1. This construction is based on facets of the runcinated 9-orthoplex. 2. The 40 triangular faces of the runcinated 5-cell can be seen as removed. 3. The 384 triangular faces of the runcinated 24-cell can be seen as removed. 4. Hence, the runcinated tesseract may be thought of as the 4-dimensional analogue of the rhombicuboctahedron. 5. The rhombicuboctahedron is also constructed from the cube or the octahedron in an analogous way to the runcinated tesseract. 6. The maximal cross-section of the runcinated 5-cell with a 3-dimensional hyperplane is a cuboctahedron. 7. These square faces can be seen on the runcinated 5-cell, using all 60 edges and 20 vertices. 8. These square faces can be seen on the runcinated 24-cell, using all 576 edges and 288 vertices. 9. The tetrahedron-first orthographic projection of the runcinated 5-cell into 3-dimensional space has a cuboctahedral envelope. 10. The maximal cross-section of the stericated hexateron with a 4-dimensional hyperplane is a runcinated 5-cell.
