Bevel gears used to only come in one size - well technically there was the "old" bevel with 14 teeth and the "new" bevel with 12 teeth, but I always considered these two varieties not to be inter-operable. The rest of this discusion will ignore the 14 tooth bevel gear completely.
In some of the newer sets (8448, 9748) we're seeing some new gears:
I haven't had much success getting any of these bevels to mesh with other non-bevel gears, but I haven't spent much time at it either. I've focussed on how these bevel gears can mesh with one another.
Two rather obvious properties of a gear are the number of teeth it has, and its in-plane meshing radius. Some new numbers are needed when talking about beveled gears since they are capable of meshing at right angles. I picked the following:
T = teeth
R = radius for in-plane meshing
P = perpendicular meshing radius
O = offset (due to thickness)
I considered the 12 tooth bevel to be the canonical case:
What this means is that if you mesh two of these together, each one's axle is 1 lego unit away from the plane behind the other gear.
Filling in the table for the other gears I get:
Consider two beams meeting at right angles. You now want to use two bevel gears to mesh at right angles, each gear backed up against one of the beams. What should the distances between the gear's axles and the corner be? Measuring from the inside corner, A's distance is P(A) + O(B). B's distance is P(B) + O(A). Notice it is asymetrical.
For example, to mesh the 12t double bevel with a 20t bevel....
The 12t double bevel should be 1.0 + 0 = 1.0 units away from the inside corner.
The 20t bevel should be 1.5 + 0.5 = 2.0 units away from the inside corner.
I had considered replacing the O() measurement with "thickness", and adjusting the P() measurement accordingly (in effect measuring from the front rather than the back of the gear) - but this leads to the following table
The values are still used the same way - distance for A = P(A) + O(B).
Either table works fine - I prefer the first because I tend to think of the 12t bevel as the base case, and having values of P() = 1.0 and O() = 0.0 seems appropriate.