Self GMD of 4-strand conductor in terms of r

The self-GMD of an $n$-strand conductor is the $n^2$-th root of the product of all inter-strand distances, with each strand's self-distance $r' = 0.7788r$ (for inductance).

                (4)
      (1)   (2)   (3)
  touching strands, centre spacing 2r

Strand 4 sits in the groove between 2 and 3, so 2, 3, 4 form an equilateral triangle of side $2r$. Take centres: 1 at $(0,0)$, 2 at $(2r,0)$, 3 at $(4r,0)$, 4 at $(3r, \sqrt3 r)$.

Distances

Product of the six mutual distances: $(2r)^4 (4r)(2\sqrt3 r) = 16 \times 4 \times 2\sqrt3\, r^6 = 128\sqrt3\,r^6 = 221.7\,r^6$. Each pair appears twice in the $n^2 = 16$ terms.

Self-GMD

$$\begin{aligned} D_s &= \left[(0.7788r)^4 \times \left((2r)^4 (4r)(2\sqrt3 r)\right)^2\right]^{1/16} \\ &= 1.8453\,r \end{aligned}$$

For capacitance (self-distance $r$): $D_s = 1.9644\,r$.

Answer: self-GMD $= 1.845r$ (for inductance).