Simplicial honeycomb

<table><tbody><tr><th>                                                                        A                ~                                                          2                                {\displaystyle {\tilde {A}}_{2}}  </th><th>                                                                        A                ~                                                          3                                {\displaystyle {\tilde {A}}_{3}}  </th></tr><tr><th><a href="/facts/Triangular_tiling/Jx8f8Krh">Triangular tiling</a></th><th><a href="/facts/Tetrahedral-octahedral_honeycomb/mX4CxSvp">Tetrahedral-octahedral honeycomb</a></th></tr><tr><td>With red and yellow equilateral triangles</td><td>With cyan and yellow <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedra</a>, and red rectified tetrahedra (<a href="/facts/Octahedron/G38q92xW">octahedra</a>)</td></tr><tr><th></th><th></th></tr></tbody></table>
<p>In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, the simplicial honeycomb (or n-simplex honeycomb) is a dimensional infinite series of <a href="/facts/Honeycomb_(geometry)/seyRBcel">honeycombs</a>, based on the 
  
    
      
        
          
            
              
                A
                ~
              
            
          
          
            n
          
        
      
    
    {\displaystyle {\tilde {A}}_{n}}
  
 affine <a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a> symmetry. It is represented by a <a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter-Dynkin diagram</a> as a cyclic graph of <i>n</i> + 1 nodes with one node ringed. It is composed of n-<a href="/facts/Simplex/0FCfNRN8">simplex</a> facets, along with all <a href="/facts/Rectification_(geometry)/PkoUdq53">rectified</a> n-simplices. It can be thought of as an n-dimensional <a href="/facts/Hypercubic_honeycomb/MQxLbDeB">hypercubic honeycomb</a> that has been subdivided along all <a href="/facts/Hyperplane/2s2ycYYd">hyperplanes</a> 
  
    
      
        x
        +
        y
        +
        ⋯
        ∈
        
          Z
        
      
    
    {\displaystyle x+y+\cdots \in \mathbb {Z} }
  
, then stretched along its main diagonal until the simplices on the ends of the hypercubes become regular. The <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a> of an n-<i>simplex honeycomb</i> is an <a href="/facts/Expansion_(geometry)/eMpoRlnm">expanded</a> n-<a href="/facts/Simplex/0FCfNRN8">simplex</a>.
</p><p>In <a href="/facts/2_dimensions/f4b397W8">2 dimensions</a>, the honeycomb represents the <a href="/facts/Triangular_tiling/Jx8f8Krh">triangular tiling</a>, with Coxeter graph  filling the plane with alternately colored triangles. In <a href="/facts/3_dimensions/KAqKWUbS">3 dimensions</a> it represents the <a href="/facts/Tetrahedral-octahedral_honeycomb/mX4CxSvp">tetrahedral-octahedral honeycomb</a>, with Coxeter graph  filling space with alternately tetrahedral and octahedral cells. In <a href="/facts/Four-dimensional_space/vFQd0M7T">4 dimensions</a> it is called the <a href="/facts/5-cell_honeycomb/YgY6eSTr">5-cell honeycomb</a>, with Coxeter graph , with <a href="/facts/5-cell/Rh6lwDcc">5-cell</a> and <a href="/facts/Rectified_5-cell/2UAMuo2P">rectified 5-cell</a> facets. In <a href="/facts/Five-dimensional_space/bcrNVzhj">5 dimensions</a> it is called the <a href="/facts/5-simplex_honeycomb/XSLrRsLz">5-simplex honeycomb</a>, with Coxeter graph , filling space by <a href="/facts/5-simplex/KIfBFfhl">5-simplex</a>, <a href="/facts/Rectified_5-simplex/H5kx0rFW">rectified 5-simplex</a>, and <a href="/facts/Birectified_5-simplex/H5kx0rFW">birectified 5-simplex</a> facets. In <a href="/facts/Six-dimensional_space/Lh23NhTa">6 dimensions</a> it is called the <a href="/facts/6-simplex_honeycomb/H7LWND7E">6-simplex honeycomb</a>, with Coxeter graph , filling space by <a href="/facts/6-simplex/mInfSIeH">6-simplex</a>, <a href="/facts/Rectified_6-simplex/b3R5YeFN">rectified 6-simplex</a>, and <a href="/facts/Birectified_6-simplex/b3R5YeFN">birectified 6-simplex</a> facets.
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Simplicial honeycomb open-in-new

Simplicial honeycomb