Originally Posted by
FBinNY
Any is a big word, so needs qualification, as in any wheel 36h or greater.
It's simple geometry. 4x 36h is full tangent. Likewise, 3x28, 2x24, and 1x12. You could do 4x32, but the spokes would leave the hole on a secant, passing inward before going out to the rim. It's possible, but annoying and pointless.
Thank you for that. Even with my geometry superpowers, the above explanation did not occur to me. I can easily visualize this. But it's not obvious if there is an easy way to determine the above numerically, versus doing math on the full geometry. Full tangent spoke (let's say a leading spoke), means the opposing spoke (so the trailing spoke, on the same side), also tangent, is parallel to the first spoke... that means the distance between those spokes at the rim, must be equal or greater than the diameter across spoke holes at the hub. 36h means rim holes are 10 degrees apart... so 20 degrees between those two rim spoke holes... no, I think doesn't matter, but rather the *distance* between those holes, versus the diameter at the hub holes, so the spokes are parallel, or get wider apart at the rim... the rim holes cannot be too close together or that is not the case... I'm seeing the geometry, just can't yet figure if there's an easy numerical relationship, like 36/4=9, but no, that won't work if, for example, the hub is very large diameter in relation to the rim, then 4X won't work on 36h, because like you said, secant (what I'll call "overtangent"
) So I think it's all about, take the diameter of the hub spoke holes, assume tangent, parallel leading/trailing spoke pairs, determine the distance between those at the rim (about half that distance is the space between adjacent rim holes, close, is actually the cosine of a low angle with the rim), and based on that, figure out how many spoke holes you have room for. OK, that distance at the rim is about the hub hole radius. Looking up... typical shimano hub rear 36h shows 45mm, so 22.5mm radius, so... also long as 22.5mm x (hole number) is *smaller* than the actual rim *circumference*, should be good to go. Let's see... My 406 wheels, about 390mm inside diameter, so 390 x pi = 1225mm circumference. 22.5 x 36 = 810, so far so good. (Wheels are actually 28 hole, I have seen 36 hole 406s, but I think they were 3X, I think 4X might have spokes over adjacent (first cross) spoke heads.) Now let's go way down, like 305, so assuming same 8mm depth on each side for a diameter of 305-16=289. 289 x pi = 908 circumference. That 810 number is awful close, given that 22.5 is an underestimate of the actual circumference distance. Let's say 100mm hub hole diameter so 50mm radius, on the 406 rims: 50x36=1800, that's way over the 1225 estimated circumfirence, so no way for 36h 4X pure tangential. So yeah, for ability for tangential spoking, it's about the relationship between the hub spoke hole radius and the distance between rim spoke holes. It's not just about number of crosses versus number of holes. (The above took me at least a half hour to think out, maybe an hour.) But the practical result is exactly what you said. I was just trying to figure it out or prove it numerically. Brain tired.