Looking at a Dura-Ace 12-speed sprocket, I got to thinking: if you kept making sprockets bigger in 1T increments without limit, what would happen to the ridge line traced by the sprocket teeth?
↑12-speed sprocket (front double setup)
The tooth counts are 13-15-16-17-18-19T. It's unusual to skip a tooth on the top end, but there's a reason for it. But I won't explain.
Anyway, from 15T to 19T it's a 1T-increment setup, and when you connect these tooth tips together, they form a straight line—or so I believe.
In the diagram above I've only drawn up to the 7th stage, but the line connecting the tooth tips (blue line) should remain straight no matter how large the sprocket gets.
Let's say the thickness of a gear is 1.65mm and the spacer is 2.35mm. That's 4mm total. At 250,000T you'd get exactly 1,000,000mm, or 1km. If you don't like it ending with a spacer, you could do 250,001T for 1,000,001.65mm.
Now, if a 1000m-tall sprocket appeared in the middle of town, how would it look? I think it would appear pyramid-shaped like in the drawing above, but
some people insist it would be tower-shaped like this. Whether it actually becomes this shape at 1000m is beside the point—apparently it eventually takes on this form. Is that really so?
First, about the sprocket tooth tips. Sprocket teeth are machined so shifting is smooth, and the distance to the tooth tip isn't constant. In the case of SRAM, the tooth design is such that one tooth is essentially missing.
This makes it impossible to define tooth tip positions consistently. So I'm going to establish a definition for "tooth tip" here.
Whether it's for a mama-chari or an 11-speed bike, the center-to-center pitch of chain rollers is standardized. Two links equal one inch (25.4mm), one link equals 1/2 inch (12.7mm).
～Digression～
Long ago, there was a Dura-Ace 10 groupset for track bikes, and
this is the lock ring remover for it. The "10" comes from the chain pitch being 10mm.
It's completely different compared to 12.7mm pitch. Such exceptions do exist.
↑Please overlook how crude the diagram is (laughs). The diagram shows a 9T tooth, but when the chain (roller section) falls cleanly between sprocket teeth and you draw a line connecting the roller centers, for an n-tooth sprocket it forms a regular n-gon. I'm defining the vertices of this regular polygon as "tooth tips," and calling the distance from the sprocket center to the vertex r as the "distance to tooth tip."
Multi-stage sprockets shift the tooth tip phase gradually to improve shifting performance.
Here I'm considering cases where the phase is aligned at one point even as sprockets increase in number.
～Another digression～
If someone asks what the most cross-ratio tooth count is for Shimano 10-speed, how would you answer? 12-21T? That might be the right answer, but
16-16-16-16-16-16-16-16-16-16T
is also a combination that exists.
It's got alternative names like "Death Corn" and "Death Face Massager" (I just made those up).
Making it taught me something different from its intended purpose.
That is: "When tooth tip phases are aligned, shifting becomes terribly poor." I'm sure the fact that neighboring gears are the same size is one cause, but the rear derailleur's slant angle becomes equally shocking with this setup.
Before moving to verification, let me present the argument from the "it'll become tower-shaped" camp.
The front gear can be any number of teeth, but when the rear shifts from 11T to 12T, it gets about 10% easier. Even with the same 1T shift, if it were 100T to 101T, it only gets about 1% easier. Furthermore, with 10,000T to 10,001T, it only gets about 0.01% easier.
When you graph gear lightness, it should look something like this. Even shifting from 100 million teeth to 100 million 1 tooth, the gear ratio barely changes.
By drawing a line mirror-symmetric to that curve, this shape is supposedly what a sprocket looks like when tooth count is sufficiently large.
So I've diagrammed the newly defined "distance to tooth tip r."
Now I need to compare the difference between r of n-tooth and r of (n+1)-tooth (the red dotted line in the diagram above). If it becomes tower-shaped, then for sufficiently large n, ((n+1)T's r) - (n-T's r) should approach 0.
The center of the chain rollers, which I've called the tooth tip, forms a regular polygon when connected. If that regular polygon's circumradius is r and its side length is 12.7mm, calling it a, then
↑you can find r using this formula. Below I'll calculate and write out the difference in r between n and n+1 (units in mm).
n=11
11T's r 22.5391061３
12T's r 24.5345159９
Difference 1.99540986
n=12
12T's r 24.5345159９
13T's r 26.5339923３
Difference 1.99947634
n=13
13T's r 26.5339923３
14T's r 28.5366409７
Difference 2.00264864
Hmm? The numbers aren't matching... which means the ridge line is definitely not straight. Now let me try much larger numbers for n.
n=10,000
10,000T's r 202,212.678105１5
10,001T's r 202,214.699372８9
Difference 2.021267774
n=1 trillion
1 trillion T's r 2,021,267,777,267.070764２6
1 trillion 1 T's r 2,021,267,777,269.092032０4
Difference 2.021267778
... This gives me a sense of the shape. From 10,000T onward, one tooth gains almost the same amount of r growth as from 1 trillion T. When n gets sufficiently large, what converges to 0 is not "r" itself, but rather "the difference in r between n and n+1." And since that difference varies slightly at each step, the ridge line isn't truly straight, but it's fair to say it appears almost straight. With smaller tooth counts, the difference between n and n+1 is larger, but this difference grows as the regular polygon is further from a perfect circle.
↑There's no such thing as a 6-tooth, but there is "surplus area" between a regular hexagon and its circumcircle. The larger this surplus area, the larger the difference between r of n-T and r of (n+1)-T.
↑This says "regular 6 million-gon," but at this point it's virtually a circle.
Since the growth in r per tooth around 10,000T is nearly identical to growth around 1 trillion T,
I draw a straight line from the 10,000T tooth tip to the 1 trillion 1-T tooth tip. This isn't the true ridge line, but it's close to it.
When you extend that line all the way to the top gear, it looks something like this (exaggerated).
Even so, if someone says
"1 trillion T isn't large enough to be called 'sufficiently large' tooth count"
"Far beyond 1 trillion T, somewhere down the line, it will eventually tower-ify"...
If r's growth per tooth eventually converges to 0, imagine a sufficiently large tooth count n where it's nearly converged. The diameter at 10n-T is about 10 times the height from 11T to n-T, but think about it: "The chain pitch is fixed at 12.7mm and the gear count differs 10-fold, yet the gear outer diameter would be nearly the same? That doesn't make sense?" If you think about that, you can imagine that tower shape isn't going to happen. "You should've just said that from the start!" you might say, but I didn't calculate r initially because I thought the ridge line was straight—so it wasn't a waste.
↑12-speed sprocket (front double setup)
The tooth counts are 13-15-16-17-18-19T. It's unusual to skip a tooth on the top end, but there's a reason for it. But I won't explain.
Anyway, from 15T to 19T it's a 1T-increment setup, and when you connect these tooth tips together, they form a straight line—or so I believe.
In the diagram above I've only drawn up to the 7th stage, but the line connecting the tooth tips (blue line) should remain straight no matter how large the sprocket gets.
Let's say the thickness of a gear is 1.65mm and the spacer is 2.35mm. That's 4mm total. At 250,000T you'd get exactly 1,000,000mm, or 1km. If you don't like it ending with a spacer, you could do 250,001T for 1,000,001.65mm.
Now, if a 1000m-tall sprocket appeared in the middle of town, how would it look? I think it would appear pyramid-shaped like in the drawing above, but
some people insist it would be tower-shaped like this. Whether it actually becomes this shape at 1000m is beside the point—apparently it eventually takes on this form. Is that really so?
First, about the sprocket tooth tips. Sprocket teeth are machined so shifting is smooth, and the distance to the tooth tip isn't constant. In the case of SRAM, the tooth design is such that one tooth is essentially missing.
This makes it impossible to define tooth tip positions consistently. So I'm going to establish a definition for "tooth tip" here.
Whether it's for a mama-chari or an 11-speed bike, the center-to-center pitch of chain rollers is standardized. Two links equal one inch (25.4mm), one link equals 1/2 inch (12.7mm).
～Digression～
Long ago, there was a Dura-Ace 10 groupset for track bikes, and
this is the lock ring remover for it. The "10" comes from the chain pitch being 10mm.
It's completely different compared to 12.7mm pitch. Such exceptions do exist.
↑Please overlook how crude the diagram is (laughs). The diagram shows a 9T tooth, but when the chain (roller section) falls cleanly between sprocket teeth and you draw a line connecting the roller centers, for an n-tooth sprocket it forms a regular n-gon. I'm defining the vertices of this regular polygon as "tooth tips," and calling the distance from the sprocket center to the vertex r as the "distance to tooth tip."
Multi-stage sprockets shift the tooth tip phase gradually to improve shifting performance.
Here I'm considering cases where the phase is aligned at one point even as sprockets increase in number.
～Another digression～
If someone asks what the most cross-ratio tooth count is for Shimano 10-speed, how would you answer? 12-21T? That might be the right answer, but
16-16-16-16-16-16-16-16-16-16T
is also a combination that exists.
It's got alternative names like "Death Corn" and "Death Face Massager" (I just made those up).
Making it taught me something different from its intended purpose.
That is: "When tooth tip phases are aligned, shifting becomes terribly poor." I'm sure the fact that neighboring gears are the same size is one cause, but the rear derailleur's slant angle becomes equally shocking with this setup.
Before moving to verification, let me present the argument from the "it'll become tower-shaped" camp.
The front gear can be any number of teeth, but when the rear shifts from 11T to 12T, it gets about 10% easier. Even with the same 1T shift, if it were 100T to 101T, it only gets about 1% easier. Furthermore, with 10,000T to 10,001T, it only gets about 0.01% easier.
When you graph gear lightness, it should look something like this. Even shifting from 100 million teeth to 100 million 1 tooth, the gear ratio barely changes.
By drawing a line mirror-symmetric to that curve, this shape is supposedly what a sprocket looks like when tooth count is sufficiently large.
So I've diagrammed the newly defined "distance to tooth tip r."
Now I need to compare the difference between r of n-tooth and r of (n+1)-tooth (the red dotted line in the diagram above). If it becomes tower-shaped, then for sufficiently large n, ((n+1)T's r) - (n-T's r) should approach 0.
The center of the chain rollers, which I've called the tooth tip, forms a regular polygon when connected. If that regular polygon's circumradius is r and its side length is 12.7mm, calling it a, then
↑you can find r using this formula. Below I'll calculate and write out the difference in r between n and n+1 (units in mm).
n=11
11T's r 22.5391061３
12T's r 24.5345159９
Difference 1.99540986
n=12
12T's r 24.5345159９
13T's r 26.5339923３
Difference 1.99947634
n=13
13T's r 26.5339923３
14T's r 28.5366409７
Difference 2.00264864
Hmm? The numbers aren't matching... which means the ridge line is definitely not straight. Now let me try much larger numbers for n.
n=10,000
10,000T's r 202,212.678105１5
10,001T's r 202,214.699372８9
Difference 2.021267774
n=1 trillion
1 trillion T's r 2,021,267,777,267.070764２6
1 trillion 1 T's r 2,021,267,777,269.092032０4
Difference 2.021267778
... This gives me a sense of the shape. From 10,000T onward, one tooth gains almost the same amount of r growth as from 1 trillion T. When n gets sufficiently large, what converges to 0 is not "r" itself, but rather "the difference in r between n and n+1." And since that difference varies slightly at each step, the ridge line isn't truly straight, but it's fair to say it appears almost straight. With smaller tooth counts, the difference between n and n+1 is larger, but this difference grows as the regular polygon is further from a perfect circle.
↑There's no such thing as a 6-tooth, but there is "surplus area" between a regular hexagon and its circumcircle. The larger this surplus area, the larger the difference between r of n-T and r of (n+1)-T.
↑This says "regular 6 million-gon," but at this point it's virtually a circle.
Since the growth in r per tooth around 10,000T is nearly identical to growth around 1 trillion T,
I draw a straight line from the 10,000T tooth tip to the 1 trillion 1-T tooth tip. This isn't the true ridge line, but it's close to it.
When you extend that line all the way to the top gear, it looks something like this (exaggerated).
Even so, if someone says
"1 trillion T isn't large enough to be called 'sufficiently large' tooth count"
"Far beyond 1 trillion T, somewhere down the line, it will eventually tower-ify"...
If r's growth per tooth eventually converges to 0, imagine a sufficiently large tooth count n where it's nearly converged. The diameter at 10n-T is about 10 times the height from 11T to n-T, but think about it: "The chain pitch is fixed at 12.7mm and the gear count differs 10-fold, yet the gear outer diameter would be nearly the same? That doesn't make sense?" If you think about that, you can imagine that tower shape isn't going to happen. "You should've just said that from the start!" you might say, but I didn't calculate r initially because I thought the ridge line was straight—so it wasn't a waste.