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.