Look at this photograph of St Clare's in Assisi, which is top in the series of photographs below, and at the Basilica of St Francis of Assisi in Santa Fe New Mexico, which is second. The first is 13th century and second was completed in 1886.
If one takes in each case the lower section (containing the door) it is bigger than the second, containing a rose window, which in turn is bigger than the third containing a smaller round window. Even though the lower section is subdivided in the Santa Fe Basilica, the main door unifies the two elements into a single larger one. In both there is a rhythmical progression upwards so that the first is to the second as the second is to the first.
Both these churches have proportions in which there are three sections of different size in consonant relationships with each other. Proportion is defined as a consonant relationship between to two ratios. As a ratio is a relationship between two magnitudes, there is a minimum of three different magnitudes needed to create two ratios.The beauty of architecture therefore is analogous to the beauty of music in which three notes are needed to define a chord. If you have just two notes you can have pleasing relationships - harmonious intervals - but the full chord needs a third not so that we know if it is participating in, for example, a major or a minor chord. We have seen this musical connection to architecture before in consideration, for example, of the octave, here.
It is not just the principle of three that is important here, but the principle of harmony is one that is derived from relationship between three distinct objects. So there is no harmony and no chord when all three notes are identical. They have to be different. One immediately thinks, of course that this might give us a sense of how the Trinity, three distinct persons, is Beautiful.
This three tiered design principle can be applied to just about anything - below are couple of buildings. The first is the grand Attingham House in Shropshire (seen before in the octave article); the second is an 18th century house in Frederick, MD; the third is in Newburyport, MA.
In each of the buildings above there is very little decoration - the elegance is derived almost exclusively from the proportions.
On a recent visit to the Cloisters Museum in New York I saw the following beaker and even a plant cut to follow the same design principle. In this way the whole culture can participate in the liturgical form which is at its root.
Now compare with this modern house below. This is in Frederick, Maryland too and it looks to me as though the architect is trying to design something to complement the colonial architecture that dominates the town. Yet because he has even sized windows and stories, it lacks this elegance. He is using the pattern of three, but because the windows are evenly sized he is not following the traditional pattern of harmonious proportion.