One belief shown on three rulers at once: probability from 0 to 1, odds from 0 toward infinity, and log-odds from minus 6 to plus 6. A slider drags the belief, and buttons double or halve the odds; each doubling is a shrinking shuffle near the ends of the squashed probability ruler, a lopsided jump on the odds ruler, and an identical stride on the log-odds ruler. Ghost ticks mark previous positions so the equal strides are visible. The sigmoid is simply the trip from the straight log-odds ruler back to the squashed probability ruler.

Un-squashing the probability ruler

One belief, three rulers. Drag it — then press double the odds a few times and watch the strides: shrinking shuffles near the ends of the top ruler, identical steps on the bottom one.

Same belief, three rulers. Odds remove the ceiling; the log removes the lopsidedness. Equal steps on the bottom ruler = equal multiplies of the odds. The sigmoid is nothing but the return trip — from the straight ruler back to the squashed one.