The Lengberg Cap with Sprang (which, by the way, has been nicely published, and you can find the article here) has the usual features of lots of medieval headwear: White linen and surprisingly narrow hems. In this case, it also features the sprang panel, and also...needle lace.

The needle lace connects the ties to the main part, and also is an embellishment of the front of the cap - so after hemming the individual pieces, and whip-stitching in the sprang panels, the next step will be making some needle lace.

Fortunately, the patterns are rather simple, and the originals don't have the same pattern on both sides of the tie connections, so I can go astray a little bit and it won't be too bad.

Meanwhile, that's the sprang panel seen from above, on a dark styrofoam head.