/Compounds_of_ten_octahedra/7JzN0h4r">UC15</a></td><td>gissi</td><td></td><td>10</td><td><a href="/facts/Octahedron/G38q92xW">octahedra</a></td><td>(20+60){3}</td><td>120</td><td>60</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i>3d</a></td></tr><tr><td><a href="/facts/Compounds_of_ten_octahedra/7JzN0h4r">UC16</a></td><td>si</td><td></td><td>10</td><td><a href="/facts/Octahedron/G38q92xW">octahedra</a></td><td>(20+60){3}</td><td>120</td><td>60</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i>3d</a></td></tr><tr><td><a href="/facts/Compound_of_five_octahedra/8M825cd5">UC17</a></td><td>se</td><td></td><td>5</td><td><a href="/facts/Octahedron/G38q92xW">octahedra</a></td><td>40{3}</td><td>60</td><td>30</td><td>Regular</td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>h</a></td></tr><tr><td><a href="/facts/Compound_of_five_tetrahemihexahedra/eIsR4qF0">UC18</a></td><td>hirki</td><td></td><td>5</td><td><a href="/facts/Tetrahemihexahedron/cloC6TBc">tetrahemihexahedra</a></td><td>20{3}<p>15{4}</p></td><td>60</td><td>30</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i></a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i></a></td></tr><tr><td><a href="/facts/Compound_of_twenty_tetrahemihexahedra/pkrKdnW5">UC19</a></td><td>sapisseri</td><td></td><td>20</td><td><a href="/facts/Tetrahemihexahedron/cloC6TBc">tetrahemihexahedra</a></td><td>(20+60){3}<p>60{4}</p></td><td>240</td><td>60</td><td>2 polyhedra per vertex</td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i></a></td><td><a href="/facts/Cyclic_symmetries/wQUCezbQ"><i>C</i>3</a></td></tr><tr><td><a href="/facts/Prismatic_compound_of_prisms_with_rotational_freedom/UrHF9cPz">UC20</a></td><td>-</td><td></td><td>2<i>n</i><p>(2<i>n ≥</i> 2)</p></td><td><i>p</i>/<i>q</i>-gonal <a href="/facts/Prismatic_uniform_polyhedron/MBJwRJxB">prisms</a></td><td>4<i>n</i>{<i>p</i>/<i>q</i>}<p>2<i>np</i>{4}</p></td><td>6<i>np</i></td><td>4<i>np</i></td><td>Rotational freedom</td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i><i>np</i>h</a></td><td><a href="/facts/Cyclic_symmetries/wQUCezbQ"><i>C</i><i>p</i>h</a></td></tr><tr><td><a href="/facts/Prismatic_compound_of_prisms/049MhzQA">UC21</a></td><td>-</td><td></td><td><i>n</i><p>(<i>n ≥</i> 2)</p></td><td><i>p</i>/<i>q</i>-gonal <a href="/facts/Prismatic_uniform_polyhedron/MBJwRJxB">prisms</a></td><td>2<i>n</i>{<i>p</i>/<i>q</i>}<p><i>np</i>{4}</p></td><td>3<i>np</i></td><td>2<i>np</i></td><td></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i><i>np</i>h</a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i><i>p</i>h</a></td></tr><tr><td><a href="/facts/Prismatic_compound_of_antiprisms_with_rotational_freedom/3BkR9f8R">UC22</a></td><td>-</td><td></td><td>2<i>n</i><p>(2<i>n</i> ≥ 2)</p><p>(<i>q</i> odd)</p></td><td><i>p</i>/<i>q</i>-gonal <a href="/facts/Prismatic_uniform_polyhedron/MBJwRJxB">antiprisms</a><p>(<i>q</i> odd)</p></td><td>4<i>n</i>{<i>p</i>/<i>q</i>} (if <i>p</i>/<i>q</i> ≠ 2)<p>4<i>np</i>{3}</p></td><td>8<i>np</i></td><td>4<i>np</i></td><td>Rotational freedom</td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i><i>np</i>d</a> (if <i>n</i> odd)<p><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i><i>np</i>h</a> (if <i>n</i> even)</p></td><td><a href="/facts/Cyclic_symmetries/wQUCezbQ"><i>S</i>2<i>p</i></a></td></tr><tr><td><a href="/facts/Prismatic_compound_of_antiprisms/dVrCqK0n">UC23</a></td><td>-</td><td></td><td><i>n</i><p>(<i>n ≥</i> 2)</p></td><td><i>p</i>/<i>q</i>-gonal <a href="/facts/Prismatic_uniform_polyhedron/MBJwRJxB">antiprisms</a><p>(<i>q</i> odd)</p></td><td>2<i>n</i>{<i>p</i>/<i>q</i>} (if <i>p</i>/<i>q</i> ≠ 2)<p>2<i>np</i>{3}</p></td><td>4<i>np</i></td><td>2<i>np</i></td><td></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i><i>np</i>d</a> (if <i>n</i> odd)<p><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i><i>np</i>h</a> (if <i>n</i> even)</p></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i><i>p</i>d</a></td></tr><tr><td><a href="/facts/Prismatic_compound_of_antiprisms_with_rotational_freedom/3BkR9f8R">UC24</a></td><td>-</td><td></td><td>2<i>n</i><p>(2<i>n ≥</i> 2)</p></td><td><i>p</i>/<i>q</i>-gonal <a href="/facts/Prismatic_uniform_polyhedron/MBJwRJxB">antiprisms</a><p>(<i>q</i> even)</p></td><td>4<i>n</i>{<i>p</i>/<i>q</i>} (if <i>p</i>/<i>q</i> ≠ 2)<p>4<i>np</i>{3}</p></td><td>8<i>np</i></td><td>4<i>np</i></td><td>Rotational freedom</td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i><i>np</i>h</a></td><td><a href="/facts/Cyclic_symmetries/wQUCezbQ"><i>C</i><i>p</i>h</a></td></tr><tr><td><a href="/facts/Prismatic_compound_of_antiprisms/dVrCqK0n">UC25</a></td><td>-</td><td></td><td><i>n</i><p>(<i>n ≥</i> 2)</p></td><td><i>p</i>/<i>q</i>-gonal <a href="/facts/Prismatic_uniform_polyhedron/MBJwRJxB">antiprisms</a><p>(<i>q</i> even)</p></td><td>2<i>n</i>{<i>p</i>/<i>q</i>} (if <i>p</i>/<i>q</i> ≠ 2)<p>2<i>np</i>{3}</p></td><td>4<i>np</i></td><td>2<i>np</i></td><td></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i><i>np</i>h</a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i><i>p</i>h</a></td></tr><tr><td><a href="/facts/Compound_of_twelve_pentagonal_antiprisms_with_rotational_freedom/dXc0CC2f">UC26</a></td><td>gadsid</td><td></td><td>12</td><td><a href="/facts/Pentagonal_antiprism/gR8aMBKN">pentagonal antiprisms</a></td><td>120{3}<p>24{5}</p></td><td>240</td><td>120</td><td>Rotational freedom</td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Cyclic_symmetries/wQUCezbQ"><i>S</i>10</a></td></tr><tr><td>UC27</td><td>gassid</td><td></td><td>6</td><td><a href="/facts/Pentagonal_antiprism/gR8aMBKN">pentagonal antiprisms</a></td><td>60{3}<p>12{5}</p></td><td>120</td><td>60</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i>5d</a></td></tr><tr><td>UC28</td><td>gidasid</td><td></td><td>12</td><td><a href="/facts/Pentagrammic_crossed_antiprism/qGztCKXJ">pentagrammic crossed antiprisms</a></td><td>120{3}<p>24{5/2}</p></td><td>240</td><td>120</td><td>Rotational freedom</td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Cyclic_symmetries/wQUCezbQ"><i>S</i>10</a></td></tr><tr><td>UC29</td><td>gissed</td><td></td><td>6</td><td><a href="/facts/Pentagrammic_crossed_antiprism/qGztCKXJ">pentagrammic crossed antiprisms</a></td><td>60{3}<p>125</p></td><td>120</td><td>60</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i>5d</a></td></tr><tr><td><a href="/facts/Compound_of_four_triangular_prisms/N1KTcayt">UC30</a></td><td>ro</td><td></td><td>4</td><td><a href="/facts/Triangular_prism/Yjp4Kw9M">triangular prisms</a></td><td>8{3}<p>12{4}</p></td><td>36</td><td>24</td><td></td><td><a href="/facts/Octahedral_symmetry/qLqhYxgm"><i>O</i></a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i>3</a></td></tr><tr><td><a href="/facts/Compound_of_eight_triangular_prisms/65GA4W1q">UC31</a></td><td>dro</td><td></td><td>8</td><td><a href="/facts/Triangular_prism/Yjp4Kw9M">triangular prisms</a></td><td>16{3}<p>24{4}</p></td><td>72</td><td>48</td><td></td><td><a href="/facts/Octahedral_symmetry/qLqhYxgm"><i>O</i>h</a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i>3</a></td></tr><tr><td><a href="/facts/Compound_of_ten_triangular_prisms/erVQUyd9">UC32</a></td><td>kri</td><td></td><td>10</td><td><a href="/facts/Triangular_prism/Yjp4Kw9M">triangular prisms</a></td><td>20{3}<p>30{4}</p></td><td>90</td><td>60</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i></a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i>3</a></td></tr><tr><td><a href="/facts/Compound_of_twenty_triangular_prisms/aLTuprLa">UC33</a></td><td>dri</td><td></td><td>20</td><td><a href="/facts/Triangular_prism/Yjp4Kw9M">triangular prisms</a></td><td>40{3}<p>60{4}</p></td><td>180</td><td>60</td><td>2 polyhedra per vertex</td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i>3</a></td></tr><tr><td><a href="/facts/Compound_of_six_pentagonal_prisms/C2GWLFfg">UC34</a></td><td>kred</td><td></td><td>6</td><td><a href="/facts/Pentagonal_prism/CtvKLxcy">pentagonal prisms</a></td><td>30{4}<p>12{5}</p></td><td>90</td><td>60</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i></a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i>5</a></td></tr><tr><td><a href="/facts/Compound_of_twelve_pentagonal_prisms/7datQNa1">UC35</a></td><td>dird</td><td></td><td>12</td><td><a href="/facts/Pentagonal_prism/CtvKLxcy">pentagonal prisms</a></td><td>60{4}<p>24{5}</p></td><td>180</td><td>60</td><td>2 polyhedra per vertex</td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i>5</a></td></tr><tr><td><a href="/facts/Compound_of_six_pentagrammic_prisms/pMh7vyGW">UC36</a></td><td>gikrid</td><td></td><td>6</td><td><a href="/facts/Pentagrammic_prism/3ofu2hVH">pentagrammic prisms</a></td><td>30{4}<p>12{5/2}</p></td><td>90</td><td>60</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i></a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i>5</a></td></tr><tr><td><a href="/facts/Compound_of_twelve_pentagrammic_prisms/LvvCVwbM">UC37</a></td><td>giddird</td><td></td><td>12</td><td><a href="/facts/Pentagrammic_prism/3ofu2hVH">pentagrammic prisms</a></td><td>60{4}<p>24{5/2}</p></td><td>180</td><td>60</td><td>2 polyhedra per vertex</td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i>5</a></td></tr><tr><td><a href="/facts/Compound_of_four_hexagonal_prisms/GbHfqVWY">UC38</a></td><td>griso</td><td></td><td>4</td><td><a href="/facts/Hexagonal_prism/oxYc0nF2">hexagonal prisms</a></td><td>24{4}<p>8{6}</p></td><td>72</td><td>48</td><td></td><td><a href="/facts/Octahedral_symmetry/qLqhYxgm"><i>O</i>h</a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i>3d</a></td></tr><tr><td><a href="/facts/Compound_of_ten_hexagonal_prisms/qCzRyxtJ">UC39</a></td><td>rosi</td><td></td><td>10</td><td><a href="/facts/Hexagonal_prism/oxYc0nF2">hexagonal prisms</a></td><td>60{4}<p>20{6}</p></td><td>180</td><td>120</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i>3d</a></td></tr><tr><td><a href="/facts/Compound_of_six_decagonal_prisms/nqx2neCv">UC40</a></td><td>rassid</td><td></td><td>6</td><td><a href="/facts/Decagonal_prism/dIjtR9zW">decagonal prisms</a></td><td>60{4}<p>12{10}</p></td><td>180</td><td>120</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i>5d</a></td></tr><tr><td><a href="/facts/Compound_of_six_decagrammic_prisms/S3cJY2lj">UC41</a></td><td>grassid</td><td></td><td>6</td><td><a href="/facts/Decagrammic_prism/cpacuabl">decagrammic prisms</a></td><td>60{4}<p>12{10/3}</p></td><td>180</td><td>120</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i>5d</a></td></tr><tr><td>UC42</td><td>gassic</td><td></td><td>3</td><td><a href="/facts/Square_antiprism/4514Q02s">square antiprisms</a></td><td>24{3}<p>6{4}</p></td><td>48</td><td>24</td><td></td><td><a href="/facts/Octahedral_symmetry/qLqhYxgm"><i>O</i></a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i>4</a></td></tr><tr><td>UC43</td><td>gidsac</td><td></td><td>6</td><td><a href="/facts/Square_antiprism/4514Q02s">square antiprisms</a></td><td>48{3}<p>12{4}</p></td><td>96</td><td>48</td><td></td><td><a href="/facts/Octahedral_symmetry/qLqhYxgm"><i>O</i>h</a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i>4</a></td></tr><tr><td>UC44</td><td>sassid</td><td></td><td>6</td><td><a href="/facts/Pentagrammic_antiprism/QRPJLAH3">pentagrammic antiprisms</a></td><td>60{3}<p>12{5/2}</p></td><td>120</td><td>60</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i></a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i>5</a></td></tr><tr><td>UC45</td><td>sadsid</td><td></td><td>12</td><td><a href="/facts/Pentagrammic_antiprism/QRPJLAH3">pentagrammic antiprisms</a></td><td>120{3}<p>24{5/2}</p></td><td>240</td><td>120</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF"><i>D</i>5</a></td></tr><tr><td><a href="/facts/Compound_of_two_icosahedra/p8z6CwDw">UC46</a></td><td>siddo</td><td></td><td>2</td><td><a href="/facts/Icosahedron/wWbpnDYp">icosahedra</a></td><td>(16+24){3}</td><td>60</td><td>24</td><td></td><td><a href="/facts/Octahedral_symmetry/qLqhYxgm"><i>O</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>h</a></td></tr><tr><td><a href="/facts/Compound_of_five_icosahedra/lrE83GC5">UC47</a></td><td>sne</td><td></td><td>5</td><td><a href="/facts/Icosahedron/wWbpnDYp">icosahedra</a></td><td>(40+60){3}</td><td>150</td><td>60</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>h</a></td></tr><tr><td>UC48</td><td>presipsido</td><td></td><td>2</td><td><a href="/facts/Great_dodecahedron/LWt3Hgg4">great dodecahedra</a></td><td>24{5}</td><td>60</td><td>24</td><td></td><td><a href="/facts/Octahedral_symmetry/qLqhYxgm"><i>O</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>h</a></td></tr><tr><td><a href="/facts/Compound_of_five_great_dodecahedra/Y3ghjEKq">UC49</a></td><td>presipsi</td><td></td><td>5</td><td><a href="/facts/Great_dodecahedron/LWt3Hgg4">great dodecahedra</a></td><td>60{5}</td><td>150</td><td>60</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>h</a></td></tr><tr><td><a href="/facts/Compound_of_two_small_stellated_dodecahedra/osAbTJvL">UC50</a></td><td>passipsido</td><td></td><td>2</td><td><a href="/facts/Small_stellated_dodecahedron/lX1zlo6C">small stellated dodecahedra</a></td><td>24{5/2}</td><td>60</td><td>24</td><td></td><td><a href="/facts/Octahedral_symmetry/qLqhYxgm"><i>O</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>h</a></td></tr><tr><td><a href="/facts/Compound_of_five_small_stellated_dodecahedra/BsjjjhLm">UC51</a></td><td>passipsi</td><td></td><td>5</td><td><a href="/facts/Small_stellated_dodecahedron/lX1zlo6C">small stellated dodecahedra</a></td><td>60{5/2}</td><td>150</td><td>60</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>h</a></td></tr><tr><td><a href="/facts/Compound_of_two_great_icosahedra/lhTzBMrQ">UC52</a></td><td>sirsido</td><td></td><td>2</td><td><a href="/facts/Great_icosahedron/jaAk3a5Q">great icosahedra</a></td><td>(16+24){3}</td><td>60</td><td>24</td><td></td><td><a href="/facts/Octahedral_symmetry/qLqhYxgm"><i>O</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>h</a></td></tr><tr><td><a href="/facts/Compound_of_five_great_icosahedra/qdNqMo25">UC53</a></td><td>sirsei</td><td></td><td>5</td><td><a href="/facts/Great_icosahedron/jaAk3a5Q">great icosahedra</a></td><td>(40+60){3}</td><td>150</td><td>60</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>h</a></td></tr><tr><td><a href="/facts/Compound_of_two_truncated_tetrahedra/LXbF8F6s">UC54</a></td><td>tisso</td><td></td><td>2</td><td><a href="/facts/Truncated_tetrahedron/K9ItvmUK">truncated tetrahedra</a></td><td>8{3}<p>8{6}</p></td><td>36</td><td>24</td><td></td><td><a href="/facts/Octahedral_symmetry/qLqhYxgm"><i>O</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>d</a></td></tr><tr><td><a href="/facts/Compound_of_five_truncated_tetrahedra/PlMQP1Ic">UC55</a></td><td>taki</td><td></td><td>5</td><td><a href="/facts/Truncated_tetrahedron/K9ItvmUK">truncated tetrahedra</a></td><td>20{3}<p>20{6}</p></td><td>90</td><td>60</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i></a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i></a></td></tr><tr><td><a href="/facts/Compound_of_ten_truncated_tetrahedra/kc1T2kEv">UC56</a></td><td>te</td><td></td><td>10</td><td><a href="/facts/Truncated_tetrahedron/K9ItvmUK">truncated tetrahedra</a></td><td>40{3}<p>40{6}</p></td><td>180</td><td>120</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i></a></td></tr><tr><td><a href="/facts/Compound_of_five_truncated_cubes/FBQmJFxF">UC57</a></td><td>tar</td><td></td><td>5</td><td><a href="/facts/Truncated_cube/XVAUG4fD">truncated cubes</a></td><td>40{3}<p>30{8}</p></td><td>180</td><td>120</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>h</a></td></tr><tr><td><a href="/facts/Compound_of_five_stellated_truncated_hexahedra/UGnwReGt">UC58</a></td><td>quitar</td><td></td><td>5</td><td><a href="/facts/Stellated_truncated_hexahedron/7AEekj9t">stellated truncated hexahedra</a></td><td>40{3}<p>30{8/3}</p></td><td>180</td><td>120</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>h</a></td></tr><tr><td><a href="/facts/Compound_of_five_cuboctahedra/7v4fv8qS">UC59</a></td><td>arie</td><td></td><td>5</td><td><a href="/facts/Cuboctahedron/hD7p0Mem">cuboctahedra</a></td><td>40{3}<p>30{4}</p></td><td>120</td><td>60</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>h</a></td></tr><tr><td><a href="/facts/Compound_of_five_cubohemioctahedra/M46nN99R">UC60</a></td><td>gari</td><td></td><td>5</td><td><a href="/facts/Cubohemioctahedron/em6nfePH">cubohemioctahedra</a></td><td>30{4}<p>20{6}</p></td><td>120</td><td>60</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>h</a></td></tr><tr><td><a href="/facts/Compound_of_five_octahemioctahedra/gerC1hJj">UC61</a></td><td>iddei</td><td></td><td>5</td><td><a href="/facts/Octahemioctahedron/wj6b88G3">octahemioctahedra</a></td><td>40{3}<p>20{6}</p></td><td>120</td><td>60</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>h</a></td></tr><tr><td><a href="/facts/Compound_of_five_rhombicuboctahedra/5Tde9XaS">UC62</a></td><td>rasseri</td><td></td><td>5</td><td><a href="/facts/Rhombicuboctahedron/TmxUmEnl">rhombicuboctahedra</a></td><td>40{3}<p>(30+60){4}</p></td><td>240</td><td>120</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>h</a></td></tr><tr><td><a href="/facts/Compound_of_five_small_rhombihexahedra/4hkH6yd3">UC63</a></td><td>rasher</td><td></td><td>5</td><td><a href="/facts/Small_rhombihexahedron/MfGRyzu1">small rhombihexahedra</a></td><td>60{4}<p>30{8}</p></td><td>240</td><td>120</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>h</a></td></tr><tr><td><a href="/facts/Compound_of_five_small_cubicuboctahedra/lkntLCjE">UC64</a></td><td>rahrie</td><td></td><td>5</td><td><a href="/facts/Small_cubicuboctahedron/319k6z43">small cubicuboctahedra</a></td><td>40{3}<p>30{4}</p><p>30{8}</p></td><td>240</td><td>120</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>h</a></td></tr><tr><td><a href="/facts/Compound_of_five_great_cubicuboctahedra/IWPefApY">UC65</a></td><td>raquahri</td><td></td><td>5</td><td><a href="/facts/Great_cubicuboctahedron/xRh9NtND">great cubicuboctahedra</a></td><td>40{3}<p>30{4}</p><p>30{8/3}</p></td><td>240</td><td>120</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>h</a></td></tr><tr><td><a href="/facts/Compound_of_five_great_rhombihexahedra/abBBYfS7">UC66</a></td><td>rasquahr</td><td></td><td>5</td><td><a href="/facts/Great_rhombihexahedron/PDA51lpE">great rhombihexahedra</a></td><td>60{4}<p>30{8/3}</p></td><td>240</td><td>120</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>h</a></td></tr><tr><td><a href="/facts/Compound_of_five_nonconvex_great_rhombicuboctahedra/5Y2hgVkF">UC67</a></td><td>rosaqri</td><td></td><td>5</td><td><a href="/facts/Nonconvex_great_rhombicuboctahedron/XSwFaVVL">nonconvex great rhombicuboctahedra</a></td><td>40{3}<p>(30+60){4}</p></td><td>240</td><td>120</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu"><i>T</i>h</a></td></tr><tr><td><a href="/facts/Compound_of_two_snub_cubes/AIPyqWLz">UC68</a></td><td>disco</td><td></td><td>2</td><td><a href="/facts/Snub_cube/pbnr40gk">snub cubes</a></td><td>(16+48){3}<p>12{4}</p></td><td>120</td><td>48</td><td></td><td><a href="/facts/Octahedral_symmetry/qLqhYxgm"><i>O</i>h</a></td><td><a href="/facts/Octahedral_symmetry/qLqhYxgm"><i>O</i></a></td></tr><tr><td><a href="/facts/Compound_of_two_snub_dodecahedra/47y8AdV1">UC69</a></td><td>dissid</td><td></td><td>2</td><td><a href="/facts/Snub_dodecahedron/oouEPVxD">snub dodecahedra</a></td><td>(40+120){3}<p>24{5}</p></td><td>300</td><td>120</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i></a></td></tr><tr><td><a href="/facts/Compound_of_two_great_snub_icosidodecahedra/bCnn5jn8">UC70</a></td><td>giddasid</td><td></td><td>2</td><td><a href="/facts/Great_snub_icosidodecahedron/n0vFdDWt">great snub icosidodecahedra</a></td><td>(40+120){3}<p>24{5/2}</p></td><td>300</td><td>120</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i></a></td></tr><tr><td><a href="/facts/Compound_of_two_great_inverted_snub_icosidodecahedra/amoLfAIS">UC71</a></td><td>gidsid</td><td></td><td>2</td><td><a href="/facts/Great_inverted_snub_icosidodecahedron/dD4RAlrl">great inverted snub icosidodecahedra</a></td><td>(40+120){3}<p>24{5/2}</p></td><td>300</td><td>120</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i></a></td></tr><tr><td><a href="/facts/Compound_of_two_great_retrosnub_icosidodecahedra/U5Gt7IoV">UC72</a></td><td>gidrissid</td><td></td><td>2</td><td><a href="/facts/Great_retrosnub_icosidodecahedron/ofggBlnC">great retrosnub icosidodecahedra</a></td><td>(40+120){3}<p>24{5/2}</p></td><td>300</td><td>120</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i></a></td></tr><tr><td><a href="/facts/Compound_of_two_snub_dodecadodecahedra/phnhfGtb">UC73</a></td><td>disdid</td><td></td><td>2</td><td><a href="/facts/Snub_dodecadodecahedron/k830QaMW">snub dodecadodecahedra</a></td><td>120{3}<p>24{5}</p><p>24{5/2}</p></td><td>300</td><td>120</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i></a></td></tr><tr><td><a href="/facts/Compound_of_two_inverted_snub_dodecadodecahedra/vducca5Q">UC74</a></td><td>idisdid</td><td></td><td>2</td><td><a href="/facts/Inverted_snub_dodecadodecahedron/vQ7Gpjtz">inverted snub dodecadodecahedra</a></td><td>120{3}<p>24{5}</p><p>24{5/2}</p></td><td>300</td><td>120</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i></a></td></tr><tr><td><a href="/facts/Compound_of_two_snub_icosidodecadodecahedra/Al74olTa">UC75</a></td><td>desided</td><td></td><td>2</td><td><a href="/facts/Snub_icosidodecadodecahedron/Pku4VL1F">snub icosidodecadodecahedra</a></td><td>(40+120){3}<p>24{5}</p><p>24{5/2}</p></td><td>360</td><td>120</td><td></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i>h</a></td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23"><i>I</i></a></td></tr></tbody></table> <ul><li>Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", <i><a href="/facts/Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society/KUUzcS7S">Mathematical Proceedings of the Cambridge Philosophical Society</a></i>, 79 (3): 447–457, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1017%2FS0305004100052440">10.1017/S0305004100052440</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0397554">0397554</a>.</li></ul>