Resistance of double infinite length resistor ladders

Looking in from the green node and from the red node, we see the same resistance $R$ . Therefore:

$\frac{(R+1)(1)}{(R+1) + 1} = \frac{R+1}{R+2} = R \\ R^2 + R - 1 = 0 \\ R = \frac{-1 + \sqrt{5}}{2}$

Then the total resistance between $A$ and $D$ is:

$R_{AD} = 1 + 2R = \sqrt{5}$