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.
6 mouths x 5 pairs of eyes = 30 different moods.
A few weeks ago I volunteered to lead a make and do session for a staff team building “fun day.” Two nights before the event I started prototyping this “Mood-O-Meter” which actually presented an interesting mathematical problem: what combination of gear teeth and multiples of eye and mouth expressions will display all possible mood options?
For example if I had two 12 tooth gears it wouldn’t matter if I put four eye expressions on one and four mouth expressions on the other because only four expressions out of a possible of sixteen would ever display.
If I printed four expressions on one 12 tooth gear and three on the other this would be even worse as only one mood would be displayed. The other eleven options would always be out of sync.
I settled upon a 18 tooth and 15 tooth with six eye expressions and five mouth expressions respectively to produce 30 individual moods.
My colleagues at Texas Performing Arts cutting out gears to make their Mood-O-Meters. They may have lacked the exuberance of my elementary aged students but when they meshed their gears for the first time they were positively giddy.