I wonder why?

Imagine a cube inscribed in a sphere. Now from the centre of the sphere project the edges of the cube on the circumsphere. The projected edges will follow great circle arcs i.e. geodesics. Each face of the cube can be divided into an orthogonal grid. Again projecting this grid from the centre of the sphere on the circumsphere will give a quadrilateral mesh with a geodesic pattern (all the lines follow great circle arcs). The number of divisions of the cube’s edge will give the frequency breakdown. The figures show a 6 frequency spherical cube.

This geodesic dome has a 4-fold symmetry, which makes it adaptable to plans with right angles, which is the norm.

Nowadays, when any imaginable configuration (sculpture) can be built, we should remind ourselves that the architectural language, for some 5000 years, is primarily based on the discipline of orthogonal ground plans! Also observe how the structures grow directly from the plans.

What is different and new?

Contemporary architects and engineers should ask: how much does their building weigh?

Of course, this design approach is not always appropriate.

Geodesic grid shell, 6 frequency spherical cube. 10 different strut lengths are required to construct this geodesic grid shell.

Geodesic grid shell, 6 frequency spherical cube. 10 different strut lengths are required to construct this geodesic grid shell.

Geodesic grid shell, 6 frequency spherical cube. 10 different strut lengths are required to construct this geodesic grid shell.

Geodesic dome plus orthogonal ground plan and elevation in concert! The grid shell is propped by a trabeated frame.

Truncated modular geodesic grid shells that can be tessellated along the X and Y axes.

The standard solutions are plane lattice structures which resist wind loads via bending. A much more efficient solution is a double curved space frame, which resist wind loads principally in an axial manner (via compression or tension). The figure show a barrel vault capped by half domes. Its geometry is a geodesic envelope!

It is not generally recognised that a series of circles (arches) parallel to one another, and a series of straight lines parallel to one another, intersecting the circles at right angles, form the simplest case of a geodesic pattern on the surface of a cylinder. Indeed, structures with geodesic patterns on single curved surfaces (Euclidean geometry) are nothing new! The geometry of the half dome is derived from the spherical cube.

Returning to lightweight roofs

The transept roof of the Crystal Palace, for the Great Exhibition of 1851, with its barrel vault construction was given lateral stability via diagonal bracing made from wrought iron rods. It was a highly efficient system. It is very instructive to note how Paxton used hierarchical systems for the transfer of loads — nature’s way. This seminal roof structure inspired engineers to develop ligtweight grid shells with rectangular meshes and diagonally pre-stressed cables. If the cladding is transparent the cables become almost invisible!

Geodesic grid shell with quadrilateral and rectangular meshes. It resembles a loaf of bread - organic.

Geodesic grid shell with quadrilateral and rectangular meshes. It resembles a loaf of bread - organic.

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!

This is a good model of the synergy between the opposing forces (tension and compression).
The working together of the two forces to produce an effect greater than the sum of their individual parts.

The constant interaction between two opposing balancing forces in this closed system can stand as a metaphor for Yin and Yang!

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!