hyperdimensional lattices. The easiest lattice to visualize is the cubic lattice, since it is a repetition without transformation (rotation, mirroring, etc) in all coordinate vectors. A 3-dimensional cubic lattice can be “extended” into 4D by extending the definition of each point on the lattice to include an additional coordinate. Thus, a 4D cubic lattice is a multiplicity of 3D cubic lattices separated by one unit each in the 4th dimension. In other words, for each point on a 3D cubic lattice, there exist a number of points in 4D with the same <X,Y,Z> coordinates, the number of which depends on the extents of the lattice.
3D projection
hypercubic.lattice # of pts: 81
2D projection
xu: 0° yu: 0° zu: 0°
a.
c.
xu: 20° yu: 60° zu: 45°
b.
d.
0.1| views of hypercubic lattice a.3D-projection b.3D-projection-rotated c.2D-projection d.2D-projection-rotated
12
Rn [ hyper ]