speedevil 

Review the radial lacing visualization and add this twist to it. Literally, the hub is "twisted" and the spokes are no longer exiting the hub straight to the rim, the spokes are moving towards tangent to the spoke hole circle.

By "exiting the hub straight to the rim" I mean that the hub axle, the spoke hole in the hub, and the spoke hole in the rim are aligned. With 2/3/4 cross lacing patterns, there is an angle between a radial spoke and the cross pattern spokes. Visually, this angle is the angle between the hub axle and the spoke hole in the hub, and the spoke hole in the rim. The greater this angle, the more required length to reach the same depth in the nipple.

By using the angle that the spoke makes, we can calculate the additional length required. We know how many increments there are around the rim based on the number of spokes per side, which gives us the angle as compared with a radial spoke position. This angle "lengthens" the long side of the right triangle, which lengthens the hypotenuse (the spoke itself) as well.

If the spoke was perfectly tangent to the spoke hole circle, it would need to be increased in length equal to the radius of the spoke hole circle. That's an extreme example but it demonstrates how the spoke lengths change as the number of crosses increases from 0.