When I look at most buildings designed in the traditional manner - this would be most built before the Second World War - it strikes me that the goal of the architect in his design is beauty and they seek visual harmony through an appropriate proportioning of the parts which will have different magnitudes. Generally, these will have been deliberately chosen to conform to a mathematical pattern considered to correspond to the pattern of the beauty of the cosmos and which is, in turn, a participation in the pattern of divine beauty.

In contrast, when I look at modern buildings built, say, since the Second World War, I discern just two simple guiding principles of architectural design. These are even spacing and random spacing. Neither, in my opinion, is a principle of beauty. The first, even spacing, generates visual monotony. The second, random spacing, generates visual cacophony.

The traditional design principle has its origins in the mathematics of the ancient Greeks and in one form or another was used, unquestioned, as the standard mode of design in art and architecture in the West until the period around the end of the 19th century when artists, architects (and musical composers) began, quite deliberately, to reject the tradition and with it all traditional forms. By the mid-20th century, it had not only been rejection but had been, with very few exceptions, all but forgotten.

Does this matter? I think so because I think beauty matters. The test for each of us to decide if it matters is to consider the buildings we would prefer to see, live and work in.

Consider first, this Georgian house built in 17th century England.

What we see here is a classic manifestation of visual harmony in which, like a musical chord which is comprised of three different notes, each story has a different magnitude and the combination is, to my eye at least, pleasing. That certainly was the intention of the architect in designing it this way.

Contrast this with the following building built more recently, in which every story is evenly spaced.

I would characterize this using another musical analogy. It is a visual manifestation of a string quartet in which four identical violins play nothing but the continuous sounding of one note. However, clean and pure that note might be, however perfectly rendered, it quickly gets dull to listen to. It is, quite literally, monotonous.

The building below is built on the same design principle but on a grander scale so that the result is the visual equivalent of a vast Mahler-scale orchestra, but once again consisting of only one instrument, say 100 violins, all playing the same note. It doesn’t matter how many times you replicate that note, it is still monotonous. If that monotone is blasted at us through a megaphone, which is the visual equivalent of what is happening here, it gets worse because we cannot escape and it obliterates all else around it that might be beautiful. In this case, it becomes offensive, a scar on the landscape.

Here’s another example displaying a different design principle. Look at this building below.

First of all, can you guess what its purpose is?

It's a church believe it or not. This random design is directed by uninformed intuition and is the visual equivalent of cacophony. It is like the effect you would get if you had an orchestra comprised of many different instruments with each musician just notes randomly and completely without any regard for what the others are playing. Here's another church. 

Does this look like a building made to house chant and polyphony and the worship of God? The piece of music that best corresponds to this design that I can think of is Stockhausen's absurd Helicopter String Quartet.

The traditional mathematics of beauty, in contrast, is an authentic analysis of the common human perception of the world around us and is richer and more varied as a result. Furthermore, it is the basis upon which a Christian architecture is built. The mathematics of harmony and proportion came from classical sources but was developed and enriched, just as instrumental music itself developed in the context of a Christian culture.

The more that we try to be different, as a deliberate statement of originality, the more, it seems that everything looks the same. The ugliness of so much modern architecture, and art and music for that matter confirms for me the truth of the principle that there is no order outside God's order, only disorder.

For those who want to know more about the mathematics of beauty, you can read my book, The Way of Beauty or for an even more detailed account, take the online class offered at www.Pontifex.University’s Master of Sacred Arts program called The Mathematics of Beauty.

The problem with this is that there is, as far as I can ascertain, little evidence that it has been used as a design principle by any artists or architects before the modern period. Furthermore, psychological studies (or art sales) don't support the assertion that people find the Golden Ratio beautiful; while scientific studies don't support the hypothesis that it exists in nature. These are all myths that arose towards the end of the 19th century which were subsequently adopted by new-age folk. From there many more, even traditional Christians, then assumed it to be true.  

In practice,  the Golden Section is awkward to construct and not easily incorporated into the design, and in the end, the effort doesn't seem worth it, because there's little evidence that its presence actually induces anyone to be more likely to commission the building or buy the art. So designers might include a token representation of it somewhere or else just ignore it altogether. Then they rely predominantly on the two fallbacks: the simplest mathematical relationship - even spacing which is expressed mathematically as the ration 1:1; or just untethered gut feeling, that is, pure subjectivity. Hence, we get either monotony or cacophony.

There is a contradiction in these two approaches arises from the fact that the validity of any mathematical analysis of beauty relies on the assumptions that there are objects around us that are universally considered beautiful; and that the beauty of those objects is a property it possesses, which can be analyzed in the same way that the scientist might examine any of its physical properties, such as its position, mass, weight or microstructure.  These assumptions are consistent with the idea that beauty is an objective truth and so contrary to the idea that it is only a matter of opinion.  

This assumption of objectivity relies in turn on the assumption that the world around us exists and therefore that through our senses we are perceiving something real. Our perceptions may be incomplete or partially distorted or affected by culture - which does give rise to some variation in judgment from one person to the next in some cases. But we are responding to a reality, nevertheless, and as a result, the responses of one person are close enough to others, generally speaking, for there to be a consensus on what is beautiful. What we find is that the greater the consensus, the more we see the same mathematical patterns permeating what is considered beautiful.

This is what was noticed by the ancient Greeks. Plato cites Pythagoras as the originator of the first theories of harmony and proportion. The Church Fathers who did most to reconcile them ideas with the Faith are St Augustine and St Boethius. Boethius lists 10 proportions, which he says come from Plato, Aristotle and later, unnamed, thinkers. These proportions are the basis of traditional harmony and proportion. In his book the Spirit of the Liturgy, Benedict XVI refers to the ancient Greek and early Christian ideas of cosmic beauty, referring directly to Pythagoras and Augustine, and notes how they permeate the culture and constitute a heavenly pattern for us to model our lives on in this world, most especially the pattern of worship. 

So, for example, it doesn't matter where, when or what cultural background people are from, most believe that creation is beautiful, and hear the fundamental harmonic intervals in music such as the octave, the perfect fifth and the perfect fourth as consonant. 

This irrationality and contradiction work their way into modern design. 

The traditional belief, going right back to the ancient Greeks, was that 

Then, there is the question of what mathematical relationship that is most commonly quoted as being fundamental to the beauty of nature is the Golden Section, followed by equal spacing.

In practice,  the Golden Section is awkward to construct and not easily incorporated into the design, and furthermore, the way that it is incorporated it doesn't generate any discernible increase in beauty, so normally designers either ignore it or they might have a token representation of it somewhere. Then they rely predominantly on the fallbacks: even spacing or just untethered gut feeling. The result is either

Quite often, in my experience, it is the same people who say both of these things and they don't seem to realise the contradiction. 

design. On the one hand  Might surprise some that there is a mathematics of beauty - but there is going right the way back to the ancient Greeks

It is an analysis of things that look beautiful which looks for common mathematical patterns in the whole (ie it is examining the emergent order). Contrast synthesis and synthetic analysis with scientific analysis, su

Existence, I am therefore I think.

We need consensus first - it begins with a human response.

The cosmos (heavenly music)

Musical harmony (instrumental music)

The beauty of man (human music)

The symmetry of relationships within number itself that match the natural order - arithmetic. (so striking is this correspondence that it was surmised that God ordered things according to numerical symmetry and it is the numbers that are the fundamental and common principle, which reflects  ‘the mind of the Creator’ [Benedict XVI]

You find this detailed in Plato (who says he got it from Pythagora), Aristotle, and other ancient Greeks. Boethius and Augustine particularly brought into the Christian world.

Boethius lists 10 mathematical formulae from which you can derive harmonious proportion. He says he got them from Plato, Aristotle and other ancient thinkers and each can be related to these source.

They are described in some detail in the Way of Beauty.

The assumption is that if these relationships underlie the pattern of beauty (which is about the emergent order), then man can order his life and work to the same order and then the culture and his life will be beautiful.

NB not the Golden Section!!!!

They were used in architecture and the design of just about anything made up to about 1900 when gradually the ideas of modernity rejected them. Still , the pattern of the calendar and sacred time follows these numbers.

For our purposes I want to discuss some key features o

The importance of Three. Proportion requires three or more different magnitudes. Look at this Georgian house The beuaty of God

Notice how there are three DIFFERENT magnitudes for each storey, as indicated by window size.

Contrast that with this modern building which has even spacing. No architect before 1900 would ever have built such a thing.

/rapahel

A chord in music requires three or more notes - music theory still acknowledges this because they can’t deny the common human response to the arrangement of notes that please most people. The second building is the equivalent of having an orchestra of many instruments, but every instrument is identical and the all play the same not over and over again. The result is dull.

The parallel with the beauty of God and Trinity as analagous to this is striking - three distinct persons and a unified whole.

Modernity rejects this idea along with the objectivity of beauty - beginning with Enlightenment thinkers eg Burke

We see the rejection of harmony in music (Schoenbergy) art - Picasso; and architecture (the Bauhaus) which are now generally accepted.

Example of the symmetry in arithmetic

The teractys. 1, 2, 3, 4

There is no order outside the divine order, only disorder.