The drawings show a well-proportioned geodesic elliptical dome, modelled on a actual melon! The ratio between its major and minor axes is 1.44:1. It has bilateral symmetry, the breakdown is a 4-frequency octahedral symmetry.

Pictorial view

Top view

However, truncated geodesic domes with a pentagonal symmetry are difficult to aggregate in two directions, since the geometry along the cross-sections, where the truncated geodesic domes join together, are messy. This problem can be neatly solved in designing domes with a 6-fold symmetry.

The drawings show a geodesic gridshell with a 6-fold hexagonal symmetry. It is derived by projecting the hexagonal face of a truncated tetrahedron, with a triangular subdivision, on a circumsphere. All the lines follow great circle arcs.

The hexagonal face of a truncated tetrahedron marked on a sphere, with a geodesic triangular pattern.

Gridshell with geodesic triangles having a 6-fold symmetry, supported on columns.

Top view. Note how the areas of the geodesic triangles vary. Being relatively large at the crown where the loads are low and smaller towards the supports where the loads increase.

An aggregate of 3 hexagonal geodesic gridshells. Note the clean geometry along the cross-sections where the truncated domes join.

Being a keen cyclist, I am fascinated by the lightweight efficiency of the bicycle wheel with tension spokes. This small and inexpensive project is about applying the structural principle of the bicycle wheel to a spatial closed system.

The starting point were the two polyhedra: the great stellated dodecahedron inscribed in a dodecahedron. For the dodecahedral cage I used Herringbone struts (manufactured by Simpson Strong-Tie) and for the great stellated dodecahedron I used galvanised wires.

Instead of the turnbuckles I used long eye bolts with two nuts. To fix the wire end to the eye bolt I threaded the wire through the eye and bent it over by 180 degrees, then pushed a washer over the two wires and bent the end again.

I began by assembling the dodecahedral cage (the ends of the struts had prepared holes). As the form had “hinged” joints and no triangulation it collapsed! To make it stand up I temporarily stabilised all the 12 pentagonal faces with thin wires (radiating from the centre to each of the five vertices).

Next, piece by piece, the 90 prepared wires, with L-shaped ends, were fixed into their correct positions.

Finally, the exciting bit could begin – prestressing the structure one vertex at a time. With each complete cycle the structure became progressively stronger and stiffer – magic. Indeed, prestressed wires resist forces like columns! The larger the structure the more efficient it becomes.

Yes, I do, and I understand!

Dodecahedron filled with a stellated dodecahedron. Straw model.

The principle of the bicycle wheel applied to a spatial closed system.

Constructional details

Constructional details

Struts and “spider webs”, the latter being almost invisible.

Applied to architecture and structure, the former is primarily an aesthetic position; the latter is a principle of economy.

Designing lightweight space frames requires rigorous discipline following a few basic rules.

• No bending
• Symmetry and geodesics
• Hierarchical systems for the transfer of loads
• Remember – God is in the construction details

No bending

Take a bamboo cane, and with your hands, subject it to tension, compression and bending. This very simple experiment demonstrates that bending is an indirect inefficient transfer of forces, as compared with the direct-action (vectors) of tension and compression.
Remember, in any structure that is capable of transmitting forces, tension (pulling) and compression (pushing) always coexist! (It is impossible to imagine one without the other).

A truss, which is an assembly of triangulated struts and ties, is much more efficient and lighter than a beam on girder.

A space frame is a three-dimensional truss, transferring forces in an axial manner. Designing highly efficient space frames requires a clear differentiation between these two states of stress, using hollow or angle steel or aluminium sections for the struts (bones) and wires or cables for the ties (ligaments).

Prestressing is a particular effective way of achieving lightness as it enables undesirable compression stress to be converted into tensile stress! The bicycle wheel is a classic example.

Symmetry and Geodesics

The optimal solution of the enclosure of space is the spherical geodesic dome (the sphere encompasses the given volume with a minimum of surface). Increasing the size of this structure will increase its efficiency!

Imagine the straight edges of the 5 regular Platonic solids and the 13 Archimedean solids expanded into curved edges, as if each polyhedron has been blown up like a balloon. These curved edges will form a grid of great circles arcs, i.e. geodesics (the shortest line between two points on a spherical surface is an arc of a great circle). Each geodesic polygon can be divided again into smaller geodesic polygons, i.e. triangles. The number of the subdivision of the polygon edge define its frequency.

Buckminster Fuller recognised that a pattern of great circles arcs with a 5-fold symmetry, based on the icosahedron, is the best solution to the problem he set himself, i.e. how to enclose space with a structure that requires the least amount of material. His iconic design for the U.S. Pavillion, Expo ‘67, Montreal, Canada, is a double grid geodesic dome, 76 meter diameter, with a 5-fold symmetry. The outer triangular grid has a 16-frequency.

Geodesic domes with a 5-fold symmetry are difficult to aggregate in two directions, since the geometry along the cross-section where they join lack order and is messy. This problem can be neatly solved in designing domes with 3-fold, 4-fold or 6-fold symmetries.

Hierarchical systems for the transfer of loads

The strategy in designing lightweight space frames is straightforward. Think of curvatures that are synclastic and anticlastic with “geodesic” triangles. Choose a geometry with symmetrical patterns.

Employ hierarchical systems for the transfer of loads – it is Nature’s way”! Differentiate between the primary structure, its secondary structure and the tertiary structure. Size their elements accordingly. Where it is not absolutely essential – remove inactive material in the structure. Employ elements that act mainly in axial tension or compression as against those acting in bending. Tensile ties are much more economical than compression struts, especially when the former are prestressed!

A minimal space frame can be defined mathematically!

At any chosen cross-sectional area of the structural element, the fraction-stress under maximum load divided by the permissible stress of the material – is equal to one.

God is in the construction details

“God is in the details”
Mies van der Rohe

Indeed, good construction details are the mark of a healthy construction. The Victorian engineers were masters in construction details, made from wrought-iron. The structural forms they used, like trusses, arches and portal frames were primarily one-directional load carrying arrangements. Space frames carry loads in a two-way action. This makes the design of the connection a challenging problem. There are numerous space frame connectors on the market. They are usually made from steel.

The most successful and also conceptually satisfying , is the MERO connectors. It was invented by the engineer and good businessman Max Mengeringhausen (1903-1988). The connector has the form of a rhombicuboctahedron (polyhedron with 26 faces). It is an ingenious spatial nut and bolt fixing. An 18-way joint at angles of 45 degrees. This did put a constraint on the possible space frame configurations that could be built with these connectors.

However, being able to vary the angles between the struts and ties and their lengths in the MERO system via computer numerical controlled (CNC) processing, any imaginable configuration can be built. The MERO connectors come in a range of different sizes (the MERO system was used for the iconic Eden Project in Cornwall). For low-tech DIY space frames my deceptively simple universal connector is hard to beat!

Model of the geodesic hemispherical dome, 4-frequency icosahedron.

The universal joint using off-the-peg materials.

Unpacking the struts and ties from the boot of the car.

Fixing the cladding to the space frame was as easy as hanging a curtain!

Curtain rods (springs) prestressed the cladding.

Interior view of the parachute shelter. The principles of ancient kite technology applied to a contemporary tent.

An instant temporary private Eden on a camping site in the south of France in 1972.

Reminiscent of the iconic photo - Earthrise - taken by Apollo 8 astronaut in 1968.

Curiosity and the pleasure of finding things out motivated this project.

In order to understand the geometry of geodesic domes and to get a good feel for its structural behaviour, one must build one!

There are two major design problems: connector and cladding.

Connector

Looking into the published images on connectors for geodesic domes, I could not find a really simple joint suitable for DIY.

Starting with the knowledge that in a triangulated frame the connector can be a ‘pin-joint’ (hinge) as it is primarily subjected to axial forces (compression or tension) – no bending!  The Eureka moment came while I was playing in the workshop joining bits of wood and metal. The universal joint (see Figure 2) takes advantage of the bendability, strength and durability of metal.. With this connector there is no need to calculate the axial angles of the struts and ties. By tightening the nut of a long bolt  the metal strips, and the ends of the members, will bend to the correct axial angle.

Struts and Ties Lengths

I wanted my dome to be a hemisphere and without dissecting the triangles at its equator. The lengths of the struts and ties should not exceed 1 metre. A 4-frequency icosahedron met this brief. The chord factors (length of members) were simply worked out using the appropriate table (Dome Book 2, published by Pacific Domes, 1971). For this 6 metre diameter geodesic dome I used birch dowels 18mm dia.

Required:

1. 93.50 cm. 30 members
2. 90.00 cm. 70 members
3. 85.90 cm. 30 members
4. 80.00 cm. 30 members
5. 84.75 cm. 60 members
6. 72.85 cm. 30 members

Total: 250

Note the very high 52/1 slenderness ratio (length divided by diameter) of the struts.

Cladding

To clad a doubly curved surface with prefabricated materials and to make it watertight is a challenge.

For my experimental, lightweight, demountable shelter I chose a parachute (price £4.50, 1971). It is dome shaped and fits loosely inside the 6 metre diameter dome. Curtain rods (springs) fixed the parachute to the joints of the dome, thus prestressing the fabric.

Erection

For the location of my experimental shelter I chose a camping site in the hills near Roquebrune-Cap-Martin on the easterly end of the Cote d’Azur (not far from the ‘hostel’ where Le Corbusier spent the summers and where he dies of a heart attack while swimming in 1965). A slightly sloping spot in a wooded area was ideal.

The assembly of my prefabricated ultra-lightweight geodesic dome resembled the ‘growth’ of cells, expanding in a centrifugal nature. The ends of the struts and ties had a colour code.

I began by assembling a triangulated pentagon on the ground, next adding 5 triangulated hexagons each sharing 1 edge with the former. A shallow dome began to form. I supported this structure on a stool in order to lift it off the ground. Adding 5 more triangulated hexagons, each sharing 2 edges with the above. Next adding 5 triangulated half-hexagons, each sharing 3 edges with the above, thus completing the top part ½ of the 4-frequency icosahedron dome.

The 5-fold symmetry of this structure dictated where the rest of the various lengths of struts and ties should go, making sure, at every stage, that the colour coded ends of the members did match. It took me about 2 hours to erect the dome single-handed.

Fixing the parachute to the space frame was easy and quick, like hanging a curtain! It was exciting to experience how this spherical double layer prestressed system increased the rigidity of the whole structure – synergy.

This project was first published in AD (Architectural Design) 02.1973.

Lighter than Air!

Having read that the geodesic domes (Biomes) at the Eden Project in Cornwall weigh less than the volume of air they enclose, I thought to check whether this is also true with my small geodesic dome?

1.2 kg (density of air per cubic metre) x 56.5 (volume of air enclosed by hemispherical dome, 6 metre diameter) = 67.8 kg.

Weight of the structure + cladding: 38 kg (struts and ties) + 3 kg (parachute) = 41 kg!

However, with my simple universal joint the strut lengths can vary, allowing easy low tech fabrication. I do not see these minimal octet trusses as sculptures. Their job is to support, train and display plants and thus they become invisible, analogous to the skeletons in animals.

1. Building blocks – octahedron and tetrahedron – of the octet truss. This example has five different strut lengths.
2. Octet truss screen with a chamfered top edge.
3. Octet truss screen on an inclined ground.
4. Octet truss forming a curved screen.
5. Octet truss forming a curved screen.
6. Octet truss forming a screen with a right angle.

Building blocks - octahedron and tetrahedron - of the octet truss. This example has five different strut lengths.Octet truss screen with a chamfered top edge.

Octet truss screen with a chamfered top edge.

Octet truss screen on an inclined ground.

Octet truss forming a curved screen.

Octet truss forming a curved screen.

Octet truss forming a screen with a right angle.