Hexagonal Geodesic Domes

Minimal stable system

Immobilization with seven cables

Here is a tetrahedron stabilised by seven cables.

Immobilization with seven cables

The sides of the triangle are in the same plane as the anchors. This doesn't lead to the greatest restoring forces - but it does make it easy to demonstrate stability by resolving forces and taking moments around the triangle's apex.

Magnetic model

The next two diagrams here illustrate another model. It is shown from two different directions.

Stability with seven cables (bottom view)

Stability with seven cables (top view)

A fairly simple argument is sufficient to demonstrate the stability of this system.

• The central point of the hexagon behaves as though constrained by four cables, leading to each anchor point. That is sufficient to fix its coordinates in space.

• Each of the three pairs of cables connected to the ground removes precisely one of the remaining degree of rotational freedom from the system.

Magnetic model

Magnetic model

Magnetic model

These models are provided to help visualise the structure.

Because the cables in this model can actually sustain compressive forces, the model isn't intended to be an accurate representation of the system.

Magnetic model

Notice that the hexagon is twisted with respect to its supports. This allows the cables to cross by each other without touching.

Removing translational degrees of freedom

Removing translational freedoms

The centre of gravity of the tetrahedron is effectively immobilised, though the tetrahedron is still free to twist on a small scale.

This is a well-known, classical configuration.

Removing rotational degrees of freedom

Please note that the red node marked "O" is considered to be fixed in space in this diagram.

The triangle would be free to rotate about it - were it not attached to the anchors (marked A-E) by the five horizontal cables.

Removing rotational freedoms

My first attempt at solving this problem remains visible here.

Minimal torque

Under some circumstances, as well as immobilising an object, it is desirable to apply as few torques as possible to the object.

Fixing an object in space with no torque at all is only possible in free fall.

Assuming free fall conditions, the following diagram attaches nine cables to four points on a white triangle - in such a way that the object is immobilised - and the resultant of the externally-applied forces at each point is zero.

If the object is not in free fall such a configuration would minimize the applied torques.

9 cables - minimal torque

Note that this configuration is a simple combination of the last two diagrams.

My first attempt at solving this problem remains visible here.

Immobilization with six struts

As far as I can tell the answer is six. The following diagram illustrates the configuration:

Immobilization with six struts

Related work

"made awkward geometrical mistakes such as how many spokes are needed on a wheel to hold it rigid (Fuller said 12 instead of 7)"

The reference given is to an interview with Walter Whiteley.

It is nice to see that someone else agrees with the figure of seven for the number of tensile elements required - and that they also reached the conclusion that this was one of Fuller's mistakes.

I published my conclusions in the area about six months before the book was published - but it seems fairly likely that these conclusions were reached independently.

Links

It contains photographs of a bicycle wheel with seven spokes.