Foundations of Data Science

y
7
3
1
1 3 7
x
Figure 5.10: Paths in a 2-dimensional lattice obtained from the 3-dimensional construction
applied in 2-dimensions.
dimensional example is
1
2 + 1 4 2 + 1 8 4 + · · · = 1 2 + 1 2 + 1 2 + · · · = ∞.
In the analogous three dimensional construction, paths go up, to the right, and out of
the plane of the paper. The paths split three ways at planes given by x + y + z = 2 n − 1.
Each time the paths split the number of parallel segments triple. Segments of the paths
between splits are of length 1, 2, 4, etc. and the resistance of the segments are equal to
the lengths. The resistance out to infinity for the tree is
1
3 + 1 9 2 + 1
27 4 + · · · = 1 3
(
1 +
2
+ 4 + · · ·) = 1 3 9 3
1
1− 2 3
= 1
The resistance of the three dimensional lattice is less. It is important to check that the
paths are edge-disjoint and so the tree is a subgraph of the lattice. Going to a subgraph is
equivalent to deleting edges which only increases the resistance. That is why the resistance
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