LiveGraphics3D views of
Jim Lehman's Vector Matrix (LVM)
Mouse to rotate,
Shift-mouse to zoom in/out,
s for stereo,
s a 2nd time for cross-eyed vs. splay-eyed stereo,
ALT + right mouse button: drag down to remove elements (drag up to
I imagine the green cube as a 2-frequency, volume 24
affair from the RBF
concentric hierarchy. For the pentagonal dodecahedron to embrace
it in this way, it has to be scaled up from the structural dual to
the icosahedron of edges 1 (these two together form the edges of the
rhombic triacontahedron -- see link).
A scale factor of Sqrt(2) accomplishes this. The blue stellate with
the orange ridges is called a concave
dodecahedron and space-fills in complement with pentagonal dodecahedra.
Jim Lehman uses a repeating
pattern of these shapes to form a lattice.
Above we see that the orange segments, known as "rybo
keels" on the synergeo
list, are actually the long edges of the icosahedron's 3 golden
rectangles. These rectangles have short:long dimensions in the ratio
of 1 : phi, where phi = (1+sqrt(5))/2.
Above, the orange segments have been removed and we
see the six spires (dark blue) which have base triangles congruent
to icosahedron faces, and tips at the corners of the 2-frequency cube.
The red octahedron is the volume 4 regular octahedron
from the concentric hierarchy. Its edges comprise the long diagonals
of the space-filler rhombic dodecahedron faces (not depicted). The
icosahedron, in order to inscribe in this octahedron's faces, had
to be scaled down from the icosahedron with unit edges (of volume
approx 18.51). A scale factor of (sqrt(2)/2))(3-sqrt(5)) is needed.
Above I've added the cuboctahedron (yellow), which has
a volume of 20, and is generated when 12 IVM
spheres close pack around a nuclear center sphere. This is equivalently
the FCC or CCP packing. The red octahedron tips end up in between-sphere
voids in this description, but form the anchor points of an alternative
IVM (i.e. if those points become spheres, and the current spheres
become points, we will still have an IVM).
For Further Reading: