Proportions derived from the square and openings

[Moon Adamant]

Hi all

To link to my previous post and the importance of the openings on this vernacular architectures of Southern Europe, I'll write a bit now on common proportions for the openings on traditional construction.
This all makes sense when you think that in most cases, the tools available would be rather simple like ropes and plumb lines, so simple geometric methods were developed for them.
A very surprising one is the tool to make a Pythagorean (right) triangle with rope.

People would take a rope and make 12 equally-spaced knots on this. Then, by folding the rope so that the segments would have 5+4+3 knots, they'd get a right triangle and so would be able to set things at right angles.

So, for example, to make a square, they would nail down a rope and draw a circle, its radius being the desired square side. Then they would drop the plumb line from the center of the circle till it met the circle itself, finding the vertical side of the square; and from it they could apply the 12-knot rope to determine the horizontal side.

Another simple construction is what I am calling the root-of-2 rectangle, as I don't know what it is called in English. It is constructed by drawing an arc of circle through the diagonal of the square. If you take a square of side=1, then the diagonal of that square is √2. This rectangle has a proportion approximately 1:1.41 -- handier to know the numbers when you are working in SL

The root-of-2 rectangle is often called the golden rectangle, but that's not correct! The true golden rectangle is more elongated. Its proportion is the Gold Ratio or φ (Phi), so about 1:1.62.

The construction is a bit more complicated: you find the median point of the side (which you can do easily with a rope; in the drawing, the dash-dotted line with the symmetry sign marks it), and from that median point you draw an arc of circle to a new diagonal and extend the side.
The golden rectangle is considered specially pleasing, and you can find it everywhere -- for example, the A-format paper-sheet is a golden rectangle, as the golden rectangle is infinitely recursive. You can also draw the logarithm spiral from it, and that spiral exists in nature from the arms of galaxies to sunflowers and mollusc shells.

Then, you have the Double Square proportion, or 1:2. This is the proportion of a 512*1024px texture It is just two equal squares

This used to be a common proportion for doors, as a 2m-tall door would be 1m-wide, making for a very comfy door to go through (most human bodies will occupy a width of 0.80m when carrying something in their arms, like a tray or a pot). Nowadays, most mass-produced doors for domestic settings will typically be produced in increments (for example, widths of 0.65, 0.70, 0.75...) and so will miss proportions altogether.

And you can have Triple Square, so three equal squares, or 1:3.

BTW, I have shown here tall rectangles, because on my other post I was talking about the vertical expression of the vernacular architecture in Tuscany, but obviously you can have wide proportional rectangles too.
I am having issues placing more pictures in this post, so I am going to finish it and do a reply to go on

[Moon Adamant]

Let's go back to that pic of the Casale in Lucca. I did a VERY quick proportional study:

So on the left, I am marking the openings. We can see that the three main openings are Double Squares, and that on the top right we have two similar windows that are Golden Rectangles. Then we have an odd window on the lower right which is misaligned (and looks like a later addition), but even that oddity is a Root-of-2.
Now, this kind of proportion analysis can be endless and maddening the purpose is not that we end up obsessing about these. But we can take this into account as we think our builds.
For example, let's say that we are now completely convinced about this vertical expression thingy! But the plot that we are building for has a wide front so it seems like it demands a wider building. Seems like a contradiction, but not really.
Taking this Lucca casale again, we can see that actually it is a rather long building. But you have a rhythm on the façade: mass, void, mass, void... that
splits the façade visually into narrower, taller portions. And the rather tall openings accent that too. Imagine that façade if the openings were square -- it would look much more squashed. So if you have a wider front for your building, a rhythm of tall openings can stretch it vertically.
A good idea here can be that to do a set of mock windows - rectangles in the different proportions -- and just play with those on the façade as markers till we find something that works.

Now, I know very well that buyers always want the bigger windows possible, and that can be a consideration that makes the whole façade thing a tad more difficult. So some tricks for that:
First, we can do large windows. For example, if you go to CN E.02, where we held the gov hours today, you'll see that those windows there almost reach the floor. A window with a bench seat can start 40-50cm above the floor, instead of the usual 90cm to 1m.

Or windows can almost reach the ceiling too (though mind that generally windows and doors align at the top; the door can then be solved if it has an extra fixed pane on top to match the window's height).
Then we can have French windows even in higher floors if there are balconies, either continuous or one tiny balcony for each window.


Or even just a close metal rail that fits completely inside the window's stone limbs.

Then you can place an unique window layout as the main opening of a room. They are not meant to be repeated throughout the building, but they can mark the importance of the space. Half-round windows, for example, or Palladian (Venetian) windows:
https://www.alamy.com/stock-photo-shutt ... archtype=0
https://www.alamy.com/stock-photo-windo ... archtype=0

Or even plainer windows that have fine details: their colour like this lovely blue tint on the shutters, or that beautiful, very understated classic pediment
https://cdn.w600.comps.canstockphoto.co ... 604889.jpg
Have fun