Certain electric circuit can be perceived as undirected graphs whose
edges are 1-ohm resistances. Ohm's law allows calculation of
equivalent single resistances between two arbitrary points on the
electric circuit. For graphs embeddable in the plane, there are four
functions that allow the implementation of Ohm's law and calculation
of equivalent resistances. Consequently, no knowledge of electrical
engineering is needed for this talk. It is a talk about interesting
properties of graphs whose edges have specific resistances and which
allow reduction to other graphs. Interesting results are possible when
the underlying graph belongs to certain families. For example the
resistance between two corners (degree-two vertices) of a graph on $n$
edges consisting of $n − 2$ triangles arranged in a line is
${n-1\over5}+{4\over5}{F_{n-1}\over L_{n-1}}$ with $F$ and $L$
representing the Fibonacci and Lucas numbers respectively. This
presentation explores triangular graphs of $n$ rows of equilateral
triangles. These triangular graphs were mentioned in passing in one
paper with a conjecture on the equivalent resistance between two
corners. In this presentation we present new computation methods,
allowing reviewing more data. It turns out that the limiting behavior
of these $n$-row triangular grids (as $n$ goes to infinity) has
unexpected simply described behavior: The sides of individual
triangles are conjectured to asymptotically equal products of
basically fractional linear transformations and $e^{-1}$. We also
introduce new proof methods based on a simple *verification*
method.

Russell Jay Hendel