Hexagonal Geodesic Domes - Close Packing

Sphere packing

In particular the following two problems have been well explored:

• How can spheres of identcal size can be best packed around a sphere;

• What is the minimum enegry configuration of charged particles arranged in a spherical shell?

As far as sphere packing goes - it is well known that struts in geodesic domes tend not to be all the same length - and so modelling using identical spheres does not seem appropriate.

As far as electro-static models go - they model forces between non-adjacent neighbours (forces which don't exist in most domes) - and tend to result in configurations where particles don't approach their neighbours in all directions - resulting in "squares" in the resulting networks. As far as dome strength goes, squares seem like bad news - it's structurally better to make sure you triangulate everything, even if there's some cost in terms of not properly minimising strut material.

So - while a lot of work has been done in the area, a lot of it is of reduced relevance to dome builders - since their problem is neither electrostatic in nature nor associated with packing identical spheres.

Some people have considered more relevant problems - as the following quote from another site illustrates:

I will refer to this problem as the electrostatic problem; it is also sometimes known as the Thomson Problem. It is alternatively possible to maximize the average Euclidean distance for a number of points [...]. I will refer to this problem as the geometric problem. This problem can alternatively be formulated by using the distances along the surface rather than the Euclidean distances, in which case I refer to it as the arclength problem. Instead of considering the distances of a given point to all others, it is alternatively possible to consider its interactions with only its nearest neighbours by maximizing the minimum distance between points but this is expected to be a much harder problem.

Which problem?

It seems generally agreed that dome builders want to maximise a function involving structural strength - while minimising material and construction costs.

However strength is not an easy variable to model. You can build spheres and smash them into walls - but that stuff is hard work - a simpler geometric problem is needed if we want to be able to actually solve it in reasonable time.

Fortunately, it seems to me that constructing the topology of a good hexagon-based dome seems closely analogous to another - simpler - physical problem.

Analogous system

The soap film would be artificially divided into a fixed number of regions - each with the same number of gas molecules in them.

Soap films try to minimise the extents of their surfaces - this corresponds well to minimising the total strut length of the load-bearing elements in a hexagon-based dome.

The duals of such structures naturally tend to form complete triangulations - since the bubble surfaces form what is essentially a Vornoi diagram - which is the product of a complete triangulation process.

There are basically two variables in this problem:

• The number of bubbles;

• The number of gas molecules.

The number of gas molecules represents how important having cells of equal size is - compared to the issue of minimising strut length. This is important - because if you make the gas too easily compressed, and the problem has degenerate solutions - corresponding to one large bubble and lots of tiny ones.

Having cells of equal areas in not usually critical. Consequently I favour eliminating the gas pressure constraint - by setting it to the smallest value that avoids degenerate solutions to the problem - where some cells have obviously collapsed under pressure from their neighbours.

Unfortunately, I'm not aware of any previous attempts to address this particular class of problems.

Progress

At least it doesn't involve anything as difficult to model as "strength".

However it is still a pretty difficult problem - and seems highly likely to be prone to suffering from local minima.

So far, my attempts to solve it have involved several more simplifications.

Rather than modelling the edges of the soap films, I've treated each soap bubble as a simple sphere - i.e. not as a bubble but as a spherical balloon.

The balloons are inflated until they press up against each other - and then are put under pressure by their neighbours.

My model for pressure is also a crude approximation - I pump material into the balloons at a constant rate. Collisions between balloons are then modelled - and the balloons are deflates a little whenever they collide with each other.

The effect of this is that those balloons that hit each other most frequently are considered to be under the most pressure from their neighbours - and are deflated the most.

So far I'm pleased with the results I've got from this highly simplified model. The balloons corresponding to the pentagons in the spheres do shrink in an appropriate fashion - and gaps between balloons are gradually eliminated as the simulation progresses - resulting in triangulation failures being relatively rare.

There is not yet any serious attempt to minimise the Vornoi length - but the results seem to be positve nontheless - and it is my hope that some simulated annealing will further improve the quality of the results before consideration of minimising Vornoi lengths is required.