11-cell

<table><tbody><tr><th colspan="2">11-cell</th></tr><tr><td colspan="2"><i>The 11 hemi-icosahedra with vertices labeled by indices 0..9,t. Faces are colored by the cell it connects to, defined by the small colored boxes.</i></td></tr><tr><td>Type</td><td><a href="/facts/Abstract_polytope/awkKfRGM">Abstract regular 4-polytope</a></td></tr><tr><td>Cells</td><td>11 <a href="/facts/Hemi-icosahedron/1PDWTJ1M">hemi-icosahedron</a></td></tr><tr><td>Faces</td><td>55 {3}</td></tr><tr><td>Edges</td><td>55</td></tr><tr><td>Vertices</td><td>11</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td><a href="/facts/Hemi-dodecahedron/44VmMjGY">hemi-dodecahedron</a></td></tr><tr><td><a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a></td><td>                    {        {        3        ,        5                  }                      5                          ,        {        5        ,        3                  }                      5                          }              {\displaystyle \{\{3,5\}_{5},\{5,3\}_{5}\}}  </td></tr><tr><td><a href="/facts/Symmetry_group/CNqeseMp">Symmetry group</a></td><td>order 660Abstract <a href="/facts/Projective_special_linear_group/y7P9rMyR">L2(11)</a></td></tr><tr><td>Dual</td><td><a href="/facts/Self-dual_polytope/rkIHbmAf">self-dual</a></td></tr><tr><td>Properties</td><td>Regular</td></tr></tbody></table>
<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the 11-cell is a <a href="/facts/Duality_(mathematics)/8jGuyQIU">self-dual</a> <a href="/facts/Abstract_polytope/awkKfRGM">abstract regular 4-polytope</a> (<a href="/facts/4-polytope/mY8vBpec">four-dimensional polytope</a>). Its 11 cells are <a href="/facts/Hemi-icosahedron/1PDWTJ1M">hemi-icosahedral</a>. It has 11 vertices, 55 edges and 55 faces. It has <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli type</a> {3,5,3}, with 3 hemi-icosahedra (Schläfli type {3,5}) around each edge.
</p><p>It has symmetry order 660, computed as the product of the number of cells (11) and the symmetry of each cell (60). The symmetry structure is the abstract group <a href="/facts/Projective_special_linear_group/y7P9rMyR">projective special linear group</a> of the 2-dimensional vector space over the finite field with 11 elements L2(11).
</p><p>It was discovered in 1976 by <a href="/facts/Branko_Gr%25C3%25BCnbaum/taVzX1MG">Branko Grünbaum</a>, who constructed it by pasting hemi-icosahedra together, three at each edge, until the shape closed up. It was independently discovered by <a href="/facts/H._S._M._Coxeter/D2l5jWGG">H. S. M. Coxeter</a> in 1984, who studied its structure and symmetry in greater depth. It has since been studied and illustrated by <a href="/facts/Carlo_H._S%25C3%25A9quin/K0t7eRmK">Séquin</a>.
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11-cell open-in-new

11-cell