To explore the mathematics, design and construction of a geodesic dome. The pupils/students will make their own analysis of the structure and mathematics of these fascinating objects. Although this is fairly simple it does require puzzling-things-out and some conceptual thinking. Finally they will attempt to make a large geodesic structure, the design, the shape and size of which they will decide for themselves (within sensible limits). Designs on paper will be required as well as constructing scale models and of course the final product.

Geodesic dome structures are extremely strong, lightweight architectural dome buildings that can be used as shelters, houses, industrial buildings as well as earth quakeproof accommodation and other extreme situations (such as those experienced in the polar regions etc.). It is claimed that there are 100,000 geodesic domes in use around the world. The main architect responsible for the development of the geodesic dome was Buckminster Fuller, the American visionary, dreamer and designer. A diagram of one of Fullers geodesic domes is shown over page.

A geodesic line is defined as the shortest possible line on the surface between two points. The struts that make up Fullers domes tend to follow the shortest lines on the spherical surface - hence the name geodesic domes. Stress and strain tends to be dispersed evenly over the surface of the geodesic domes making them very strong.

Last summer I ran an activities day at the Angmering school where we built geodesic domes out of rolled up newspaper! This was based on a very simple design found on the web (see end). In the Smithkline Beecham W/S I would like to extend this work, not only by considering the simple maths involved, but also by extending the number of possible designs considered. Most importantly by using bamboo instead of paper we will build something that could actually be used as a shelter. Such a structure should be more impressive, stronger and require pupil/student input and creativity. To date I have built scale models of the domes and started to think in detail about the planning of larger structures (NESTA has supported this work so far).

The first day of the workshop will consist of an introductory talk with some simple practical work. The talk will discuss the dome structures, their use and also the recent spin-off developments in science with the discovery of the fullerenes. A model kit will be given out that allows the pupils/students to make a molecular model of one of these fullerenes called C60, Buckminsterfullerene. C60 is a newly discovered, third form of carbon that not only has a structure based on the concepts of the geodesic domes (hence its name) but also looks like it will revolutionise carbon chemistry and material science.

After the talk the main body of the next workshop will consist of thinking, discussing and designing how they are going to make a geodesic dome (based on the information provided in the introduction talk). This will require them to puzzle out the symmetry and structure of the domes. They will also have to think about the practical problems of what materials they will use, the number of struts, for example, how they will join them together, how they will work together to construct each stage of the design, where should they start etc.

Technical Details
In this workshop we will look at the mathematics of shapes and how it can help us understand the structure of the geodesic domes. Eulars law holds for a great number of different shapes: the numbers of corners (C) plus the number of faces (F) minus the number of Edges (E here called the struts) equals 2 (i.e. C + F - E = 2). We can also expand on this and get a formula in terms of just the type of faces (e.g. number of triangles, squares, etc). From this we will explore how simple maths lets us predict and also how it shows the links between the mathematics and symmetry of structures found in the animal world to architecture and down to the tiny molecules of the atomic world.

We will discover that squares, triangles and hexagons can cover a flat surface perfectly. However introduction of pentagons leads the flat surface to curve. For example when hexagons are considered the formulas shows that exactly 12 pentagons are required to curve the sheet into a ball or cage. 6 pentagons are needed for a dome (half a cage, sphere or ball etc). By introducing different numbers of triangles or hexagons different types of domes can be constructed. Further, by joining up the pentagons and hexagons in the middle to form triangles, a very rigid structure can be obtained - this is our ultimate aim. Three different structures will be explored and there will be scope for discussing the merits of each as well as their own designs/ideas.

By building scale models from card (or small sticks) it becomes clear that not every strut (or edge) in the dome structure is the same length. This is easily shown by considering a regular pentagon. Here all the sides are of equal length, however when we draw lines to the centre (to form the triangles needed for strength) these lines are found to be shorter (unlike in the situation in the hexagon where they are all the same length). The pupils/students can use trigonometry to work out the two lengths or can draw an accurate diagram and measured off. If this 'ideal' length is used, for the short struts, a faceted dome will result. For the dome to look good (i.e. to look nice and round) these 'shortest' struts need to be a little larger than the calculated shortest length, but not too long. This, to a certain extent, is subjective and so they will need to discuss and chose for themselves.

There is therefore a great deal of planning, calculating, drawing-up, discussion and teamwork required for this workshop. I think this fits well with the brief of the Masterclass'. They will also make a very stunning dome as a product of all there hard work which I feel (and know from previous experience) will be very satisfying.

Scope for pupil/student activity and initiative
There is a lot of scope for ingenuity and creativity in this W/S. They will need to work out/discuss a number of issues:
1) well drawn designs on paper, sizes and lengths of struts etc.
2) scale models to compare
3) which design to finally go for
4) decide on the size of the dome (i.e. if it might be raining outside how big can they make one inside ?) and the lengths of each part
5) What materials to use
6) What constructional problems they might have (joints, cutting lengths, holes etc)
7) How they will go about the first steps
8) Problems they might face as the dome starts to get bigger (and higher)
9) Improvements on the final construction
10) What they have learned, how they can do it better another time etc.
11) Any other issues that arise

Likely outcome
Although it would be nice to let the pupils/students have full reign to design and build what they wanted to we will have to limit the size and complexity of the designs. At least we will need to keep an eye on the development. I suggest showing three different designs of dome, letting them decide which they are to go for and then to let them explore and calculate the details. As a 'rule of thumb' the dome diameter will be about 3 times the length of a single strut. If the weather is fine we could build the final design outside, in which case 1.5m bamboo canes would produce a dome of about 4-5m diameter. If the weather is bad a smaller 2m diameter dome could be made inside using 60 cm green garden sticks.

Materials needed (aprox. for each dome)
50 green garden sticks (smaller dome, ca. 2m diameter) 50 bamboo canes (large dome, ca. 5m diameter) Paper, cardboard etc String, copper wire, cutters, tape, hand drill, hacksaw, bench vice, bench, battery electric drill

Further information:
see information about geodesic and other dome structures


Dr Jonathan Hare, The University of Sussex
Brighton, East Sussex. BN1 9QJ.

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