Even as I posted it I wasn’t satisfied with my Mood-O-Meter post from a few days ago. I knew intuitively that in order for the gears to display all possible “moods” the ratio of eye expressions to mouth expressions must be in its simplest form but I couldn’t quite describe mathematically why this was the case. The images below attempt to convey this relationship between the expression ratio and the gear ratio both mathematically and visually.

Essentially, to find how many “moods” will be displayed the numerator is multiplied by the simplified denominator. If the ratio is already in its simplest form, then this is obvious. For the ratio of 6:5, the viewable moods will be 6 x 5 = 30. But in the case of 6:4 (which can be reduced to 3:2), the viewable moods will be 6 x 2 = 12. In the more extreme example a ratio of 4:4, this is reduced to 1:1 so the viewable moods are limited to 4 x 1 = 4.

Now I’m satisfied.