Re: [math-fun] How many spokes does a bicycle need?
This seems to be the extent of Amy C. Edmondson's discussion of the minimum number of spokes needed to fix the hub of a bicycle wheel relative to its rim -- sans 4 illustrations -- after Buckminster Fuller. The idea of reducing the degrees of freedom to 0 makes sense. But I can't say I follow her argument, which seems thoroughly non-rigorous, and especially mentions of "negative" degrees of freedom. (In fact, there are indeed 6 degrees of freedom (in the usual sense) for the position and orientation of the hub relative to the rim, which would suggest the possibility that only 6 spokes might work -- if their constraints were independent. Which may explain why Coxeter is cited as claiming that 7 is the correct minimum. What happens if -- as in one of Edmondson's illustrations -- 3 spokes are on one side of the hub at 120 degrees apart, and another three in the same configuration on the other side of the hub . . . but rotated 60 degrees with respect to the first 3 spokes? I don't see intuitively why that won't work, though both books agree that it won't. --Dan << Degrees of Freedom The subject of twelve fundamental directions of symmetry, with their six natural positive-negative pairs, leads directly to a discussion of the "twelve degrees of freedom" inherent in space. The term is almost self-descriptive, but can be best explained in reverse. That is, we explore the number of degrees of freedom inherent in space (and thus affecting every system) in terms of how many restraining forces are necessary to completely inhibit a system's motion. What is the minimum number of applied forces necessary to anchor a body in space? Fig. 7-9 Click on thumbnail for larger image. Again, we can start with a planar analogy. Imagine a fiat circular disk, such as a coaster, lying on a table and held in place by two taut strings pulling in opposite directions. The disk looks stable, but actually is free to move back and forth, at 90 degrees to the line of the two restraints (Fig. 7-9a). So, we try applying three tension forces, 120 degrees apart (Fig. 7-9a), and observe that the circle's position is fixed. Actually, it turns out that only the location of the exact center of the circle is fixed, for the disk is free to rotate slightly in place. Because rotation involves motion directed at 90 degrees to all three strings, there is nothing to restrain the circle from twisting back and forth, as shown in Figure 7-9b. Three additional strings to counteract each of the original restraints would have to be added to prevent all motion, for a total of three positive and three negative vectors. Fig. 7-10. Minimum of twelve spokes needed for stability (a) wheel with six spokes (b) wheel with twelve spokes. Click on thumbnail for larger image. In space, a similar procedure involves a bicycle wheel. Suppose that our goal is to anchor the hub with a minimum of spokes. At first glance this may appear to be the same problem as the previous planar example; however, in this case, the hub has both width and length. Both circular ends of the narrow hub—typically about a half inch wide and 3 inches long—must be stabilized. With only six spokes attaching the hub to the rim (three fixing the position of each end, as shown in Figure 7-l0a), the system feels quite rigid; force can be applied to the hub from any direction—up, down, back or forth—without budging it. However, the hub has no resistence to an applied torque, the effect of which occurs at ninety degrees to the spokes, and is therefore able to twist slightly about its long axis. Three more spokes at each end, to counterbalance the original six, remove the remaining flexibility. All twelve degrees of freedom are finally accounted for, with a minimum of twelve spokes (Fig. 7-l0b). This experiment is quite rewarding to experience—well worth trying for yourself. You don't need to go as far as dismantling a bicycle wheel; just find a hoop of any material and size and a short dowel segment, and then connect the two with radial strings added one at a time until the hub suddenly becomes rigidly restrained. It is enormously satisfying to feel the hub become absolutely immobile (all "freedom" taken away), with the surprisingly low number of twelve spokes. (4) What both the planar and spatial procedures indicate is that degrees of freedom are both positive and negative. In anchoring the hub of the bicycle wheel, there at first appear to be six degrees of freedom; however, each has a positive and negative direction. In conclusion, degrees of freedom measure the extent of a system's mobility: how many alternative directions of motion must be impeded before the body in space is completely restrained. (5)
Those who sleep faster are more rested sooner.
I don't follow the ACE argument at all, I'm afraid. Since motion of a rigid body in 3-space has freedom 6, 3 spokes at each end of the hub suffice to fix it with respect to the rim: they remain in tension since each triplet opposes the other. This arrangement is mechanically unsound, since torque applied to the hub is not transmitted to the rim. To counter this the spokes may be lengthened, causing them to spiral (say) clockwise: now a fourth spoke on each side spiralling anticlockwise stabilises the torque. More elegantly (and practically), the 4 spokes may be re-arranged in opposing pairs. The above argument shows that 6 are necessary and 8 sufficient: off the top of the head, I don't see how to improve either figure to 7. Fairly recently, a fashion emerged for bicycle wheels having only a few thickened, purely radial spokes: how well these lasted in practice I don't know, but in principle one might manage using only 6 such ... Fred Lunnon On 10/23/10, Dan Asimov <dasimov@earthlink.net> wrote:
This seems to be the extent of Amy C. Edmondson's discussion of the minimum number of spokes needed to fix the hub of a bicycle wheel relative to its rim -- sans 4 illustrations -- after Buckminster Fuller.
The idea of reducing the degrees of freedom to 0 makes sense. But I can't say I follow her argument, which seems thoroughly non-rigorous, and especially mentions of "negative" degrees of freedom.
(In fact, there are indeed 6 degrees of freedom (in the usual sense) for the position and orientation of the hub relative to the rim, which would suggest the possibility that only 6 spokes might work -- if their constraints were independent. Which may explain why Coxeter is cited as claiming that 7 is the correct minimum.
What happens if -- as in one of Edmondson's illustrations -- 3 spokes are on one side of the hub at 120 degrees apart, and another three in the same configuration on the other side of the hub . . . but rotated 60 degrees with respect to the first 3 spokes? I don't see intuitively why that won't work, though both books agree that it won't.
--Dan
<< Degrees of Freedom
The subject of twelve fundamental directions of symmetry, with their six natural positive-negative pairs, leads directly to a discussion of the "twelve degrees of freedom" inherent in space. The term is almost self-descriptive, but can be best explained in reverse. That is, we explore the number of degrees of freedom inherent in space (and thus affecting every system) in terms of how many restraining forces are necessary to completely inhibit a system's motion. What is the minimum number of applied forces necessary to anchor a body in space?
Fig. 7-9 Click on thumbnail for larger image.
Again, we can start with a planar analogy. Imagine a fiat circular disk, such as a coaster, lying on a table and held in place by two taut strings pulling in opposite directions. The disk looks stable, but actually is free to move back and forth, at 90 degrees to the line of the two restraints (Fig. 7-9a). So, we try applying three tension forces, 120 degrees apart (Fig. 7-9a), and observe that the circle's position is fixed. Actually, it turns out that only the location of the exact center of the circle is fixed, for the disk is free to rotate slightly in place. Because rotation involves motion directed at 90 degrees to all three strings, there is nothing to restrain the circle from twisting back and forth, as shown in Figure 7-9b. Three additional strings to counteract each of the original restraints would have to be added to prevent all motion, for a total of three positive and three negative vectors.
Fig. 7-10. Minimum of twelve spokes needed for stability (a) wheel with six spokes (b) wheel with twelve spokes. Click on thumbnail for larger image.
In space, a similar procedure involves a bicycle wheel. Suppose that our goal is to anchor the hub with a minimum of spokes. At first glance this may appear to be the same problem as the previous planar example; however, in this case, the hub has both width and length. Both circular ends of the narrow hub—typically about a half inch wide and 3 inches long—must be stabilized. With only six spokes attaching the hub to the rim (three fixing the position of each end, as shown in Figure 7-l0a), the system feels quite rigid; force can be applied to the hub from any direction—up, down, back or forth—without budging it. However, the hub has no resistence to an applied torque, the effect of which occurs at ninety degrees to the spokes, and is therefore able to twist slightly about its long axis. Three more spokes at each end, to counterbalance the original six, remove the remaining flexibility. All twelve degrees of freedom are finally accounted for, with a minimum of twelve spokes (Fig. 7-l0b).
This experiment is quite rewarding to experience—well worth trying for yourself. You don't need to go as far as dismantling a bicycle wheel; just find a hoop of any material and size and a short dowel segment, and then connect the two with radial strings added one at a time until the hub suddenly becomes rigidly restrained. It is enormously satisfying to feel the hub become absolutely immobile (all "freedom" taken away), with the surprisingly low number of twelve spokes. (4)
What both the planar and spatial procedures indicate is that degrees of freedom are both positive and negative. In anchoring the hub of the bicycle wheel, there at first appear to be six degrees of freedom; however, each has a positive and negative direction. In conclusion, degrees of freedom measure the extent of a system's mobility: how many alternative directions of motion must be impeded before the body in space is completely restrained. (5)
Those who sleep faster are more rested sooner.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Update: 7 spokes are sufficient --- 3 radial on one side of the wheel, 2 pairs of spiralling on the other side (preferably next to the rear cogs!). But are 7 also necessary, to transmit torque under tension? WFL On 10/24/10, Fred lunnon <fred.lunnon@gmail.com> wrote:
I don't follow the ACE argument at all, I'm afraid.
Since motion of a rigid body in 3-space has freedom 6, 3 spokes at each end of the hub suffice to fix it with respect to the rim: they remain in tension since each triplet opposes the other.
This arrangement is mechanically unsound, since torque applied to the hub is not transmitted to the rim. To counter this the spokes may be lengthened, causing them to spiral (say) clockwise: now a fourth spoke on each side spiralling anticlockwise stabilises the torque.
More elegantly (and practically), the 4 spokes may be re-arranged in opposing pairs. The above argument shows that 6 are necessary and 8 sufficient: off the top of the head, I don't see how to improve either figure to 7.
Fairly recently, a fashion emerged for bicycle wheels having only a few thickened, purely radial spokes: how well these lasted in practice I don't know, but in principle one might manage using only 6 such ...
Fred Lunnon
On 10/23/10, Dan Asimov <dasimov@earthlink.net> wrote:
This seems to be the extent of Amy C. Edmondson's discussion of the minimum number of spokes needed to fix the hub of a bicycle wheel relative to its rim -- sans 4 illustrations -- after Buckminster Fuller.
The idea of reducing the degrees of freedom to 0 makes sense. But I can't say I follow her argument, which seems thoroughly non-rigorous, and especially mentions of "negative" degrees of freedom.
(In fact, there are indeed 6 degrees of freedom (in the usual sense) for the position and orientation of the hub relative to the rim, which would suggest the possibility that only 6 spokes might work -- if their constraints were independent. Which may explain why Coxeter is cited as claiming that 7 is the correct minimum.
What happens if -- as in one of Edmondson's illustrations -- 3 spokes are on one side of the hub at 120 degrees apart, and another three in the same configuration on the other side of the hub . . . but rotated 60 degrees with respect to the first 3 spokes? I don't see intuitively why that won't work, though both books agree that it won't.
--Dan
<< Degrees of Freedom
The subject of twelve fundamental directions of symmetry, with their six natural positive-negative pairs, leads directly to a discussion of the "twelve degrees of freedom" inherent in space. The term is almost self-descriptive, but can be best explained in reverse. That is, we explore the number of degrees of freedom inherent in space (and thus affecting every system) in terms of how many restraining forces are necessary to completely inhibit a system's motion. What is the minimum number of applied forces necessary to anchor a body in space?
Fig. 7-9 Click on thumbnail for larger image.
Again, we can start with a planar analogy. Imagine a fiat circular disk, such as a coaster, lying on a table and held in place by two taut strings pulling in opposite directions. The disk looks stable, but actually is free to move back and forth, at 90 degrees to the line of the two restraints (Fig. 7-9a). So, we try applying three tension forces, 120 degrees apart (Fig. 7-9a), and observe that the circle's position is fixed. Actually, it turns out that only the location of the exact center of the circle is fixed, for the disk is free to rotate slightly in place. Because rotation involves motion directed at 90 degrees to all three strings, there is nothing to restrain the circle from twisting back and forth, as shown in Figure 7-9b. Three additional strings to counteract each of the original restraints would have to be added to prevent all motion, for a total of three positive and three negative vectors.
Fig. 7-10. Minimum of twelve spokes needed for stability (a) wheel with six spokes (b) wheel with twelve spokes. Click on thumbnail for larger image.
In space, a similar procedure involves a bicycle wheel. Suppose that our goal is to anchor the hub with a minimum of spokes. At first glance this may appear to be the same problem as the previous planar example; however, in this case, the hub has both width and length. Both circular ends of the narrow hub—typically about a half inch wide and 3 inches long—must be stabilized. With only six spokes attaching the hub to the rim (three fixing the position of each end, as shown in Figure 7-l0a), the system feels quite rigid; force can be applied to the hub from any direction—up, down, back or forth—without budging it. However, the hub has no resistence to an applied torque, the effect of which occurs at ninety degrees to the spokes, and is therefore able to twist slightly about its long axis. Three more spokes at each end, to counterbalance the original six, remove the remaining flexibility. All twelve degrees of freedom are finally accounted for, with a minimum of twelve spokes (Fig. 7-l0b).
This experiment is quite rewarding to experience—well worth trying for yourself. You don't need to go as far as dismantling a bicycle wheel; just find a hoop of any material and size and a short dowel segment, and then connect the two with radial strings added one at a time until the hub suddenly becomes rigidly restrained. It is enormously satisfying to feel the hub become absolutely immobile (all "freedom" taken away), with the surprisingly low number of twelve spokes. (4)
What both the planar and spatial procedures indicate is that degrees of freedom are both positive and negative. In anchoring the hub of the bicycle wheel, there at first appear to be six degrees of freedom; however, each has a positive and negative direction. In conclusion, degrees of freedom measure the extent of a system's mobility: how many alternative directions of motion must be impeded before the body in space is completely restrained. (5)
Those who sleep faster are more rested sooner.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On 10/24/10, Fred lunnon <fred.lunnon@gmail.com> wrote:
Update: 7 spokes are sufficient --- 3 radial on one side of the wheel, 2 pairs spiralling on the other side (preferably next to the rear cogs!).
But are 7 also necessary, to transmit torque under tension? WFL
Suppose we idealise the wheel rim as a large square S [occasionally manifested in actual hardware!], the spokes as lengths of string [compliant in flection or compression], the hub as a small equilateral triangle T. The centres of S and T coincide, their planes are perpendicular; and one edge E of T is parallel to two opposite edges of S.
From each end of E in T, two strings run straight to nearby corners in S; from the third corner of T, two strings run straight to mid-points of edges parallel to E in S. Under tension, the motion of T wrt S is restricted by the former four strings to rotation about E; the tension is maintained, and the rotation prevented, by the latter two strings.
So pace Fuller and Coxeter, 6 spokes are both necessary and sufficient! The discrepancy between this result and Fuller's / Edmondson's 12 makes one wonder whether we're all discussing the same problem ... There is a minor difficulty with the model above, in that it assumes two spokes may attach to the same point of the hub; however, I doubt this would invalidate the result. Maybe somebody out there could run up a hardware implementation, just to check? Or we could track down a wheel-builder --- all the ones I used to know are alas long gone ... Fred Lunnon
On 10/24/10, Fred lunnon <fred.lunnon@gmail.com> wrote:
I don't follow the ACE argument at all, I'm afraid.
Since motion of a rigid body in 3-space has freedom 6, 3 spokes at each end of the hub suffice to fix it with respect to the rim: they remain in tension since each triplet opposes the other.
This arrangement is mechanically unsound, since torque applied to the hub is not transmitted to the rim. To counter this the spokes may be lengthened, causing them to spiral (say) clockwise: now a fourth spoke on each side spiralling anticlockwise stabilises the torque.
More elegantly (and practically), the 4 spokes may be re-arranged in opposing pairs. The above argument shows that 6 are necessary and 8 sufficient: off the top of the head, I don't see how to improve either figure to 7.
Fairly recently, a fashion emerged for bicycle wheels having only a few thickened, purely radial spokes: how well these lasted in practice I don't know, but in principle one might manage using only 6 such ...
Fred Lunnon
On 10/23/10, Dan Asimov <dasimov@earthlink.net> wrote:
This seems to be the extent of Amy C. Edmondson's discussion of the minimum number of spokes needed to fix the hub of a bicycle wheel relative to its rim -- sans 4 illustrations -- after Buckminster Fuller.
The idea of reducing the degrees of freedom to 0 makes sense. But I can't say I follow her argument, which seems thoroughly non-rigorous, and especially mentions of "negative" degrees of freedom.
(In fact, there are indeed 6 degrees of freedom (in the usual sense) for the position and orientation of the hub relative to the rim, which would suggest the possibility that only 6 spokes might work -- if their constraints were independent. Which may explain why Coxeter is cited as claiming that 7 is the correct minimum.
What happens if -- as in one of Edmondson's illustrations -- 3 spokes are on one side of the hub at 120 degrees apart, and another three in the same configuration on the other side of the hub . . . but rotated 60 degrees with respect to the first 3 spokes? I don't see intuitively why that won't work, though both books agree that it won't.
--Dan
<< Degrees of Freedom
The subject of twelve fundamental directions of symmetry, with their six natural positive-negative pairs, leads directly to a discussion of the "twelve degrees of freedom" inherent in space. The term is almost self-descriptive, but can be best explained in reverse. That is, we explore the number of degrees of freedom inherent in space (and thus affecting every system) in terms of how many restraining forces are necessary to completely inhibit a system's motion. What is the minimum number of applied forces necessary to anchor a body in space?
Fig. 7-9 Click on thumbnail for larger image.
Again, we can start with a planar analogy. Imagine a fiat circular disk, such as a coaster, lying on a table and held in place by two taut strings pulling in opposite directions. The disk looks stable, but actually is free to move back and forth, at 90 degrees to the line of the two restraints (Fig. 7-9a). So, we try applying three tension forces, 120 degrees apart (Fig. 7-9a), and observe that the circle's position is fixed. Actually, it turns out that only the location of the exact center of the circle is fixed, for the disk is free to rotate slightly in place. Because rotation involves motion directed at 90 degrees to all three strings, there is nothing to restrain the circle from twisting back and forth, as shown in Figure 7-9b. Three additional strings to counteract each of the original restraints would have to be added to prevent all motion, for a total of three positive and three negative vectors.
Fig. 7-10. Minimum of twelve spokes needed for stability (a) wheel with six spokes (b) wheel with twelve spokes. Click on thumbnail for larger image.
In space, a similar procedure involves a bicycle wheel. Suppose that our goal is to anchor the hub with a minimum of spokes. At first glance this may appear to be the same problem as the previous planar example; however, in this case, the hub has both width and length. Both circular ends of the narrow hub—typically about a half inch wide and 3 inches long—must be stabilized. With only six spokes attaching the hub to the rim (three fixing the position of each end, as shown in Figure 7-l0a), the system feels quite rigid; force can be applied to the hub from any direction—up, down, back or forth—without budging it. However, the hub has no resistence to an applied torque, the effect of which occurs at ninety degrees to the spokes, and is therefore able to twist slightly about its long axis. Three more spokes at each end, to counterbalance the original six, remove the remaining flexibility. All twelve degrees of freedom are finally accounted for, with a minimum of twelve spokes (Fig. 7-l0b).
This experiment is quite rewarding to experience—well worth trying for yourself. You don't need to go as far as dismantling a bicycle wheel; just find a hoop of any material and size and a short dowel segment, and then connect the two with radial strings added one at a time until the hub suddenly becomes rigidly restrained. It is enormously satisfying to feel the hub become absolutely immobile (all "freedom" taken away), with the surprisingly low number of twelve spokes. (4)
What both the planar and spatial procedures indicate is that degrees of freedom are both positive and negative. In anchoring the hub of the bicycle wheel, there at first appear to be six degrees of freedom; however, each has a positive and negative direction. In conclusion, degrees of freedom measure the extent of a system's mobility: how many alternative directions of motion must be impeded before the body in space is completely restrained. (5)
Those who sleep faster are more rested sooner.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On 10/24/10, Fred lunnon <fred.lunnon@gmail.com> wrote:
... So pace Fuller and Coxeter, 6 spokes are both necessary and sufficient! The discrepancy between this result and Fuller's / Edmondson's 12 makes one wonder whether we're all discussing the same problem ...
There is a minor difficulty with the model above, in that it assumes two spokes may attach to the same point of the hub; however, I doubt this would invalidate the result. Maybe somebody out there could run up a hardware implementation, just to check? Or we could track down a wheel-builder --- all the ones I used to know are alas long gone ...
The 8 spoke arrangement suggested earlier has been verified experimentally: http://bobwb.tripod.com/synergetics/photos/bikerim.html WFL
participants (2)
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Dan Asimov -
Fred lunnon