The stereoptic images below, are 4-space rotations of the 24-Cell projected into 3-Space.  The edges shown in color are the boundaries of the surfaces that enclose the 3-D form.  See the Hypercube envelope page for instructions on how to view this figure.

Cuboctahedron Envelope


This enclosing form, or envelope is a cuboctahedron.  You may pick out the 24 octahedral cells that bound the 4-D figure as follows:
  • There are 2 superimposed octahedra associated with each of the 8 triangular faces.  (You only see 8 in this projection.)
  • There is 1 octahedra associated with each of the 6 square faces (they are flattened out to a single plane)
  • There are 2 superimposed octahedra in the center of the figure with their vertices at the center of each of the square faces.  (You will only see 1 in this projection.)

Rhombic Dodecahedron Envelope


The envelope of this projection is the rhombic dodecahedron (the dual of the cube octahedron).  Notice this envelope is the same as one of the hypercube envelopes.  Picking out the 24 octahedral cells from this figure is a challenge because a large number of edges are superimposed.
  • There are 6 octahedra associated with the 6 apparent cube faces seen in the interior of the figure.  Another way to look at these 6 octahedra is to look for the 6 vertices of the rhombic dodecahedron that are surrounded by 4 faces.  These are the vertices of 6 octahedra who's opposite vertices are all touching at the center of the figure.  Now that you have located the 6 octahedra, I need to tell you that they are really 12, 2 superimposed on each other.
  • Each of the 12 rhombic faces of the dodecahedron are also flattened out octohedra.

You Name It Envelope


The figure that encloses this 3-D projection might be described as 2 hexagonal anti-prisms joined at their bases.  Each anti-prism is composed of 2 inrregular hexagons in parallel planes and joined by 12 triangular faces.  The envelope itself is bounded by 24 triangular faces and 2 hexagonal faces.  I have included this rotation because all the 24 octahedral cells are visible, and its 2-D projection will be familiar to many of you.  A guide to finding all the octagonal cells can be found on the 24-Cell page.
Hyperspace
Last revised: 10/20/02