## A Wheel Dilemma

If the 2-dimensional treatment of the bodal wheel engenders any reservations about its practical viability, there is another, deeeper consideration that goes to the heart of the wheel as employed at least since the beginning of the industrial revolution: powering the wheel as opposed to merely spinning it.

With regard to the latter, the radially spoked wheel has, since antiquity and will continue to be, hugely successful in supporting the central hub and loads placed on the rims. With such wheels, the hexagonally triangulated spoke pattern afforded by the most simplistic of the bodal wheel interpretations poses maximum strength in supporting loads with the minimum number of spokes.

To lessen the requirement of a deeper rim to retain their curvature, more spokes must necessarily be added. With 12 comes the important introduction of orthogonality at the hub, a geometric relationship that complements the configuration of equilateral triangles.

But to actually turn the wheel by applying force to it is a different matter altogether. From the point of view of the rim, its motion is always directed tangentially which is as distant (angle-wise) from the direction of the radial spoke as possible.

The same goes for force applied to the hub. From its perspective, the closer the application of force is to the wheel’s center, the more torque must be applied there to turn the wheel (think lever).

Thus in the gap between hub and rim lies the crux of the matter. In contrast to the radially-spoked wheel, consider the solid disc. Its torque is transferred from wherever the force is applied relative to the center and, like the radial wheel, the farther from the center the less force is required. Unlike the radial wheel, however, transfer of force is not restricted to a spoke or spokes.

With the solid disc, force transmission follows the electro-magnetic knitting of the disc’s material – with a tangential pull at the point of application to a tangential pull at the rim. In this sense, the solid disc is ideal. But on the other hand, because this form is also characterized by solid mass – both inside and outside the circle of leverage – the attribute of tangential optimality is significantly compromised.

Thus the advent of the wheel spoked tangentially at the hub’s outer realm, most of the disc’s mass is eliminated.

Schemes to find the perfect balance between the virtues of the 2 extremes continue to this day. The code addresses the dilemma by first appropriating bode geometry to the hub itself, a choice validated by reason of the geometry’s infinite divisibility. In the example (right), the hub is sized such that lines extended from its outer hexagon – which constitute tangents attuned to the circle of rotation – meet points on the rim that coincide with the 12-spoke radial wheel. Furthermore, the rim itself is hexagonally inscribed so that lines extended from the hub meet it at 60º . Such is one portrayal of how the bodal wheel deals with torsional forces.

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