The Omnitruncated 5-cell


The omnitruncated 5-cell is a uniform polychoron consisting of 10 truncated octahedra and 20 hexagonal prisms. It has 150 faces, 240 edges, and 120 vertices.

Structure

The following image shows the parallel projection of the omnitruncated 5-cell into 3D:

Parallel projection
of omnitruncated 5-cell

The central yellow truncated octahedron is the nearest cell to the 4D viewpoint. Four of its hexagonal faces are joined to 4 other truncated octahedra, as shown in the next images:

Parallel projection
of omnitruncated 5-cell, with 1 cell surrounding nearest cell shown Parallel projection of omnitruncated
5-cell, with another cell surrounding nearest cell shown Parallel projection of omnitruncated
5-cell, with 3rd cell surrounding nearest cell shown Parallel projection of omnitruncated
5-cell, with 4th cell surrounding nearest cell shown

The other 4 hexagonal faces of the nearest cell are joined to 4 hexagonal prisms, as shown in the following image:

Parallel projection
of omnitruncated 5-cell, with 4 hexagonal prisms surrounding nearest cell
shown

The 6 square faces of the nearest cell are joined to 6 hexagonal prisms straddling these 4 hexagonal prisms. These 10 hexagonal prisms are shown in the next image:

Parallel projection
of omnitruncated 5-cell, with all 10 hexagonal prisms surrounding nearest cell
shown

The 5 truncated octahedra and 10 hexagonal prisms we have seen so far are the cells lying in the “northern hemisphere” of the omnitruncated 5-cell. On the opposite side of the omnitruncated 5-cell are another 5 truncated octahedra and 10 hexagonal prisms, laid out in dual orientation to these cells. These are shown in the following images.

The antipodal truncated octahedron (the “south pole” cell, shown here in brown) and its surrounding 4 truncated octahedra (yellow):

Parallel projection
of omnitruncated 5-cell, with 5 truncated octahedra on the far side
shown

And the 10 hexagonal prisms surrounding the south pole cell:

Parallel projection
of omnitruncated 5-cell, with the 10 hexagonal prisms on the far side
shown

Perspective Projection

The following image shows the perspective projection of the omnitruncated 5-cell into 3D:

Perspective
projection of omnitruncated 5-cell

This projection is almost the same as the parallel projection, except that the foreshortening of the truncated octahedra surrounding the nearest cell is more pronounced.

Coordinates

The Cartesian coordinates for the omnitruncated 5-cell, centered on the origin and having edge length 2, are:

  • ±(0, 4/√6, −2/√3, ±4)
  • ±(0, 4/√6, −5/√3, ±3)
  • ±(0, 4/√6, 7/√3, ±1)
  • ±(0, 8/√6, −1/√3, ±3)
  • ±(0, 8/√6, −4/√3, ±2)
  • ±(0, 8/√6, 5/√3, ±1)
  • ±(±10/√10, 2/√6, −1/√3, ±3)
  • ±(±10/√10, 2/√6, −4/√3, ±2)
  • ±(±10/√10, 2/√6, 5/√3, ±1)
  • (±10/√10, ±6/√6, 0, ±2)
  • (±10/√10, ±6/√6, ±3/√3, ±1)
  • ±(5/√10, 1/√6, −2/√3, ±4)
  • ±(5/√10, 1/√6, −5/√3, ±3)
  • ±(5/√10, 1/√6, 7/√3, ±1)
  • ±(5/√10, −3/√6, 0, ±4)
  • ±(5/√10, −3/√6, ±6/√3, ±2)
  • ±(5/√10, −7/√6, −1/√3, ±3)
  • ±(5/√10, −7/√6, −4/√3, ±2)
  • ±(5/√10, −7/√6, 5/√3, ±1)
  • ±(5/√10, 9/√6, 0, ±2)
  • ±(5/√10, 9/√6, ±3/√3, ±1)

Simpler coordinates can be obtained in 5D as all permutations of coordinates of:

The 4D coordinates are derived by projecting these 5D coordinates back into 4D using a symmetric projection.


Last updated 13 Sep 2011.

Powered by Apache Runs on Debian GNU/Linux Viewable on any browser Valid CSS Valid XHTML 1.1! Proud to be Microsoft-free