The 24 Cell

The following printout had been produced by Shapeways from our STL file which was generated by us from Mathematica 8. Two different materials are used.

About the object: The 24 cell is a regular 4-polytope. It can be realized in four dimensional space as the 24 points (a,b,0,0),(a,0,b,0),(a,0,0,b),(0,a,b,0),(0,a,0,b),(0,0,a,b) where a,b are either -1 or 1. The polytope is selfdual. It has also 24 chambers, which is the reason for the name 24 cell.

The current picture was obtained by rotating in 4D space with the SO(4) matrix exp(A), where A in so(4) is the skew symmetric matrix
   0    t1   t2   t3 
  -t1    0   t4   t5
  -t2  -t1    0   t6 
  -t3  -t4  -t5    0
In this case the parameters t=(t1,t2,t3,t4,t5,t6) = (0.3,0.5,0.4,0.3,0.3,0.4) were used. The points are then projected into three dimensional space with P(x,y,z,u)=(x,y,z).