Configuration
Each 24-cell in the 24-cell honeycomb has 24 neighbors with which it shares an octahedral cell. It has 24 more neighbors with which it shares a single vertex. It has no neighbors with which it shares only an edge or only a face.
The vertex figure of a the 24-cell honeycomb is a tesseract (4-dimensional cube). So there are 16 edges, 32 triangles, 24 octahedra, and 8 24-cells meeting at every vertex. The edge figure is a tetrahedron, so there are 4 triangles, 6 octahedra, and 4 24-cells surrounding every edge. Finally, the face figure is a triangle, so there are 3 octahedra and 3 24-cells meeting at every face.
Read more about this topic: 24-cell Honeycomb