When I wrote about left-right asymmetrical lacing the other day
I drew a diagram like this, which contains a note saying "the angle is the same."
This refers to "regardless of how many times the spokes cross in an X-lacing pattern, the angle between the wheel centerline and the spokes remains constant"—
but I received a comment saying "I don't believe that."
To be clear from the start: I was wrong.
Thank you for pointing that out.
↑Let me consider a rectangular solid like this.
When cut diagonally in half, I add the plane AFGD.
I draw straight lines from A to F, and from A to G.
Furthermore, I draw several arbitrary straight lines from A toward line segment FG.
Since all these red lines lie on plane AFGD,
when viewing this rectangular solid straight on
all the red lines appear on the same line—and that's where my misconception came from.
Let me reconsider using as a reference "the extension direction of X-cross spokes emerging from a hub flange hole when that hole is positioned at the vertex phase."
Below, I'll discuss the direction of the spokes colored red.
I'm using counter spokes because it's easier to see the spoke head.
For radial lacing, it looks like this:
It's perpendicular to the horizontal plane (ground), so it's 90°.
With a certain 32-hole 4-cross lacing, it's about this angle.
Even though it's called "4-cross," the angle isn't constant—it varies with flange diameter, rim depth,
and most importantly, the number of spokes.
In an extreme case, with something like 1000 holes,
the angle would become nearly the same as radial lacing.
With a certain 32-hole 8-cross lacing, it's about this angle.
32-hole 8-cross is the most tangential lacing, but
it's only slightly less than horizontal.
When illustrated together, they look like this:
"The angles are not the same."
Next, I'll consider using "when the spoke tension direction is straight up" as the reference.
Regarding spoke tension direction: for radial lacing it's the spoke extension direction itself,
while for tangent lacing it's the resultant direction of the two spokes' final cross.
If we assume a 24-hole hub and don't consider crosses beyond the hub bisector,
tangent lacing can be 2, 4, or 6-cross.
Here I've added 0-cross (radial lacing).
Since we're considering the spoke tension direction as the reference,
only 0-cross has a different hub hole phase.
When I translate that into a diagram like this...
it looks like this.
"The angles are not the same."
In the comment, someone wrote something to the effect of "radial lacing has a blunt angle, and most tangential lacing has a sharp angle"—
to express that idea,
I think the second diagram here is better than the first one.
Your comment was very helpful.
Thank you very much.
My policy is generally not to change or delete past articles except for typos,
so I'll link this correction article from the original post.
As a side note, the angle also changes between drive-side and non-drive-side spokes.
This applies to radial lacing, but with tangent lacing, you typically weave both drive-side and non-drive-side spokes, so which spoke is further outside the wheel
changes depending on position.
When the wheel is laid flat and the outer side is described as "up,"
at the hub flange area, the drive-side spoke is up, but
at the spoke crossing (when woven), the non-drive-side spoke is on top.
However, at the nipple area, I think the top-bottom relationship converges.
The angles are the same, or nearly so.
Also, regarding the comment "spoke length has never been mentioned," that's because it touches on a Meshino Taneko Code.
It's not the spoke length itself that's the issue, but there's "something" that would be noticed if I wrote it—
something I can't reveal. Watch out, watch out!
I drew a diagram like this, which contains a note saying "the angle is the same."
This refers to "regardless of how many times the spokes cross in an X-lacing pattern, the angle between the wheel centerline and the spokes remains constant"—
but I received a comment saying "I don't believe that."
To be clear from the start: I was wrong.
Thank you for pointing that out.
↑Let me consider a rectangular solid like this.
When cut diagonally in half, I add the plane AFGD.
I draw straight lines from A to F, and from A to G.
Furthermore, I draw several arbitrary straight lines from A toward line segment FG.
Since all these red lines lie on plane AFGD,
when viewing this rectangular solid straight on
all the red lines appear on the same line—and that's where my misconception came from.
Let me reconsider using as a reference "the extension direction of X-cross spokes emerging from a hub flange hole when that hole is positioned at the vertex phase."
Below, I'll discuss the direction of the spokes colored red.
I'm using counter spokes because it's easier to see the spoke head.
For radial lacing, it looks like this:
It's perpendicular to the horizontal plane (ground), so it's 90°.
With a certain 32-hole 4-cross lacing, it's about this angle.
Even though it's called "4-cross," the angle isn't constant—it varies with flange diameter, rim depth,
and most importantly, the number of spokes.
In an extreme case, with something like 1000 holes,
the angle would become nearly the same as radial lacing.
With a certain 32-hole 8-cross lacing, it's about this angle.
32-hole 8-cross is the most tangential lacing, but
it's only slightly less than horizontal.
When illustrated together, they look like this:
"The angles are not the same."
Next, I'll consider using "when the spoke tension direction is straight up" as the reference.
Regarding spoke tension direction: for radial lacing it's the spoke extension direction itself,
while for tangent lacing it's the resultant direction of the two spokes' final cross.
If we assume a 24-hole hub and don't consider crosses beyond the hub bisector,
tangent lacing can be 2, 4, or 6-cross.
Here I've added 0-cross (radial lacing).
Since we're considering the spoke tension direction as the reference,
only 0-cross has a different hub hole phase.
When I translate that into a diagram like this...
it looks like this.
"The angles are not the same."
In the comment, someone wrote something to the effect of "radial lacing has a blunt angle, and most tangential lacing has a sharp angle"—
to express that idea,
I think the second diagram here is better than the first one.
Your comment was very helpful.
Thank you very much.
My policy is generally not to change or delete past articles except for typos,
so I'll link this correction article from the original post.
As a side note, the angle also changes between drive-side and non-drive-side spokes.
This applies to radial lacing, but with tangent lacing, you typically weave both drive-side and non-drive-side spokes, so which spoke is further outside the wheel
changes depending on position.
When the wheel is laid flat and the outer side is described as "up,"
at the hub flange area, the drive-side spoke is up, but
at the spoke crossing (when woven), the non-drive-side spoke is on top.
However, at the nipple area, I think the top-bottom relationship converges.
The angles are the same, or nearly so.
Also, regarding the comment "spoke length has never been mentioned," that's because it touches on a Meshino Taneko Code.
It's not the spoke length itself that's the issue, but there's "something" that would be noticed if I wrote it—
something I can't reveal. Watch out, watch out!