A Study of the geometry of bicycle wheel spoke lacing patterns

To begin with it will be helpeful to consider only the spokes on one side of the wheel, and to assume that the wheel has no dish - that is the spokes are all perpedicular to the axis of the wheel.

Below are some different spoke lacing patterns for one side of a 32 spoke wheel:

The radial pattern is simple. Each of the 16 spokes connects hub hole n to rim hole n. The same hub and rim hole numbering system will be used in the explanations of the other spoke lacing patterns

In the 1x pattern, spokes laced through hub hole n, where n is an odd number, are laced to the rim hole 1 place clockwise from rim hole n. A spoke laced through hub hole n, where n is even, will be laced through the rim hole one place anticlockwise from rim hole n.

In the 2x pattern, spokes laced through hub hole n, where n is an odd number, are laced to the rim hole 2 places clockwise from rim hole n. A spoke laced through hub hole n, where n is even, will be laced through the rim hole 2 places anticlockwise from rim hole n.

In the 3x pattern, spokes laced through hub hole n, where n is an odd number, are laced to the rim hole 3 places clockwise from rim hole n. A spoke laced through hub hole n, where n is even, will be laced through the rim hole 3 places anticlockwise from rim hole n.

The calculation of the spoke length for the radial laced wheel is easy. It is the hub radius subtracted from the rim radius. To calculate the spoke length for a crossed spoke lacing pattern, a bit of trigonometry can be used. In each case we have a non right angle triangle where lengths b and c and angle A are known, but where length a (the spoke length) is unknown. If the angle from the axis of the wheel between two adjacent holes on the hub is B, and the spoke pattern is an x-cross pattern then the angle A will be xB. So for a wheel with n spokes, there will be n/2 spokes on one side of the wheel and the angle B will be

Therefore the angle A will be

(eq. 1).

The length a can be calculated using the cosine law:

(eq. 2)

When considering the length of a spoke in a real wheel, which will have have one of the hub flanges offset lateraly by a distance f from the centre of the axle, then we have a right angle triangle where one of the short sides is f and the other is a.

Pythagoras' theorem can now be used to calculate the length of the long side of the triangle (which is the actual spoke length). If d is the actual spoke length then we have:

By substituting the RHS of eq. 2 for in this equation we have

If we substitute the RHS of eq 1 for A in this equation we have:

So the actual spoke length d is: