The regular octahedron is one of the Platonic solids. It is bounded by 8 equilateral triangles joined 4 to a vertex. It has 8 faces, 6 vertices, and 12 edges.
The octahedron can be bisected to form two square pyramids.
In order to be able to identify the octahedron in various projections of 4D objects, it is useful to know how it appears from various viewpoints. The following are some of the viewpoints that are commonly encountered:
Vertex-first parallel projection.
Edge-first parallel projection. Four of the faces project to the edges of the projection envelope.
Face-first parallel projection.
The canonical coordinates of the regular octahedron are all permutations of:
These coordinates give an octahedron of edge length √2. An octahedron with edge length 2 has the coordinates:
with all permutations of coordinates thereof.
- The 24-cell, a special 4D regular polytope;
- The rectified 5-cell, a uniform polychoron in the 5-cell family;
- The cantellated 5-cell;
- The cantellated tesseract;
- The truncated 16-cell;
- The runcinated 24-cell;
- The cantellated 120-cell;
- The rectified 600-cell.
Some CRF polychora also contain octahedral cells, including (but not limited to):