Ildefons Cerdà’s formula and the Barcelona Superblocks

A few months ago I read a fascinating article from Harvard University on Ildefons Cerdà’s 1860 Plan for the Urban Expansion of Barcelona, and specifically, how and why it was conceived in such a unique way.

If you have ever been to Barcelona, or looked at a satellite image of the city, or recently heard about their new Superblock initiative, you are probably aware of the Eixample neighbourhood in the City. It looks like this –

Image

Those beloved and revolutionary blocks, as well as the detailed design behind them was engineered by Ildefons Cerdà in the late 19th century. The high mortality rates of the working‑class population and poor health and education conditions during the early and mid 1800s pushed Cerdà to design a new type of city, in which his main objectives were to obtain a high degree of wellbeing for the population through rational housing conditions and provision of services.


What stood out in the Harvard article published in 2011 for me was a formula that the authors identify Cerdà used to arrive at the ideal length/width of the blocks themselves –

My obsession over cities, math, design and planning seemed to converge perfectly here, and I decided to test out the formula myself. Below are some explanations from Cerdà on the variables and assumptions he used and how he arrived at them:

  • Three of his main concepts were used to determine the side length of the blocks (113 metres):
    1. the number of square metres per person
    2. the number of inhabitants per house, and
    3. the width of streets
  • His aim was to achieve optimal living standards of 6 cubic metres per person and room, and 40 square metres per person in housing: “Nowadays, one would need 6 cubic metres of atmospheric air per person and per hour in order to breathe correctly. Scientific studies establish a minimum of 40 square metres per person within towns” (Cerdà 1859)

In the end, the blocks measured 113m x 113m, while street widths ranged from 20 to 30 m, with 50m for wide boulevards (and 5m for sidewalks). Buildings on each block were about 10 to 20m deep with yards and green spaces in between each block, intertwined with pedestrian paths that linked the inner open spaces together independent of the street network. Furthermore, the corners of each block were cut at 45° angles (chamfered) and made 20 metres long to create small squares between the octagonal blocks. Heights of the building podiums were also limited to 20m to allow for sun penetration at an angular plane of 45°.

Chamfered
Intersection
Block dimensions
113 m x 113 m
Only 3 Sides of
Intevia could be
built- 1863
3 Street Types -
Façade...
Natural Light and Air
Intervia, translated as ‘block’, is a multi-functional
descriptor on what happens between streets.
2...

It is simply unbelievable how these “standardized” urban planning principles formulated by Cerdà (and others) from 150 years ago we still apply in our cities today. But what may be more unbelievable is that the formula and its explanation are incorrect! Yes I just said that about an article published from Harvard.


After plugging in the variables in the formula, my answer was nowhere near the 113m. I researched other articles and found one (from the Universities of Barcelona and Maastricht published in 1997) referencing a different formula. In fact, the 1997 article is a source in the 2011 Harvard article.

Below is the formula shown in the Barcelona and Maastricht Universities 1997 article:

I plugged in the variables in this formula and still was not getting the desired result, but it did get me closer to the 113m than the previous formula.

And then I read this:

“Few people in Barcelona know that this formula accounts for one of the most important features of the city. Still, no one knows where it comes from. Cerda did not write a single word to explain or clarify its meaning. It can be interpreted as a rhetorical device to black-box a particular technical detail, by appealing to the scientific and objective character associated with mathematical representation.”

Ok that was not the most promising thing to read – that the key principles responsible for much of Barcelona’s highly regarded and globally recognized urban fabric, of which many have been adopted by cities around the world, are based off a black-box of a formula.

But I did not stop there – I told my parents about this over lunch a few weeks back and my dad decided to help me unpack this formula (he is a civil engineer, so his math skills are certainly above mine), and he solved it! What we found out was that the formula actually checks out (whether it makes sense however is a topic for another discussion), and the issue lies in the presentation of the variables (and an incorrect formula in the 2011 Harvard article). Understandably, these academics were working with century old documents, so we can sympathize with them.

So our findings are as follows:

  • the formula in the 2011 Harvard article is incorrect
  • the formula in the 1997 University of Barcelona and Maastricht article is correct
  • b = 10 (2b is not a variable; b is the variable)
  • once you use b as 10, your answer is 113.28m

The system works.


Aside from the formula, the articles themselves are quite fascinating as they not only outline the approach and thinking Cerdà (and others) applied to design and shape the blocks (and the rest of the city), but the history behind it all, how key services would be accessed by the residents, and ultimately whether the vision for the City was achieved. (Spoiler alert – Apparently not, as economic pressures forced densification of the blocks beyond what Cerdà anticipated. As early as 1890, buildings occupied 70% of the block area on average, instead of the original 50%. In 1958 the building volume of the block, which according to Cerdà’s bylaws should not have exceeded 67,000 cubic metres, reached 295,000 cubic metres).

If you enjoy learning about the history of cities, planning and public works, I definitely recommend reading the two academic articles referenced in this post and the others shown below:

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