Hello everyone, this is Andrés Cervantes and welcome to my channel ArquiCultura. When talking about beauty in art, the golden ratio is a topic as recurring as it is incomprehensible. And if we search on the internet we will find images like these, other forced ones like this one and even one that makes no sense like this one, well not that last one.
But what is this graph supposed to be? What does it have to do with the golden ratio? And what is he doing appearing everywhere every time this topic is brought up?
Let's go by parts, first of all, what does the word proportion mean? According to the RAE, proportion is: The disposition, conformity or due correspondence of the parts of a thing with the whole or between things related to each other . In other words, proportion is the correct one.
relationship of a part of an element to the whole or of the parts to each other. For example, this cat has a correct relationship between all its parts, but this other one obviously does not, very large legs, very small head and very long tail. Now, two elements of different sizes can maintain the same proportion and now mathematics will explain how.
In mathematics, proportion is the equality of two ratios or divisions, for example 2/4 is identical to 4/8, graphically it is the same and if we solve the divisions, the result will be the same. Let's go back to the cats, this one is 30 cm high and its legs are 10 cm, and this one is 45 cm high, and its legs are 15 cm, and now mathematically we have verified that these two cats have the same proportion. The same can be seen more simply with geometric figures.
With the issue of proportions clear, let's see what the issue of the golden ratio refers to. It turns out that when the ratio of two parts of a whole is equal to the next irrational number, that whole has the golden ratio and this number is usually represented with the Greek letter phi in honor of Phidias, the famous Greek sculptor. It was said that any shape that met this proportion was a shape of great beauty.
But what is special about this number and this proportion that cannot be found in other numbers and other proportions? If we have a line segment and we divide it into two parts, so that the length of part A divided by the length of part B is equal to the golden ratio, it will automatically be true that the complete length, between the longest part, also will result in the golden ratio. And that is only true in this case, let's see an example with numbers.
We have a line segment of 100 cm and we divide it right here, so that on one side there are 61,803 cm and on the other 38,197, if we divide it, as a result we get gives 1,618, but also if we divide it by 61,803, it also results in the same number. If we divide the segment so that on one side there are 60 and on the other side there are 40, when dividing them, the result will be 1. 5, but when dividing the total by the largest number, it will never give the same number.
This will only be true when the golden ratio is present. There is another property even more interesting and with practical uses, we have A, B and C, we know that if A and B fulfill the proportion, C with A will also fulfill it. Now let's forget about the existence of B and treat Segment A as the new small segment.
This means that segment C will be treated as the large segment and now we have a new complete segment, which we will call D and will meet the proportion with segment C and so we could continue like this to infinity. Let's see what we can do with this. If we have a rectangle with the golden ratio, the longest side will be A and the shortest side will be B, if we put side B on side A and split the rectangle there, we will have another golden rectangle and so on to infinity.
If we now take and draw a quarter of a circle in the squares, we will have this famous spiral that we all know. So it's time to start playing. If we repeat the spiral several times radially, we will have this figure, but if we also duplicate it, invert and superimpose it, we will have this interesting pattern, which although not identical, is very similar to the center of sunflowers, which is why it is common to find this .
Let's now see what happens with the architecture. More precisely the case of the Parthenon, The width of the building related to the height complies with the Golden ratio, that is, the façade of the Parthenon is framed within a golden rectangle. But this is just the beginning.
After placing the first square, a new golden rectangle appears. Once the stairs are located, the second square marks exactly the level where the columns end and the entablature begins. The number of columns and their separation is not just because they are found two by two framed in golden rectangles.
We locate another square and continue with the next rectangle. If we add its square to this rectangle, we will have the level where the pediment begins. We add another square and we will have a new golden rectangle.
And now we divide it on this other side with a square and it gives us a golden rectangle with the height of the architrave, and above it a square where the frieze and the cornice are framed. Finally we go to the frieze, this is made up of triglyphs and metopes, the triglyphs are golden rectangles with three vertical stripes and the metopes are square areas occasionally decorated with moldings, the set of a triglyph and a metope makes up a golden rectangle, and This pattern is repeated throughout the entire frieze. And the Parthenon is not the only one, the Cathedral of Notre Dame also follows this rule, but a little differently, since it is not like the Parthenon, it is enough to put this graph on it and observe the Magic.
The façade of Notre Dame is made up of different modules in the shape of a golden rectangle, and when dividing these modules with squares, we find that the dividing line passes through strategic points of the façade, for example, the keystone of the pointed arches of the lateral access, the starting point of the openwork gallery or also the starting point of the pointed arches of the towers. And in this way it is possible to find many coincidences of this façade with the golden ratio. These principles are not only found in the elaborate and ornate architecture of ancient times, but can also be found in more recent constructions, and not necessarily in large and complex monuments.
Next, I will try to make a simple facade design following these proportions. I will start with this graph as a starting point, by now we all know perfectly well what it is, I will use this line as the starting point of the cover, and this rectangle on the right will be the width of a volume in this part of the house, and This line will be the top of the overhang cover on this side. Now let's go with the access porch, for this I place the graph in this area here, this will be the thickness of the roof and I place a couple of columns, to locate the windows I will put a golden rectangle and a square, and I will repeat it, This will give the place for the windows, which will also be golden rectangles and I will align them with the edge of the porch roof, and I will also place another window in the projecting volume.
Having finished this, I gave it some volumetry, color and rendering, and thus a house was left, which although simple, complies with the golden proportions. And well, this has been all for today, if you liked it, don't forget to like it and share with your friends, and so you don't miss out on future content, subscribe to my YouTube channel, and follow me on the social networks of Facebook and Instagram. Andrés Cervantes spoke to you, see you next time!
