Hexagonal Geodesic Domes

Wheel stability

Since then Bob Burkhardt has also investigated the issue. Bob has published his results [here].

I recently decided to take a look at the issue.

I constructed a number of six, seven and eight spoke wheels.

Here are photographs of some of them:

Eight spoke wheel

Eight spoke wheel

Close up of where the spokes meet the hub

The eight spoke wheel appears to be fully stable.

There are eight-spoke wheels which can be proven to be fully stable. However, no proof of stability currently exists for the configuration pictured here. This configuration seems stable enough - and looks relatively attractive in the context of a drive wheel.

Seven spoke wheel

Seven spoke wheel

Close up of where the spokes meet the hub

According to my page on the subject, seven cables is the minimum necessary to produce linear stability when stabilising a hub in space.

My research indicates that the same figure applies to bicycle wheels - so here is a seven spoke bicycle wheel.

The design was inspired by a six spoke wheel with radial spokes.

The design has five radial spokes.

While I believe this wheel is stable in theory, demonstrating its stability is actually quite complex - and unfortunately an exposition on why this wheel is stable is beyond my scope at the moment.

Six spoke wheel

Six spoke wheel

Close up of where the spokes meet the hub

This six spoke wheel has radial spokes. Like other wheels with radial spokes, it lacks linear stability.

However, it is still usable as a front bicycle wheel - provided that disc brakes are not used.

Some other of the wheels I have constructed are visible here.

Degrees of freedom arguments

Fuller presents his supporting argument for this on:

Synergetics 401.05.

Unfortunately, the argument is incoherent.

There seems to be a hand-waving argument which suggests that you need two tensile elements to restrain each degree of freedom present in a dynamical system.

A single solid object in space has six degrees of freedom, three translational, and three rotational; therefore - one might think - it needs twelve tensile elements to stabilise it.

However the argument doesn't seem to withstand close inspection. In fact, you can prevent a point in space from moving using four tendons, arranged in the directions of the points of a surrounding tetrahedron - not the six which the "degrees of freedom" argument would suggest.

Another argument producing the "twelve" figure suggests that it takes four forces to immobilise a point.

Then - to stop any sort of movement - you need to immobilise three points - and three times four is twelve.

If proceeding by immobilising points, this argument is correct. However there's no constraint that the forces needed to immobilise an object should be able to be broken down into groups of forces, each of which immobilise a point.

There are other ways of immobilising an object.

Consider an object which has already had its translational degrees of freedom taken away from it - perhaps with four tensile elements. Are another eight forces really needed to stop it rotating? The answer is clearly "no". To illustrate this, rotation can be prevented by preventing movement around each possible axis of rotation, of which there are three; resulting in six more forces - and not eight.

Each axis could be stabilised by two tendons - as shown in this diagram:

Force diagram showing a way of preventing sphere from rotating

Notice that these forces are not arranged in such a way that they can be broken down into groups of four forces - where each group has the effect of immobilising a single point. Instead the forces act together synergetically.

Cube stabilised with eight cables

Symbolic representation of stabilised cube

This is a more abstract diagram. The cube in this configuration can be fairly easily shown to have linear stabilty by a process involving resolving forces and moments.

One more diagram

Here's one more attempt at illuminating the mistake:

A simple diagram

Here is a simple diagram of a wheel rim, a strut and six cables - all of which are the same length.

Because this structure is so simple it should be fairly easy for people to understand its behaviour intuitively.

The strut in this photograph has lost five of its original six degrees of freedom.

It can't move up down, left, right or in and out.

Two of its rotational degrees of freedom are also gone.

The only remaing way in which it is free to move is by rotating about an axis along its length.

Now if Fuller was right about twelve cables being needed to immobilise an object in space, it should take another six cables to finally immobilise that central strut. I hope that conclusion is clearly mistaken to everyone.

Links

The stability essay is also relevant.