After a long hiatus, some thoughts on post-rationalization have percolated. This may be a very brief post. The essence of the thoughts is that it’s narrow-sighted to condemn post-rationalization. It’s an important skill in design work to be able to do something, look at it, and then think about it. You can learn a lot in that process, and it can yield results that you couldn’t have expected if you had only proceeded through the design by building a rationale for every move, and used others’ rationale where your personal experience comes up short. Ultimately, it’s part of an iterative process, but while the iterative process is lauded, the subprocess of post-rationalizing is somewhat demonized.

Post-rationalization can also preclude the “I don’t know what to do” crisis or “I don’t know where to start.” Everyone gets idea-block now and again, but I think it’s much better to do something than to wait for inspiration or an idea to happen to you (fully formed & rationalized), or even to force-rationalize an idea. Sometimes what’s really needed is to get going, and then pause, “post-rationalize” what you’ve done (what you think was successful and why), and proceed from there.

A fear of post-rationalization seems also like a fear of experimentation, of diving into a process where you don’t necessarily know the results.

It is a stepping stone, and not the final step of design; leaving it until a design is finalized is probably what gets hackles up about the very idea.

Even in architecture, there are multiple ways to test ideas at multiple stages in design, post-rationalize them (ie. learn from the test), and move on to scrap the idea or choose to refine it. Of course there is the digital model, but there are also scaled models, and full-scale mock-ups. The full-scale mock-up may be the toughest to do, but is likely to provide the richest learning experience and the greatest opportunity for innovation in something other than volumetric formality.

Inspired by this blog post: “Production makes perfect” and this TED talk by Elizabeth Gilbert: “Your elusive creative genius.”


Thinking about fractals and ordering systems… Click on image to go to artist’s gallery (links to DeviantArt).

“Architects love to discuss how much sleep they have gotten. One will say how he was at the studio until five in the morning, only to return again two hours later. Then another will say, guess what, I have never slept ever. My dear architects, the measure of how hard you’ve worked and how much you’ve accomplished is not related to the number of hours you have not slept. Have you heard of Rem Koolhaas? He is a famous architect. I know this because you tell me he is a famous architect. I hear that Rem Koolhaas is always sleeping. He is, I presume, sleeping right now. And I hear he gets shit done. And I also hear that in a stunning move, he is making a building that looks not like a glass cock, but like a concrete vagina. When you sleep more, you get vagina. You can all take a lesson from Rem Koolhaas.”
– Annie Choi, “Dear Architects, I am sick of your shit (an open letter)”

Click here for the source!!

It’s time for some silliness and humour, with a bit of wisdom sandwiched in there in the middle. This is a subject that really bothers me when people (read: architects) bring it up. It’s especially troublesome when people associate long hours with steadily increasing productivity in a corporate office setting. I’m pretty sure if you made a graph of people’s productivity over time, especially after the 10-hour day mark, it goes kerplunk as it hits rock bottom. Get some sleep. You’ll work better.

“Notice that this series is called ‘Evolutionary architecture and emergent design.’ Why the distinction between evolutionary and emergent? Emergent architecture, as one of my colleagues pointed out to me, isn’t such a hot idea. If you accept the premise that architecture is about things hard to change later, it becomes difficult to allow an architecture to emerge. Architecture concerns infrastructure elements that must exist before you can start the application. However, just because you can’t allow architecture to emerge doesn’t mean that it can’t evolve. If you have created a flexible architecture and taken care not to create an irreversible decision, you can allow it to evolve over time as new concerns appear. […] One of the core ideas behind evolutionary architecture is to defer decisions as late as you can, which allows you to substitute alternatives that recent experience has shown are superior.”
– Neal Ford

From an article on software design from, here.

Self-Supporting Structures

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.

1. First module is unique; 2. every module thereafter; 3. two modules put together; 4. three modules put together

1. First module is unique; 2. every module thereafter; 3. two modules put together; 4. three modules put together

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:

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: 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!

One of Rinus Roelofs' full-sized da Vinci dome structure. Many more on his website – click the image.

One of Rinus Roelofs’ full-sized da Vinci dome structure. Many more on his website – click the image.

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: 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: 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.

"Open Air Theater" from Gramazio and Kohler's website. Click image to go to page.

“Open Air Theater” from Gramazio and Kohler’s website. Click image to go to page.

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: (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.

Rinus Roelofs' sphere sculpture from the study of an octahedron. Click on the image to go to his explanation.

Rinus Roelofs’ sphere sculpture from the study of an octahedron. Click on the image to go to his explanation.

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:

  1. How to cite a word document that I found through Google? It’s called “the_da_vinci_self.docx”

“Over the years I’ve seen numbers of fanciful plans proposed by architects which have yet to convince me there is any advantage to using tensegrity over other methods of design. Usually the philosophy is akin to turning an antique coffee-grinder into the base for a lamp: it’s there, so why not find a way to put it to some use. No, I see the richness of the floating compression principle to lie in the way I’ve used it from the beginning, for no other purpose than to unveil the exquisite beauty of structure itself. Consciously or unconsciously we respond to the many aspects of order in nature. For me, these studies in forces are a rich source for an art which celebrates the aesthetic of structure, of physical forces at work; force-diagrams in three-dimensional space, as I describe them.”
– Kenneth Snelson

The inventor of “tensegrity” structures was not Bucky Fuller. Fuller stole Snelson’s idea. See letter here: