A while ago I was browsing through the work of Gramazio & Kohler when I came across a project called “Open Air Theater.” The theater is based on a “reciprocal frame structure,” otherwise known as a self-supporting structure, and as if that wasn’t enough to catch my attention, the idea for this type of structure comes from Leonardo da Vinci himself (well, there are other contenders, but more on that later). Basically, the system works by using gravity to hold itself together – support itself. Notches in the members make it easier to construct, but strictly speaking aren’t necessary as long as there is some friction. Gravity will do the rest.

Da Vinci explored two forms of the structure – a bridge and a dome. His work was commissioned by the Borgia family, with the mandate to design light and strong structures which could be built and taken down quickly. This was to aid them in their constant struggle for power with the Medici family in Renaissance Italy. (1) The bridge could be used for crossing rivers, and the dome could function as a military camp. He made several sketches to this effect and apparently (2) a reproduction can be found in Carlo Pedretti’s book “Leonardo, Architect” (which can now only be bought used for rather large sums of money… too bad).

I did a search on the bridge first. I found a helpful forum page (3) from 2006 which included a diagram of how the bridge may be built. I decided I had to try it out with whatever I had available, which happened to be a bunch of pens. I quickly learned that although the structure isn’t supposed to need any connections, it’s much easier to build the bridge in prefabricated modules, raising one section at a time. The first module will be different from the rest.

Someone’s take on the bridge with some elaborate joints(click on the image for the source):

A different take using planks and eliminating a redundant member (again, click for the source):

Some great instructions about how to make a bridge out of notched dowels can be found here: http://drsamson.wikispaces.com/file/view/Instructions.pdf.

Then I also discovered that such a bridge is pictured in a 900-year-old Chinese painting, which inspired a Chinese history professor – Tang Huan Cheng – to figure out how it could have been built and to reconstruct the bridge in a town called Jinze. The article that documents the construction of this bridge rightly points out the need for strong abutments on either end of the bridge. (4) The physics behind this are similar to any arch construction – the ends are forced outward, and if there isn’t something to push back the arch will fail. This is also true for dome structures, but since domes aren’t usually used as transportation infrastructure, the strength of the abutments only has to account for the dead load of the structure itself. I think the finished bridge looks quite elegant.

A note to self: looking at the underside of the bridge makes me think of the alternating peak and valley of a folded plate structure. An image can be found in The Function of Form, or also here.

One really interesting interpretation of da Vinci’s bridge is that the design may have been for a folding bridge. This YouTube video shows how this might have been possible, although I wonder how the connection points would work:

So that brings us to da Vinci’s domes. The person who seems to have done the most work on understanding the domes is Rinus Roelofs, a Dutch sculptor. He discovered this system in 1989 by himself, and only then found the same concept in da Vinci’s work (5). He found that there is one Rule (capital R) that governs all the possible “grids” for the domes: it is the “plus-minus-minus-plus” rule, which can be rephrased as the “over-under-under-over” rule. (6) If you look at any one member in da Vinci’s dome structure, it always follows this rule. An exception may be the end conditions; members at the edges may have one or two unfilled notches.

Rinus Roelofs has gone through dozens of grid variations that work for building the domes. The homepage for these variations is here: http://www.rinusroelofs.nl/structure/davinci-basic/davinci-100.html. He divides them into square, rotational, and translational grids. Some look very similar, but all have their differences, and all follow the Rule. He has also looked at how two patterns can be combined in one structure, and at introducing distortions into the pattern (perhaps it’s better to call it an anomaly rather than a distortion; the building block of the structure – the equally spaced notches on a member – are never distorted in his studies). He also extrapolated the same principles onto polyhedra which makes for some interesting sculptural works. He has also built quite a few of the dome structures in full scale!

It’s interesting that regular pentagons don’t appear to work as a building block for this type of structure. The only time pentagons appear is as hexagons that are missing a side. Unrelated to this topic, but Roelofs also has done and posted a lot of other geometric research, that explains the logic behind designs such as (for example) Erwin Hauer’s famous screens. See his projects page: http://www.rinusroelofs.nl/projects/projects-00.html. His website also introduced me to Kenneth Snelson and his research on weaving and “tensegrity” (fun fact: he came up with the concept, Fuller coined the buzzword and “appropriated” the concept for himself; apparently Fuller liked to appropriate a few things without giving credit – that’s a jerk move I must say). Check out Snelson’s work: http://kennethsnelson.net/tensegrity/. Maybe I should do a post some time just on weaving and the work it has inspired.

Coming back to the “Open Air Theater” project from Gramazio & Kohler’s website. It differs from da Vinci’s original designs in that the structure adapts itself to a desired form, as well as (apparently) accounting for structural loads. This means that distortions to the “grids” are allowed, and this in turn means that members can be of different lengths, with differently spaced notches, and even different thicknesses to handle varying loads. There seems to be a regular grid in some places but in others it is very distorted – octagons, heptagons (7 sides; had to look it up), and pentagons take the place of hexagons in the “Open Air Theater.” One of the advantages of the structures designed in this way is the steeper slope at the sides, which makes it possible for the space to be occupied as opposed to Roelofs’ structures which have a very shallow slope and are barely tall enough to shelter a person at their tallest height.

The one thing that always remains the same is The Rule: plus-minus-minus-plus/over-under-under-over. Looks like the work resulted in some interesting full-scale tests as well: http://www.spiro.arch.ethz.ch/de/research/reciprocal-frame/workshop.html (I think these are really impressive).

Unfortunately, if the “Open Air Theater” was made with the computational aid of a script or Grasshopper definition, that definition isn’t shared. If I was to try to produce such a script or definition, I would start by looking at the work of Roelofs again – he figured out how to transfer the logic of the da Vinci dome onto a sphere by approximating it as an octahedron and folding the pattern into that shape. Perhaps a similar simplifying logic is a key to generating the grids for more free-form shapes. It also probably starts to speak to the appearance of a larger variety of -gons in the grid. For example, Roelofs used a hexagon-based grid to get a full sphere (although he also used curved members instead of straight). The amount of sides on a polygon may be related to the desired curvature at that spot.

So what are my main interests in this? Fast and easy to install and uninstall has potential for soft architecture – temporary structures that redefine a space and can be used as a tool to test public response with a low budget. I could image these being built from dimensional lumber or perhaps bamboo. I also like the sheer elegance of the way the structure can be put together based on one simple rule, and the beautiful pattern possibilities.

I wonder if this structural system has ever been used in a building (more so than the installations I’ve linked to)? A preliminary search hasn’t yielded anything.

Some citing of sources:

How to cite a word document that I found through Google? It’s called “the_da_vinci_self.docx” http://www.rinusroelofs.nl/structure/davinci-bargrids/davinci-003.html http://ask.metafilter.com/38289/How-to-build-da-Vincis-selfsupporting-bridge http://www.pbs.org/wgbh/nova/lostempires/china/builds.html http://www.rinusroelofs.nl/structure/davinci-sticks/introduction/introduction.html http://www.rinusroelofs.nl/structure/davinci-sticks/introduction/introduction.html