The golden case: equal resistors

A particularly elegant case of the infinite ladder is obtained when all the resistors have the same value $R$, i.e. $R_A = R_B = R$. The self-similarity equation becomes:

$Z_\infty^2 - R\,Z_\infty - R^2 = 0$

Applying the quadratic formula and keeping only the positive root:

$Z_\infty = \frac{R + \sqrt{R^2 + 4R^2}}{2} = \frac{R(1 + \sqrt{5})}{2} = R\,\varphi$

where

$\ev{\varphi = \frac{1 + \sqrt{5}}{2} \approx 1{,}618}$

is the celebrated golden ratio. The infinite ladder of identical resistors “knows” the golden section.

The result is surprising: the circuit has nothing geometrically special about it, yet it spontaneously generates the most famous irrational number in geometry. The golden root appears whenever something is defined through its own self-similarity — and that is exactly what the infinite ladder does.

Why exactly $\varphi$

The golden ratio is the positive solution of $x^2 = x + 1$, i.e. $x = 1 + 1/x$: a number equal to “one plus its own reciprocal”. The infinite ladder physically builds this relation, with $R$ in series ($+R$) and $R$ in parallel (the reciprocal), hence $Z_\infty = R\varphi$.

Links

Topics: Electric circuits Concepts: Resistors in series and parallel

Related exercises: Worked exercise — mixed series-parallel network · Problem — true or false on series and parallel · Problem — series-parallel equivalent resistance