The Octahedron
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 dual of the octahedron is the cube. The octahedron occurs as cells in many 4D polytopes, most notably the regular 24cell. The full list of occurrences is given below.
The octahedron can be bisected to form two square pyramids.
Projections
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:
Projection  Envelope  Description 

Square  Vertexfirst parallel projection. 

Rhombus  Edgefirst parallel projection. Four of the faces project to the edges of the projection envelope. 

Hexagon  Facefirst parallel projection. 
Coordinates
The canonical coordinates of the regular octahedron are all permutations of:
 (±1,0,0)
These coordinates give an octahedron of edge length √2. An octahedron with edge length 2 has the coordinates:
 (±√2,0,0)
with all permutations of coordinates thereof.
Occurrences
The octahedron occurs as cells in many 4D polytopes, both regular and uniform:
 The 24cell, a special 4D regular polytope;
 The rectified 5cell, a uniform polychoron in the 5cell family;
 The cantellated 5cell;
 The cantellated tesseract;
 The truncated 16cell;
 The runcinated 24cell;
 The cantellated 120cell;
 The rectified 600cell.
Some CRF polychora also contain octahedral cells, including (but not limited to):
 Cube antiprism (K4.15) (aka octahedron antiprism);
 Octahedron atop rhombicuboctahedron (K4.107);
 The biparabigyrated cantellated tesseract;
 The tetrahedral ursachoron;
 The octahedral ursachoron;
 The octaaugmented runcitruncated 16cell.