Two mirrors

Drag the angle θ: two mirrors reach out from the centre of the eyepiece, with a few beads of glass in the space between them. Starting straight below the centre, the mirrors repeat that space piece after piece all the way round, turned over every other time; at angles such as 60°, 45°, 36° and 30°, which go a whole number of times into 180°, the last piece meets the first without a seam when the round comes back to the bottom, 360° ÷ θ pieces in all. Angles such as 40° or 72°, which do split the circle into whole pieces, still leave a seam: the pieces are odd in number, so front and back cannot alternate all the way round. At any other angle the last piece and the first fail to meet there: a seam runs straight down from the centre, the beads on either side of it do not match, and the corner reads how many degrees apart the two sides are. Press play to let the angle open by itself.