Question

Determine the resistance R_AB between points A and B of the frame made of thin homogeneous wire (Fig. 85), assuming

that the number of successively embedded equilateral triangles (with sides decreasing by half) tends to infinity. Side AB is equal to a, and the resistance of unit length of the wire is ρ.

Answer 1

It follows from symmetry considerations that the initial circuit can be replaced by an equivalent one (Fig. 215). We replace the "inner"

triangle consisting of an infinite number of elements by a resistor of resistance R_AB/2, where the resistance RAB is such that R_AB = R_x and R_AB = aρ. After simplification, the circuit becomes a system of series- and parallel-connected conductors. In order to find R_x , we write the equation 

Solving this equation, we obtain