Interesting field pattern?

If these arrows were showing the shape of a vector field, it would be a really weird field.

Series circuits with resistors

One thing that was evident with the fairy lights when attempting repair was that the leads had no indication of live or neutral. And that's because it doesn't matter. Resistors like a filament have no preferred direction for current. The fuse will work either way round and there was no Earth lead so it is double insulated.

Filament bulb

This is an old-fashioned outside Christmas fairy light bulb. The filament is the loop shown by the orange arrow. The pink arrow shows a wire strand that holds up the filament. The green arrow is a glass bead to provide insulation. The aim is to keep the wires apart to stop them shorting out.

It is a 12V 3W bulb. Using P=V^2 /R gives resistance of the filament as 48 Ohms. It looks like the filament loop is about 12mm long so resistance per unit length is 4000 Ohms per metre. Tungsten has a resistivity of 5.6 x 10^-8 Ohm.m at 20oC. Using R=rho.L/A and ignoring the temperature rise when turned on, cross-sectional area of such a filament made of tungsten would be 1.4 x 10^-11 square metres so diameter would be 2 x 10^-6 metres. That is surely much thinner than the wire in this bulb.