Simplectic honeycomb

In 2 dimensions, the honeycomb represents the triangular tiling, with Coxeter graph CDel node 1.pngCDel split1.pngCDel branch.png filling the plane with alternately colored triangles. In 3 dimensions it represents the tetrahedral-octahedral honeycomb, with Coxeter graph CDel node 1.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png filling space with alternately tetrahedral and octahedral cells. In 4 dimensions it is called the 5-cell honeycomb, with Coxeter graph CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel branch.png, with 5-cell and rectified 5-cell facets. In 5 dimensions it is called the 5-simplex honeycomb, with Coxeter graph CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png, filling space by 5-simplex, rectified 5-simplex, and birectified 5-simplex facets. In 6 dimensions it is called the 6-simplex honeycomb, with Coxeter graph CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png, filling space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets.

The (2n-1)-simplex honeycombs and 2n-simplex honeycombs can be projected into the n-dimensional hypercubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

These honeycombs, seen as tangent n-spheres located at the center of each honeycomb vertex have a fixed number of contacting spheres and correspond to the number of vertices in the vertex figure. For 2 and 3 dimensions, this represents the highest kissing number for 2 and 3 dimensions, but fall short on higher dimensions. In 2-dimensions, the triangular tiling defines a circle packing of 6 tangent spheres arranged in a regular hexagon, and for 3 dimensions there are 12 tangent spheres arranged in a cuboctahedral configuration. For 4 to 8 dimensions, the kissing numbers are 20, 30, 42, 56, and 72 spheres, while the greatest solutions are 24, 40, 72, 126, and 240 spheres respectively.