The platform can rotate. It has a gear on it that has 40 teeth. (this can NOT be changed).
What I would like to do is, have an encoder that will turn 360:1, yes to work out how many degrees the platform has moved. Speed is not an issue here as long as it doesn't take all day to complete 1 turn.
What I need help with is the maths so the I can make the encoder turn 360 times. The only fixed number is the 40 teeth on the plaform, the rest I haven't finished designing yet so is open to suggestions.
If you want a gear wheel rotating 360 times as fast as the end gear, this is one way to do it:
Dissolve the number 360 into integer divisors (I'm not sure of the right English term) by dividing with the least prime numbers that gives a division without a reminder.
360 /2 = 180, no remainder, so OK
180/2 = 90, still no remainder, so OK as well
45/2 =22.5, it has got a remainder - so no go, try the next higher integer 3
5/3=1.67, no go, raise to next integer divider (5)
Now you've reached one, gather all the good ones: 2x2x2x3x3x5 or more commonly expressed: 2^3x3^2x5
There's your gear ratios. you could perhaps find a set with a ratio of 4, 6, 8, 9 or so. As long as they can fit into the found ratios they can be used.
Remember, if you use a 10-tooth wheel on the 40-tooth wheel, you've got a ratio of 4 (2^2) in that alone and if you can find a "worm" to drive the 40-t, you will get a ratio of 40 (2^3x5) and need less gear wheels.
The encoder can not be mounted on the platform gear, so I can not just have the encoder with 360 division !! sorry, very limited space where the platform gear is. so it would need a gear train to drive it at another location.
You could mount it on the wheel driving the 40-t. In that case you wouldn't need such a large total ratio. A disk with say 8 divisions (easy to draw) would lower the needed amount of gear wheels as well. Using a dual reader, you will get twice the ticks as there are divisions.
All that said, I really cannot imaging that you need a resolution of one entire revolution for each degree. The backlash and imprecision in the gear train alone gets you nowhere near the resolution that your method points towards and I don't believe you need a resolution of less than 0.3% even if using an entire rotation a tick.
Reality/sanity check: Most resistors you use is 5%. Do you think you can control or use a tolerance 18 times tighter (and don't get me started about the imprecision of a cheap motor).
If you describe what you are building, perhaps you could get some guesses to what's realistic.