Alright, let's get moving!
Today I'm talking about the limit on how many spokes you can lace when doing tangent lacing.
↑It's exactly as written in that text. That's it.
Well, not quite that simple, so...
First, the part about "when the hub and rim have the same number of holes with evenly-spaced phase"—basically that just means a normal rim and spokes.
There was one thing I left out: the number of spokes on the left and right sides must also be equal.
I'll proceed with that assumption.
If you want to lace y crossings on a normal rim with x total spokes, then wheels can be built within the range y≦x/4.
Now that I think about it, I've never touched on the concept of how many "crosses" a wheel has, so I'll write it down.
The figure above is a hub flange.
When you look from the side and count which holes in the hub flange the spokes that finally cross each other are coming from, a 2-cross is when the 1st and 2nd spokes cross.
Normally, crossing spokes are one on the drive side (J-spokes) and one on the non-drive side (反J-spokes), but in the figure above they're both J-spokes.
When viewed from the side, making the crossing parts contact—I'll call this "weaving" from here on.
Even if you try to weave on 2-cross, it's impossible because of the hub flange thickness.
However, if you align both spokes of the 2-cross lacing to be either J-spokes or anti-J-spokes, you can actually weave the spokes even in 2-cross.
↑This is J-spoke 2-cross lacing
A brief tangent—I'll also cover "weaving" while I'm at it.
This is a Mavic Ksyrium (high-end aluminum wheelset) ceramic rim, but I don't need to brag about rims (laugh).
See how the spoke crossing points are in contact? That's "weaving."
When you weave spokes, the wheel becomes more resistant to lateral twisting and less prone to wobble.
The downside is that if spoke tension is low, you get a squeaking noise.
I really hate the black spokes that mysteriously get used on factory-built wheels,
and one reason is that "black spokes are more prone to squeaking."
Another reason is "you basically can't solder them."
↑With aluminum spokes like Ksyrium,
even though spokes appear to cross when viewed from the side
from the front-to-back direction, they're not woven.
Aluminum spokes are vastly more susceptible to wear compared to steel spokes, so it's a necessary measure, but many factory-built wheels don't weave even with steel spokes.
The current Ksyrium Elite has aero steel spokes under absurdly high tension, and they're woven.
Looking at that, I think "Ah, I can't compete with factory-built wheels in this respect."
Back to tangent lacing.
↑This is 4-cross. The 1st and 4th spokes cross each other.
4-cross and higher use J-spokes and anti-J-spokes to cross.
↑Next up is 6-cross.
↑And finally, 8-cross.
For example, imagine doing tangent lacing on a 24H hub.
Since it's 24H, each flange has 12 holes.
↑I tried 6-cross. No problems.
I tried 8-cross. Something's off.
In tangent lacing, the phase angle of the holes from which the last crossing spokes exit cannot exceed 180°.
（Technically it's not impossible if the spokes don't contact the hub body, but there's no mechanical advantage）
When doing normal tangent lacing, the limit for the hole positions where you can lace spokes is half of half the total number of hub holes.
Half of half is one quarter.
And there's the formula from the beginning.
Substituting the number of spokes for x and the cross count for y,
you can instantly calculate the maximum cross count for tangent lacing with x spokes or the minimum number of spokes needed for y-cross lacing.
For example, with 24H, you can do a maximum of 6-cross.
Since the left and right sides are connected by "≦",
lower crosses—4-cross, 2-cross, and 0-cross (which is radial lacing)—also work.
8-cross doesn't work because the left side becomes larger.
8-cross becomes possible from 32H and up.
It's essentially extinct as a specification now (materials aren't available), so I've never actually built one, but from 40H you can do 10-cross.
This formula is quite handy—for example, if you imagine a hypothetical 200H hub,
you can instantly calculate that "the limit is 50-cross."
I often do thought experiments imagining what kind of stress changes would occur if you selectively removed spokes regularly or irregularly from something like a 120H wheel (I'm a dangerous person, aren't I), and this formula comes in handy for that.
In real wheels, this leads to the practical point that "you can't do 6-cross on a 20H rim, but you can just barely do it on 24H".
That's a story for another day.
Today I'm talking about the limit on how many spokes you can lace when doing tangent lacing.
↑It's exactly as written in that text. That's it.
Well, not quite that simple, so...
First, the part about "when the hub and rim have the same number of holes with evenly-spaced phase"—basically that just means a normal rim and spokes.
There was one thing I left out: the number of spokes on the left and right sides must also be equal.
I'll proceed with that assumption.
If you want to lace y crossings on a normal rim with x total spokes, then wheels can be built within the range y≦x/4.
Now that I think about it, I've never touched on the concept of how many "crosses" a wheel has, so I'll write it down.
The figure above is a hub flange.
When you look from the side and count which holes in the hub flange the spokes that finally cross each other are coming from, a 2-cross is when the 1st and 2nd spokes cross.
Normally, crossing spokes are one on the drive side (J-spokes) and one on the non-drive side (反J-spokes), but in the figure above they're both J-spokes.
When viewed from the side, making the crossing parts contact—I'll call this "weaving" from here on.
Even if you try to weave on 2-cross, it's impossible because of the hub flange thickness.
However, if you align both spokes of the 2-cross lacing to be either J-spokes or anti-J-spokes, you can actually weave the spokes even in 2-cross.
↑This is J-spoke 2-cross lacing
A brief tangent—I'll also cover "weaving" while I'm at it.
This is a Mavic Ksyrium (high-end aluminum wheelset) ceramic rim, but I don't need to brag about rims (laugh).
See how the spoke crossing points are in contact? That's "weaving."
When you weave spokes, the wheel becomes more resistant to lateral twisting and less prone to wobble.
The downside is that if spoke tension is low, you get a squeaking noise.
I really hate the black spokes that mysteriously get used on factory-built wheels,
and one reason is that "black spokes are more prone to squeaking."
Another reason is "you basically can't solder them."
↑With aluminum spokes like Ksyrium,
even though spokes appear to cross when viewed from the side
from the front-to-back direction, they're not woven.
Aluminum spokes are vastly more susceptible to wear compared to steel spokes, so it's a necessary measure, but many factory-built wheels don't weave even with steel spokes.
The current Ksyrium Elite has aero steel spokes under absurdly high tension, and they're woven.
Looking at that, I think "Ah, I can't compete with factory-built wheels in this respect."
Back to tangent lacing.
↑This is 4-cross. The 1st and 4th spokes cross each other.
4-cross and higher use J-spokes and anti-J-spokes to cross.
↑Next up is 6-cross.
↑And finally, 8-cross.
For example, imagine doing tangent lacing on a 24H hub.
Since it's 24H, each flange has 12 holes.
↑I tried 6-cross. No problems.
I tried 8-cross. Something's off.
In tangent lacing, the phase angle of the holes from which the last crossing spokes exit cannot exceed 180°.
（Technically it's not impossible if the spokes don't contact the hub body, but there's no mechanical advantage）
When doing normal tangent lacing, the limit for the hole positions where you can lace spokes is half of half the total number of hub holes.
Half of half is one quarter.
And there's the formula from the beginning.
Substituting the number of spokes for x and the cross count for y,
you can instantly calculate the maximum cross count for tangent lacing with x spokes or the minimum number of spokes needed for y-cross lacing.
For example, with 24H, you can do a maximum of 6-cross.
Since the left and right sides are connected by "≦",
lower crosses—4-cross, 2-cross, and 0-cross (which is radial lacing)—also work.
8-cross doesn't work because the left side becomes larger.
8-cross becomes possible from 32H and up.
It's essentially extinct as a specification now (materials aren't available), so I've never actually built one, but from 40H you can do 10-cross.
This formula is quite handy—for example, if you imagine a hypothetical 200H hub,
you can instantly calculate that "the limit is 50-cross."
I often do thought experiments imagining what kind of stress changes would occur if you selectively removed spokes regularly or irregularly from something like a 120H wheel (I'm a dangerous person, aren't I), and this formula comes in handy for that.
In real wheels, this leads to the practical point that "you can't do 6-cross on a 20H rim, but you can just barely do it on 24H".
That's a story for another day.