Notes

Chapter 9: Fundamental Physics

Section 7: Space as a Network


3D network

The 3D network (c) can be laid out in space using Array[x[8 {##}]&, {n, n, n}] where

x[m:{_, _, _}] := {x1[m], x1[m + 4], x2[m + {4, 2, 0}], x2[m + {0, 6, 4}]} x1[m:{_, _, _}] := Line[Map[(# + m)&, {{1,0,0}, {1,1,1}, {0,2,1},{1,1,1}, {3,1,3}, {3,0,4}, {3,1,3}, {4,2,3}}]] x2[{i_, j_, k_}] := x1[{-i-4, -j-2, k}] /. {a_, b_, c_} -> {-a, -b, c}

The resulting structure is a cubic array of blocks with each block containing 8 nodes. The shortest cycle that returns to a particular node turns out to involve 10 edges. The structure does not correspond to the way that chemical bonds are arranged in any common crystalline materials, probably because it would be likely to be mechanically unstable.


From Stephen Wolfram: A New Kind of Science [citation]