It appears the shortest overall length of spinneret thread would be ≈ 6.196152 meters.
The red lines in Plainglazed figure would each be √3/3 meters in length, the green lines would each be √3/6 meters in length, and the central yellow line would be (1 - √3/3) meters in length. This should total (3 + 9√3)/3 meters.

DejMar's post in Simple Math was marked as the answer
March 14, 2016