Conventional bicycle wheels are spoked using a semi-tangent, crossed spoking pattern. Each spoke crosses one or more other spokes on the same side of the wheel, and there are leading and trailing spokes at the hub. To transmit power, the leading spokes' tension decreases and the trailing spokes' tension increases; vice versa when a hub brake is applied.

If a wheel has a large hub, though, crossed spokes may approach the rim at a sharp angle which the spoke holes and spoke nipples cannot accommodate. This is a common problem with electric-bike hub motors.

On the other hand, with radial (cross zero) spoking, (spokes pointing directly outward from the hub), a wheel which transmits torque "winds up," as described in the main wheelbuilding article. Excess motion leads to spoke failure, and to the risk of pulling apart hub flanges due to excessive spoke tension. Wheels with large hubs are sometimes laced radially with extra-thick spokes, but still, spoke breakage has occurred.

Within limits, the problem of excessive spoke angles can be solved by reducing the cross number of a conventional spoking pattern. The wheel shown below is pushing the limit with a cross-one pattern, the most forgiving conventional pattern which can transmit torque . Even in this fully-tensioned wheel, the spokes are bending as they leave the spoke nipples.

Another approach has been to place leading spokes on one side of the wheel, and trailing spokes on the other, so the spokes can be more nearly radial. In this spoking pattern, none of the spokes on either side of the rim cross over each other, but spokes on opposite sides of the wheel do.

(Image from Golden Motor: fair use as commentary under U.S. copyright law)

This pattern converts the wind-up problem into other problems:

• Because all the spokes on one side of the wheel are trying to twist the hub one way, and all those on the other side, the other way, this pattern is only practical if the hub has a strong barrel between the flanges.
• Torque from pedaling or a hub brake loosens all of the spokes on one side of the wheel, and tightens all of the spokes on the other side, pulling the rim to one side. The rim will rub on a brake shoe of a rim brake, and the excess motion may lead to spoke failure.

What is to do?

An answer is found in any of several unconventional spoking patterns. All have cross numbers less than cross one but more than cross zero (radial): the angle of the spokes as they approach the rim is a fraction of what it would be for cross one. Cyclist and engineer Damon Rinard has described on this site how to calculate spoke lengths for fractional cross numbers, and provided a spoke-length calculator which allows these combinations. Several spoke patterns are described below, suitable for different rims and hubs.

Warning, though: All of these patterns use near-radial spokes, and some use hubs with high spoke counts. These patterns should only be used with large hubs with strong flanges. There is no reason anyway to use them with smaller hubs, and also, the resistance to torque may be too low with these patterns and small hubs.

Some of these patterns have all radial spokes on one side. These patterns should not be used when torque is applied at both sides (for example, with a freewheel on one side and a disc brake on the other) unless the hub barrel is stout and strong.

Also: the math for some of the patterns needs checking and testing by building wheels. Please recalculate before using the information here. sheldonbrown.com cannot be held responsible for incorrect spoke lengths.

The patterns are listed in the table below, with links to full descriptions later in this article. I suggest that you read through the first section below the table before selecting a pattern, to learn how these patterns work.

Type of pattern	Spoke holes in hub	Spoke holes in rim
5:3 ratio	40	24
4:3 ratio	32	24
	48	36
3:2 ratio	36	24
	48	32
5:4 ratio	40	32
9:7 ratio	36	28
2:1 ratio	48	24
Four more holes in hub than in rim	28	24
	32	28
	36	32
	40	36
Spokes paired at hub	See description	Any mult. of 4
Dished, 2 right:1 left	24 drilled to 48	36
	36 drilled to 72	36
	32 drilled to 64	48
	48 drilled to 96	48

Mismatched spoke counts are practical when the number of spokes in the hub and rim are in small ratios of whole numbers, so the spokes form a repeating pattern. In the patterns described in this article, the hub has more spoke holes than the rim. All of the spoke holes in the rim are used, to build a strong wheel. These patterns require no modification to the hub or rim. They all use more than one length of spokes.

With most of these patterns, if the rim is of opposite handedness to what is shown, the valve hole should go into the next gap, left or right, between groups of spokes.

5/3 ratio: 40-hole hub, 24-hole rim

A 40-hole hub and 24-hole rim offer a good example. In the drawing below, we are looking at the right side of the wheel. Spokes and spoke holes shown in white are on the right side of the wheel, and those in dashed blue, on the left side. The valve is shown in green. The left- and right-side spokes may be switched if the handedness (see explanation here) of the spoke holes in the rim would place the key spoke -- just to the right of the valve in the image -- on the left side at the rim.

 

With 5 spoke holes in the hub for every 3 in the rim, groups of 3 spokes at the hub are separated by two empty holes, one in each flange. Two holes at a time must be skipped so the next occupied spoke hole is on the opposite side from the previous one, as shown in the drawing.

The middle spoke of each group of 3 is radial, and the adjacent spokes are near-radial -- not quite radial, thanks to the smaller angle between spoke holes in the hub, than in the rim.

With this pattern, two of every three spokes -- all the non-radial ones -- transmit torque. A leading spoke on the right side is adjacent to a trailing spoke on the left side, and this combination pulls the rim to the left at that pair of spokes. The next pair of spokes which transmits torque pulls the rim to the right. Eight pairs of spokes in all transmit torque, and so the rim tends to develop four "waves", compared with the six it would have with conventional spoking. Somewhat greater rim stiffness is needed, then, to avoid the rim's zig-zagging from side to side under a torque load.

The fractional cross numbers for the near-radial spokes can be entered into Rinard's Spocalc spoke-length calculator. The resulting spoke length will be the same, whether calculated based on the 24-spoke rim or the 40-spoke hub.

The cross numbers are:

40/24	Radial	Near-radial
Count	8	16
Angle	0.000°	6.000°
X, 24	0.000	0.200
X, 40	0.000	0.333

4/3 ratio: 32/24, 48/36

Next, let's look at a wheel which has 4 spokes in the hub for every 3 in the rim. We'll work with a 32-hole hub and a 24-hole rim.

Because we must skip two spoke holes in a row in the hub, the spokes are in four groups of 5, 7, 5 and 7, as shown in the image below. The central spoke in each group is radial, and there are three different lengths of near-radial spokes. The odd number of spokes in each group cancels torque. If the spokes were in four groups of six, one flange would be tending to rotate the hub shell forward, and the other, backward, twisting the hub shell. Also torque from pedaling would pull the rim sideways, as described earlier.

Potential zigzagging of the rim is a bit irregular because of the different numbers of spokes in the groups, but there are four major waves. The greatest excursions are between the pairs of spokes which are farthest apart.

The angles and cross numbers for the three different spoke lengths are given in the table below.

32/24	dist. from midline of 5- or 7-spoke pattern
	radial	closest	middle	farthest
Count	4	8	8	4
Angle	0.000°	3.75°	7.5°	11.25°
X, 24	0.000	0.125	0.250	0.375
X, 32	0.000	0.167	0.333	0.500

48/36 also uses 4 spoke holes at the hub for every 3 at the rim.

48/36	dist. from midline of 5- or 7-spoke pattern
	radial	closest	middle	farthest
Count	6	12	12	6
Angle	0.000°	2.5°	5°	7.5°
X, 36	0.000	0.125	0.250	0.375
X, 48	0.000	0.167	0.333	0.500

With the 36 spokes, there are six waves. This pattern works with a conventional 48-spoke hub where the spoke holes are offset between the two flanges -- not with a 24-hole hub drilled with 24 additional holes. We have a pattern for that hub too, later in this article.

3/2 ratio: 36/24, 48/32

Combinations with 3 spokes in the hub to every 2 in the rim are also possible. Because we must skip two spoke holes at a time, we need alternating groups of 3 and 5 spokes to get 8 spokes in the rim for every 12 spoke holes in the hub. The uneven number of spokes in each group cancels torque on the hub shell, as described earlier. The drawing below shows the spoking pattern for a 36-hole hub and 24-hole rim.

The angles and cross numbers, calculated as in the previous examples, are

36/24	dist. from midline of pattern
	radial	middle	farthest
Count	6	12	6
Angle	0.00°	5.00°	10.00°
X, 24	0.000	0.167	0.333
X, 36	0.000	0.250	0.500

A 48-hole (conventional, not re-drilled 24-hole) hub and 32-hole rim also can use this pattern.

48/32	dist. from midline of pattern
	radial	middle	farthest
Count	8	16	8
Angle	0.00°	3.75°	7.50°
X, 32	0.000	0.167	0.333
X, 48	0.000	0.250	0.500

3 out of 4 spokes transmit torque, but those at the ends of the 5-spoke groups are tensioned or slackened more when transmitting torque. This difference is reduced somewhat by the flexibility of the rim, and these spokes are on the right side, where the spokes are under higher tension in a dished wheel and so less likely to go slack due to torque.

5/4 Ratio: 40/32

This laces a 32-hole rim to a 40-hole hub, as shown in the illustration below.

The spokes are in groups of 7 and 9, with a radial spoke at the center of each as has already been described above for the 4/3 ratio. Five different lengths of spokes are required:

40/32	dist. from midline of 7- or 9-spoke pattern
	radial	closest	second	third	fourth
Count	4	8	8	8	4
Angle	0.000°	1.125°	3.375°	5.625°	7.875°
X, 32	0.000	0.05	0.15	0.25	0.35
X, 40	0.000	0.062	0.187	0.313	0.438

As with the 32/24 pattern, there are four waves.

9/7 Ratio: 36/28

A 9/7 ratio laces a 28-spoke rim to a 36-spoke hub, as shown in the illustration below.

Each quadrant of the wheel has 7 spokes with a central, radial spoke. Four different spoke lengths are required. Angles and cross numbers are as in the table below.

36/28	dist. from midline of 7-spoke pattern
	radial	closest	middle	farthest
Count	4	8	8	8
Angle	0.00°	2.86°	5.71°	8.57°
X, 28	0.000	0.111	0.222	0.333
X, 36	0.000	0.143	0.286	0.429

With torque from pedaling or a hub brake, the two topmost spokes and the other three pairs which are spaced most widely pull the rim sideways the most, and in alternate directions.

2/1 Ratio: 48/24

A 2/1 ratio results in a pattern as shown in the image below. This is for a 48-spoke hub and 24-spoke rim. All of the left-side spokes are radial. If the rim is of the opposite handedness, the valve moves clockwise by 3 spokes, rather than reversing the sides for the spokes -- to keep the radial spokes on the left side.

Angles are as in the table below:

48/24	Left	Right
Count	12	12
Angle	0°	7.5°
X, 24	0.0	0.250
X, 48	0.0	0.500

 

Right-side spokes alternate between leading and trailing, and so the rim tends toward 6 waves, same as with conventional spoking. Tension changes on the leading and trailing spokes due to torque are like those on any half-radial wheel.

4 more spoke holes in the hub than in the rim

36-hole hubs and 32-hole rims are common. The solution shown below uses two groups of spokes, one with 17 spokes, the other with 15. This wheel looks odd because the two groups have different numbers of spokes, but as with some of the patterns already described, groups of different sizes are necessary to avoid twisting the hub barrel. Each group is symmetrical around a radial spoke at its center. The spoke pattern results in a very slight wheel imbalance, but less than is probably produced by the valve.

The illustration below is for a wheel with a 36-hole hub and 32-hole rim. This pattern theoretically requires nine different spoke lengths, but the length of the radial spokes and adjacent ones may be close enough that the same length may be used.

The same principle may be applied to other combinations where the hub has 4 more spoke holes than the rim. All common spoke counts are covered in the tables below.

28/24	Radial	Next					Last
Count	2	4	4	4	4	4	2
Angle	0°	2.14°	4.29°	6.43°	8.57°	10.71°	12.86°
X, 24	0.0	0.071	0.143	0.214	0.286	0.357	0.429
X, 28	0.0	0.083	0.166	0.250	0.333	0.417	0.500

32/28	Radial	Next						Last
Count	2	4	4	4	4	4	4	2
Angle	0°	1.61°	3.21°	4.82°	6.43°	8.04°	9.64	11.25°
X, 28	0.0	0.063	0.125	0.179	0.250	0.313	0.375	0.438
X, 32	0.0	0.071	0.143	0.214	0.286	0.357	0.429	0.500

36/32	Radial	Next							Last
Count	2	4	4	4	4	4	4	4	2
Angle	0°	1.25°	2.5°	3.75°	5°	6.25°	7.5°	8.75	10.0°
X, 32	0.0	0.056	0.111	0.167	0.222	0.278	0.333	0.389	0.444
X, 36	0.0	0.063	0.125	0.179	0.250	0.313	0.375	0.438	0.500

40/36	Radial	Next								Last
Count	2	4	4	4	4	4	4	4	4	2
Angle	0°	1.00°	2.00°	3.00°	4.00°	5.00°	6.00°	7.00°	8.00°	9.00°
X, 36	0.0	0.050	0.100	0.150	0.200	0.250	0.300	0.350	0.400	0.450
X, 40	0.0	0.056	0.111	0.167	0.222	0.278	0.333	0.389	0.444	0.500

People who frequent this Web site know that we are not great fans of "boutique" wheels with paired spokes at the rim. Pairing spokes at the hub, however, offers an elegant way to avoid excessive spoke angles with a large hub and conventional rim.

The spoking pattern is essentially a radial pattern, except that uneven spacing of the spoke holes at the hub produces leading and trailing spokes. This pattern can produce any desired cross number by adjustment of the location of spoke holes in the hub.

This pattern is especially useful because it starts out with a hub and rim which have the same number of spoke holes. This can be any multiple of 4, as it is with all common hubs and rims.

Let's start our exploration of this pattern by first considering a 36-spoke wheel, spoked radially. The spokes all point directly outward from the hub, as shown in the image below.

The wheel in the photo has radial spoking on a small hub. Our analysis for large hubs works best if the valve hole goes at zero degrees, halfway between the spokes at 355 and 5 degrees as shown in the table below.

Right flange			355				15				35				55				75				etc.
Spoke			|				|				|				|				|
Rim		345		355		5		15		25		35		45		55		65		75		85		etc.
Spoke	|				|				|				|				|				|
Left flange	345				5				25				45				65				85					etc.

Let's now drill new spoke holes in the hub, halfway in between the pre-drilled ones. This is only safe -- and only of any advantage anyway -- if the hub is very large so the spoke holes are far apart. The hub flanges must be meaty enough to withstand a near-radial pull. There will now be 36 holes in each flange, including holes directly in line with those in the opposite flange.

Now, we rotate the hub by 5 degrees, so the angles of the spoke holes in the hub are between those in the rim, and we spoke the wheel as shown in the table and drawing below. The two spokes nearest the top are now 20 degrees apart at the hub, at 350 and 10 degrees, but they are still 10 degrees apart at the rim. We pair the remaining spoke holes in the hub as shown in the table below.

Left flange			350				20				30				60				70				etc.
Spoke			\				/				\				/				\
Rim		345		355		5		15		25		35		45		55		65		75		85		etc.
Spoke	/				\				/				\				/				\
Right flange	340				10				20				50				60				90					etc.

The result is the pattern shown below.

36/72	Near-radial
Offset	10°
Count	36
Angle	5°
X, 36	0.250

If you drill the hub with new spoke holes halfway between the all of the original ones, rim handedness is never a problem: use whatever spoke holes give the correct handedness. If the spoke holes in the rim are close to the centerline, matching handedness probably doesn't matter anyway, because a near-radial pattern is forgiving of misalignment of the spoke nipples.

As noted already, this pattern will work with a conventional rim, as the pattern repeats every four spokes. In fact, the spoke alignment is the same as with the outer ends of spokes in a conventional crossed pattern. Here's the same pattern for a 24-hole rim, and showing only the holes which are used.

Left flange			345				30				45				90				105				etc.
Spoke			\				/				\				/				\
Rim		337.5		352.5		7.5		22.5		37.5		52.5		67.5		82.5		97.5		112.5		127.5		etc.
Spoke	/				\				/				\				/				\
Right flange	330				15				30				75				90				135					etc.

24/48	Near-radial
Offset	15°
Count	24
Angle	7.5°
X, 24	0.250

If the hub is pre-drilled with paired spoke holes, don't assume automatically that they will work with this pattern. The pairs on one flange need to be offset from those on the other, not directly opposite and not halfway in between the pairs on the other flange. The counterclockwise hole of a pair on one flange must be halfway between the clockwise holes of the adjacent pairs in the other flange (and vice versa). (Note how this is the case in the 24/48 drawing above. In each group of four spokes, the rightmost spoke goes to the left flange, and the leftmost one, to the right flange. The holes in the hub flanges for these spokes are evenly spaced.)

The photo below shows a 36-spoke wheel with paired spoke holes of the hub close together, producing a cross number greater than 0.25. Though the holes nearer the middle of the group are not directly opposite each other as in our earlier examples, the hub meets our requirement for the outermost spoke holes in each group, The handedness of this wheel places the pairs of left-side spokes counterclockwise of the right-side ones, because the first spoke hole clockwise of the valve hole is toward the left side of the rim. The valve is visible at the lower right in the photo.

Image from electricbicyclesmagazine.com

Once having determined that the positions of spoke holes in a pre-drilled hub is correct, you need to measure and calculate the cross number depending on the spacing between paired spokes in the hub:

Measure the angle or distance between the two spoke holes of a pair in one hub flange (holes which are closer together). We'll call that A.

Also measure the angle (or distance, that's accurate enough) between two holes which are farther apart. We'll call that B.

The cross number x is as

x = (B - 2A)/2B

So, we see, for example, for a purely radial 36-spoke wheel, A is 20 degrees and B is 40 degrees, so the cross number is

x = [40 - (2 * 20)]/(2 * 40) = 0

For our 36-hole wheel where we drilled additional spoke holes, so the holes are half as far apart as in a radial wheel, A is 10 degrees, so the cross number is

x = [40 - (2 * 10)]/(2 * 40) = 0.25

If the paired spoke holes in the hub are brought closer and closer together, they approach being directly on top of one another. Then A would be 0 degrees, and the cross number would be

x = [40 - (2 * 0)]/(2 * 40) = 0.5

And as a check on the math, for a normal cross 1 pattern, where A is -20 degrees,

x = [40 - (2 * -20])/(2 * 40) = 1

Most wheels with a sprocket cluster must be dished to center the rim. Dishing reduces the tension of the spokes on the left side of a conventional wheel which has the same number of spokes on both sides. If the left hub flange is about twice as far from the centerline as the right flange, as is common, using one spoke on the left for every two on the right will nearly equalize the tension. The rim must have a number of spoke holes divisible by 3. The practical, available choice is a 36-hole or 48-hole rim. A 24-hole rim would have only 8 spokes on the left, not enough with most rims.

The 36-hole rim works with a 36-hole hub drilled to have 72 spoke holes, 36 in each flange. There are then holes directly in line with one another in the two flanges. 24 spokes go to the the right flange of the hub, using paired spoking with spokes in 2 adjacent holes of every 3 in the right hub flange. The 12 spokes from the left flange alternate leading and trailing to avoid twisting the hub barrel. The difference in angle at the hub and rim for the spokes on both sides is 5 degrees, (compared with the 20 degrees of cross 1) so the spoke length is for a 0.25 cross, 36-spoke wheel. This pattern is shown in the image below. The spokes are in repeating groups of 6, so this pattern only works for hubs and rims with a multiple of 6 spokes. Rim handedness doesn't matter, because it will always be wrong for one of the two spokes closest to the valve.

36/72	Near-radial
Count	36
Angle	5°
X, 36	0.250

A 48-hole rim may also be used, similarly.

48/96	Near-radial
Count	48
Angle	3.75°
X, 48	0.250

The 2/1 pattern also works with a hub that starts with 2/3 as many spoke holes as the rim. You need to drill new spoke holes halfway between the original ones on both sides of the hub, and alternate leading and trailing spokes on the left side to avoid twisting the hub shell.. The image below shows a 36-spoke wheel with a 24-spoke hub modified by doubling the number of spoke holes. Rim handedness will always be wrong for some spokes.

36/48	Left	Right
Count	12	24
Angle	7.5°	2.5°
X, 36	0.375	0.125
X, 24	0.250	0.083

A 48-hole rim and 32-hole hub may also be used, similarly.

48/64	Left	Right
Count	16	32
Angle	5°	1.67°
X, 48	0.375	0.125
X, 32	0.250	0.083

With the pattern above, most of the torque is taken by the left-side spokes, which are fewer. Crossing the right-side spokes increases their cross number. The crossed spokes make the wheel slightly harder to build and require a different spoke length -- or, if the flange positions are just right, the higher cross number on the right and greater bracing angle at the left may result in all spokes' being of the same length. The wheel looks crossed and is less likely to scare a customer.

36/48	Left	Right
Count	12	24
Angle.	7.5°	12.5°
X, 36	0.375	0.625
X, 24	0.250	0.417

This pattern also may be used with a 48-spoke rim and a modified 32-spoke hub:

48/64	Left	Right
Count	16	32
Angle.	5°	8.33°
X, 48	0.375	0.625
X, 32	0.250	0.417

All of the 2 right/1 left patterns send some spokes to the "wrong" side of the rim, and so they work better if the left and right spoke holes in the rim are near the centerline.

A wheel with a large hub, especially with a small rim, may need spokes shorter than any which are available at most bicycle shops.

There are three main ways to solve this problem.

• Some bicycle shops have a machine to cut and thread spokes at custom lengths. The Phil Wood Spoke Cutting and Threading Machine is the most well-known of such machines.
• Vendors on eBay sell spokes in unusual or custom lengths. I have found a vendor who can supply spokes in any length from 75mm (just under 3 inches) up to the more usual sizes, using a Phil Wood Spoke Cutting and Threading Machine. There are also vendors on eBay from time to time who sell new old stock spokes in an assortment of lengths.
• If you are a manufacturer, the economy of scale works for you, and you order through your usual sources!