Hexagonal Geodesic Domes - Close Packing
The problems associated with packing small particles around
a sphere so that they are more-or-less evenly spaced has
been well studied.
In particular the following two problems have been well explored:
Unfortunately, the results of both of these classes of problems
are not as relevant to dome builders as they could be.
- 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
When trying to solve a problem, the problem first needs
to be stated clearly.
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.
That system consists of a soap film divided into many
regions - while being suspended between two transparent
spheres of slightly different sizes.
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 first variable corresponds to how strong you want the
resulting dome to be.
- 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
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.
This soap bubble problem seems almost tractable.
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
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
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
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
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.
Tim Tyler |