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Description
These images show different ways of arranging 92 bevel gears to rotate freely along the surface of a sphere. Each sphere contains 12 large blue gears, 20 large yellow gears, and 60 small red gears. The gears are slightly off-centered from the vertices of a geodesic dome / sphere.

When I first attempted this on 2/7/11, I used a 25:30:12 gear tooth ratio, but unfortunately, I couldn't get the teeth to mesh together:
[link]

However, on 7/26/11, Taff Goch demonstrated that it is possible to get the teeth to mesh reasonably well using a 20:24:13 gear tooth ratio:
[link]

As it turns out, there are other gear tooth ratios that can be made to mesh together reasonably well, as you can see here. I found these solutions by writing a program that analyzed many different ratios. For a given ratio, the location and radius of each gear was calculated so that there are no gaps between mating pitch circles. Then the gear phase angles were calculated and solutions with large phase errors were rejected. Finally, I did a visual inspection of the remaining solutions and presented some of my favorite ones here. Two of my favorites are 15:15:8 and 30:30:16 because they only require two gear sizes. It is interesting to note that the number of blue gear teeth must always be divisible by 5 and the number of yellow gear teeth must always be divisible by 3, but the number of red gear teeth does not have to be divisible by anything (it can be a prime number).

Description
These images show different ways of arranging 92 bevel gears to rotate freely along the surface of a sphere. Each sphere contains 12 large blue gears, 20 large yellow gears, and 60 small red gears. The gears are slightly off-centered from the vertices of a geodesic dome / sphere.

When I first attempted this on 2/7/11, I used a 25:30:12 gear tooth ratio, but unfortunately, I couldn't get the teeth to mesh together:
[link]

However, on 7/26/11, Taff Goch demonstrated that it is possible to get the teeth to mesh reasonably well using a 20:24:13 gear tooth ratio:
[link]

As it turns out, there are other gear tooth ratios that can be made to mesh together reasonably well, as you can see here. I found these solutions by writing a program that analyzed many different ratios. For a given ratio, the location and radius of each gear was calculated so that there are no gaps between mating pitch circles. Then the gear phase angles were calculated and solutions with large phase errors were rejected. Finally, I did a visual inspection of the remaining solutions and presented some of my favorite ones here. Two of my favorites are 15:15:8 and 30:30:16 because they only require two gear sizes. It is interesting to note that the number of blue gear teeth must always be divisible by 5 and the number of yellow gear teeth must always be divisible by 3, but the number of red gear teeth does not have to be divisible by anything (it can be a prime number).