The world has an order, a logic, a pulse. Some call it Science and some call it God. Regardless of wording, there undeniably exists a framework by which we operate. Within this logic there appears an environmental cohesion made up of self-autonomous forces. While those forces can be intrinsically self-sufficient, they are not oblivious of their context; they work together to create a more perfect whole.
These laws that create all things big and small are blueprints for ideal things. That which is beautiful conforms to these blueprints. Like a mathematical asymptote, the closer the object of study approaches the x-axis (being Natural Laws, Godliness, Truth) the closer it achieves beauty. As natural beings we are predisposed to the preference for aesthetics dictated by nature. Therefore in order to create beauty in the man-made world, we must conform our designs to these natural dictates. In order to do so we must first evaluate what those laws are and what they look like. It is my belief that the natural world behaves the in the following ways: the world as a whole is a fractal; it simultaneously has both autonomous and dependent forces; there exists a recurring proportion in the design of nature; and nature seeks balance.
As a fractal, the natural world’s individual parts mirror the behavior and/or look of the whole — both as identical replicas and approximate similarities. There exists a singleness of thought, a consistency throughout; the larger picture and the minutia all have the same theme that supports the whole story. This existence of fractals allows us to understand the world by looking with both a macro and micro lens. The universe is made up of galaxies and galaxies are made up of stars. Our solar system and our earth fit within this structure and we, therefore, abide by the same logic that governs the larger system—laws of nature. An example of this can be seen in studying the frond of a fern; on each frond are leaves which replicate smaller and smaller ad infinitum. The branching of a blood vessel also highlights this continual duplication at smaller scales.
While in fractals similar stories are being repeated consistently over and over to create a larger story, each story is whole in and of itself. Take again that fern leaf, for example. Each leaf within the larger frond is a whole. A bug or parasite may affect the leaf next to it, but because they are separate it does not immediately affect the one leaf in study. Despite this autonomy, the effects on the leaf adjacent to it do in fact impact our particular leaf and in fact impact the frond and the whole plant as it has to fight off the parasite. In this sense, despite the individual autonomy, there is still an inter-dependence that takes place. On a larger scale we can look to plants in general, and animals. Each is an autonomous entity and yet there is a co-dependence between them: plants produce oxygen and breathe carbon dioxide; animals produce carbon dioxide and breathe oxygen.
When we start to examine how things actually look in nature we will find recurring patterns. Certainly things do not all look the same, however appearances can often be broken down into proportions and balance, and in those we find similarities. Proportionally, the Golden Ratio/ Golden Rectangle/ Fibonacci’s number all approach 1.618… We see examples of this ratio all around us in nature: the nautilus shell spirals per the Golden Ratio and the florets within a head of a sunflower’s spiral. Every day when we look in the mirror we continue to see versions of this proportion. As was studied by Plato (of the 4th century BC) and the ancient architect Vitruvius (of the 1st century BC), and which Leonardo da Vinci (of the 15th century CE) expanded upon in his drawing “The Vitruvian Man”, the proportions that are found in nature and which are embodied in the Golden Ratio are found in the natural human body. Hands, for example, display this in the proportion of the bones in the index finger; from the tip of the finger to the joint at the wrist, those four bones increase with a ratio of ~1.618. Similarly, the length of our hands in relation to our forearms approaches this ratio as well. Not surprisingly, the example of bone proportions continues to be proof of our fractal-ed world in that the ratio of smaller bones are the same as the ratio of larger body parts that the smaller bones make up.
Of course the other common proportion we find throughout nature is 1:1, visibly appearing as balance. Balance can be seen through symmetry in leaves, birds, and humans—a mirror image across one axis—or balance can be seen through a three-dimensional axiality as seen in the spiral of a DNA strand or tree. Yin yang has become synonymous with “opposite” (and therefore not symmetrical) but less attention and credit is given to the fact that the two sides are actually the same; they are just rotated mirrors of each other. So while this may defy symmetry we must keep in mind that balance and symmetry are not transferable. Where there is symmetry there is balance; however where there is balance there is not always symmetry. And therefore the beauty of yin yang is due to its satisfaction of balance.
On a small scale, we see natural occurring balance through opposition when we study electrodynamics: the behavior of atoms. Atoms most often have equal numbers of positive charged particles (protons) and negative charged particles (electrons). Molecules, being made up of atoms, seek to create combinations of equally balanced positive and negative charges and ideally the displacement of those charges should be uniform. When the total charge is not balanced, or when the particles are distributed non-uniformly, the matter is polarized; it is just not right! It is the scientific equivalent to a chicken running around with its head cut off—seeking something! These microscopic, natural occurring elements seek stability (balance) through equal and opposite distribution. On a larger scale this embodiment of balance occurs in basic principles of structures: tree branches are only as stable as the root structure is deep and strong. Again, we see this need for balance as a fractal: the same requirements for both large and small.
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It is unequivocal that the consistencies of dependency/independency, symmetry, and balance exist. As natural beings we live within, and therefore operate within, these natural laws. Therefore when we pass judgment on the world, we are doing so from the perspective of Nature. Subliminally we recognize the ratios and laws and we then judge all things based on their adherence to these tenets. The closer the object of study is in form to natural proportions the more we find it beautiful. As Karsten Harries elucidates in his chapter, “The Aesthetic Approach” in The Ethical Function of Architecture, “asked to pick out the most pleasing rectangle, most people will choose the golden rectangle. Psychologists have thus been able to show that there is a considerable cross-cultural agreement about aesthetic preferences of this sort.” If we assume that Science/God/Logic has an ideal proportion, it is not unrealistic to expand upon that and suggest that that which adheres to the principles are healthier than that which does not. Since shape and form tend to be a manifestation of the genetic makeup, the best health for a natural object is embodied in the perfect Golden proportion. This means that we then gravitate towards genes that indicate health. We recognize healthy and fit things as good and call them beautiful to encapsulate our appreciation of its perfection. As Theodore Cooke said in 1917, “Beauty connotes humanity. We call a natural object beautiful because we see that its form expresses fitness; the perfect fulfillment of function.”
In addition to rewarding objects that adhere to natural proportions, the occurrence of symmetry in healthy and fit natural things pre-disposes us to that aesthetic as well. As Stanford University’s Charles Feng wrote in his article, Looking Good: the Psychology and Biology of Beauty, “Scientists say that the preference for symmetry is a highly evolved trait seen in many different animals. Female swallows, for example, prefer males with longer and more symmetric tails, while female zebra finches mate with males with symmetrically colored leg bands… The rationale behind symmetry preference in both humans and animals is that symmetric individuals have a higher mate-value; scientists believe that this symmetry is equated with a strong immune system. Thus, beauty is indicative of more robust genes, improving the likelihood that an individual's offspring will survive. This evolutionary theory is supported by research showing that standards of attractiveness are similar across cultures.”
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We now know that the world contains these common rules and we understand that we judge the world based on its adherence to those rules. Since the world in which we live is a fractal, it follows that we should then want to extend the rubrics by which we define beauty in the organic world to things that are in-organic. If our aim is to create beauty in the man-made world, we then must adhere to and stay consistent with those nature-existing guidelines. Just as the natural world is a fractal with repeated constants, so too, then, must the man-made world consist of fractals in order to achieve beauty. Beauty is present when the object of study maintains an internal consistency throughout its structure while maintaining an external coherency with the world around it. Accomplishing this internal and external harmony is what causes judges to register the elementary feeling “it just feels right”; its being is an obvious answer to an un-posed question. This goal of beauty is certainly desired in art (including architecture, painting, music, etc), but I would like to use the word loosely to describe the sense of perfection in life as a whole. Given the three constants we found existing within the world, namely simultaneous autonomy and dependence, consistent proportions, and an overall balance, I’d like to study three areas that are near and dear to my heart and mind: architecture, music, and relationships. It is my belief that to obtain beautiful architecture, music, and relationships, each must maintain those same dictates that already exist in the natural world.
A beautiful building is not solely self-focused; rather it is dependent on its larger context. Just as the poet John Donne wrote of humanity, “No man is an island, entire of itself; every man is a piece of the continent, a part of the main” so, too, does architecture need to seek cohesion with the world around it. In architecture this entails designing a building that relates to the larger world, its site, and its inhabitants. This progression of connectedness results in a form that is no longer decipherable from the land. Its figures become entrenched in the landscape just as the landscape becomes entrenched in the building. Via windows, planar extrusions, materials, masses, and orientation every building necessitates inclusion of land. Stephen Holl, in his book Anchoring, acknowledges the site as the physical and metaphysical foundation of a building. He says, “Architecture does not so much intrude on the landscape as it serves to explain it.”
In architecture we have experienced these buildings that seem like an obvious creation. One of my favorite architects, Peter Zumthor, in his book Thinking Architecture, understands this achievement in creating the obvious through design:
The presence of certain buildings has something secret about it. They seem to simply be there… and yet it is virtually impossible to imagine the place where they stand without them. They make the impression of being self-evident part of their surroundings and they seem to be saying, ‘I am as you see me and I belong here.’
One might recall architect Howard Roark’s buildings from the fictional Ayn Rand book The Fountainhead. Although his buildings were quite controversial, they all catalyzed a zen-like reaction. This reaction stems from an adherence to nature. His buildings were not just an obvious answer, but were a solution to a problem that made the pre-existing condition better. Rand describes the sight of one of Roark’s creations:
…the ledges had not been touched… no artifice had altered the unplanned beauty of these graded steps. Yet some power had known how to build on these ledges in such a way that the houses became inevitable, and one could not imagine the hills as beautiful without them—as if the centuries and the series of chances that produced these ledges… had waited for their final expression, had been only a road to a goal—and the goal was these building, part of the hills, shaped by the hills, yet ruling them by giving them meaning.
I believe that buildings only create this reaction by seeking harmony with the world. To obtain harmony with the outside world does not solely demand respect for and inclusion of the site from a physical perspective, but it also requires history, mood, and memories be addressed. This means buildings must acknowledge and be dependent on all external factors. However, while beautiful architecture demands cohesion with the world around it, it must also maintain an inward cohesion. Alexander Baumgarten coined the term “aesthetics” in 1735, thinking it to be a description of perfection, which to him meant a perfect whole. He necessitates that this perfection is to “insist on the self-sufficiency of the beautiful. The artwork is autonomous: its point is not to refer beyond itself.” Similarly, Aristotle once said of architecture, it “is a world within a world, complete, integral, whole, a world where there is no contradiction.” While I agree that architecture must strive to create a perfect whole autonomously, I do not think that should come at the expense of disengagement from the world. There needs to be a seamless transition from without to within. It must balance self-sufficiency with dependency.
Due to the prevalence of proportion and balance in the natural world, architecture must physically incorporate those factors for it to achieve outward and inward harmony. Since a structure is ultimately for the use of humans, architecture must therefore relate to human proportions. Both Vitruvius and Le Corbusier reference the human scale as the dictator of proportions within the built world. Using the human proportions (which align with the Golden Ratio), a building and a space has an automatic appeal to its users because of their subconscious familiarity and comfort with the scale. The Parthenon is an example of a building which adheres to these proportions. Furthermore, the Parthenon also retains the desired balance in its built form in the manifestation of symmetry.
Just as beautiful architecture adheres to the standards deemed beautiful in nature, so too does beautiful music follow those tenets. Beautiful music contains a logical structure. In Thinking Architecture Peter Zumthor references composer John Cage. When composing, Cage does not sit and write an entire ballad. Rather, he “creates a skeleton of concepts and structures and has the music fill itself in depending on its sound.” The logic behind Cage’s work lies in its rationality; it can only follow that one thing will lead to another under these guidelines. This translates to a coherent sound or theme in a song, or even in a larger composition. There is an over-arching theme so that each piece—even each chord – fits among the larger composition, and yet each is self-contained and complete. The composition becomes its own fractal.
Although a singularity of thought and direction is laudable in design that does not translate to one repeated method of actualizing it. The strength of many classical compositions lies in the composer’s ability to present an idea and then repeatedly offer that same idea through different approaches (chords, intervals, transformations, octaves, etc.) As Bruce Director explains in his review of Haydn’s Opus 33, No.3 in the article “What Mathematics can learn from Classical Music” from the magazine Fidelio, “Haydn takes an extremely simple idea and subjects it to transformations across the musical space. The effect is to enable the listener to hear a greater density of change, while maintaining a unity of effect.” German poet and philosopher, Friedrich Schiller, expands upon this thought of coherent complexity in “The Aesthetic Lectures”:
The perfect, presented with freedom is immediately transformed into the beautiful. It is, however, presented with freedom, when the nature of the thing appears harmonizing with its technique, when it looks as if it were flowing forth voluntarily from the thing itself. One can also briefly express the preceding so: An object is perfect, when everything manifold in it accords with the unity of its concept; it is beautiful, when its perfection appears as nature. The beauty increases, when the perfection becomes more complex and the nature suffers nothing thereby; for the task of freedom becomes more difficult with the increasing number of compounds and its fortunate resolution therefore, even more astonishing.
This continual cohesion, despite compounding complexities is what characterizes a beautiful piece of music. Additionally, though, the relationship of the individual chords and notes must also work together and must follow proportional patterns that the human ear recognizes from nature and has associated with beauty. It is found that attractive sounding notes can be paired together in chords in ways that conform to nature. Bruce Director explains this further in his article “What Mathematics can learn from Classical Music”, saying:
The specific frequencies of these tones, centered on middle C = 256 Hz, are revealed by discovering the bio-physical properties of the bel canto qualities of the human singing voice. These discoveries show that there are six separate species of the human singing voice: soprano, alto (mezzo-soprano), contralto, tenor, baritone, bass. Each species has a characteristic set of physiologically determined regions where the singer, in order to maintain a beautiful tone, must shift from one register to another… Familiar relationships between sets of musical notes, such as transposition between chords, directly translate into geometrical structures... Composers understand these geometries without realizing it…
Music theorist Dmitri Tymoczko of Princeton University and his collaborators have found subtle relationships between progressions of chords. There is a mathematical coherence to the sequence of intervals between octaves. The divisions between octaves can be graphed in a form which results in a perfect spiral, implying a recurring pattern and mathematical symmetry.
Audibly there also exists balance through the use of speed and tension/resolution in a composition. Typically if one were to graph the speed and tension/resolution of a piece it would look like a bell curve, where the beginning and end have a similar disposition and the middle contains a complex set of iterations on the original disposition. While it may not look like a perfect bell curve (symmetrically so), it achieves balance in the same way that yoga does: through thoughtful attention to appropriate push/pull.
Creating a beautiful relationship requires allegiance to the same forces that create beauty elsewhere: individuals must operate as a part of a whole (be co-dependent) while meanwhile retaining autonomy and independency. And the individuals within the relationship must have some characteristics that are the same (symmetrical), yet also have a share of contrasts to challenge each other and balance the other.
The need for mutual dependence and independence is an age-old challenge, however it is one which must be sought after to achieve a beautiful relationship. Being in a relationship means you are joining a team—a group in which you collectively work for the betterment of the whole. Entering this mentality means forgoing the self. In best scenarios “the whole is greater than its parts” is a saying that applies to the most beautiful relationships. The “parts” are the amalgam of jigsaw-puzzle pieces that each partner brings. Each piece of the puzzle is self-standing but it is in the connection of the pieces that makes the puzzle whole.
There is symmetry in a beautiful relationship to the extent that these “puzzle pieces” are of the same material or of the same picture, so that there is a common ground to meet upon. Determining the “symmetry” variable in a relationship is one of the largest challenges, I believe. Do you meet on common ground that both partners are Christians, and that is the cloth that you are cut from? Or are you an avid outdoorsman and therefore seek an equally outdoorsy partner? What is that larger picture that your many puzzle pieces say you are? There must exist a symmetry, an equality, in the relationship for it to be beautiful.
And yet, as the Cambodians were frequent to say, “same same but different” is imperative. The exemplary relationships I have seen all exhibit a healthy balance: complimented strengths and weaknesses. No one person is perfect in all things (that I have met, at least!) and therefore the best combination is that of equal and opposite. Not only from a daily lifestyle perspective but in a biological perspective this makes sense. The couple that has varying strengths may have one partner with strength in theoretical thinking and the other in concrete thinking. This is beneficial for the wellbeing of the couple as a whole because it provides a broad perspective with which the world and life is approached. And it is also beneficial to their offspring, should they chose to have children, because they are reared in a more balanced environment. Balance also translates in relationships to roles and behaviors: historically seen by women fulfilling domestic role of child rearing and men the role of the family provider. Regardless of how the responsibilities are divvied, a beautiful relationship is one in which both partners have a balanced share of what they bring to the table.
It may be impossible to ever achieve perfect beauty in the manmade world, just as it is so rare to find that perfect leaf or the perfect snowflake, however because we are natural I believe the closer we stay to the rules and regulations of the natural order the closer we will be to achieving the beauty which we seek. With that said, though, architect and poet, Moshe Safdie once wrote:
He who seeks truth shall find beauty. He who seeks beauty shall find vanity. He who seeks order shall find gratification. He who seeks gratification shall be disappointed. He who considers himself the servant of his fellow beings shall find the joy of self expression. He who seeks self expression shall fall into the pit of arrogance. Arrogance is incompatible with nature. Through the nature of the universe and nature of man we shall seek truth. If we seek truth we shall find beauty.
So while we may seek to find beauty in the man-made world, perhaps we should first take a step back and seek truth, for only then will beauty be found.
February 25th, 2010
we're quickly stopping for internet in phnom penh before we head into the rural village of kampong plok where we are staying with a host family. not much time to update yall, but oh the stories i have...
so, our second day in siem reap allowed us to visit the remote stilted village on the edge of tonle sap. tonle sap is a fresh water lake (the largest in all of SE Asia) and is known for it's floating villages and markets, and it's community on stilts. our guide, sarou, lived en route to the lake and so, caving to our pleas, he took us past his house for a surprise visit to see his mom, sister, and home. it was a great experience to just pop in on a typical day of a cambodian family. his mom had had no education and can't write (even in khmer) and cannot speak any english, and yet the hospitality she displayed was easily translated and understandable. their family are farmers, and have rice paddies in the back, as well as corn, pigs, chicken, kapok trees and banana palms, as do most other families that live in that village. it was a fabulous impromptu visit (and we were the first and only group he has ever taken there!)
we arrived at the canal leading to tonle sap after having passed many bright green, vermilion rice paddies (which is rare since most of the fields have been arid due to it being the dry season) and boarded our boat. making our way out onto the grand lake, we past locals fishing by cashing large nets (while in the water) and families along the canal mashing dried fish for fish paste. the men were long limbed and sinewy, toned from daily labor and lack of meals. the children were even helping in the daily tasks. because it was early evening the sun was that warm, rich light (i have probably mentioned this quality of light before, but i am obsessed with it-- in any culture!) and the shadows were perfect. the colors were just perefct, if i could call it that. nothing was needed. despite the fact that the water was a muddy brackish color (like lake Shasta or the Chesapeake bay), and the banks the same muddy red/brown, made the colors of their clothes sing in contrast. that, coupled with being on a boat with warm breezes tickling our skin, made the experience surreal. man i loved it. but i guess i just love being on the water!
the sunset from the lake (once we made through the maze of canals) was peaceful and all was right with the world.
coming back we walked through the village. sanitation does not exist, so fires of collected trash that had been swept by families, provided the only illumination for our return.
darnnnnnnnn so much to say! but my bus is leaving NOW! hate to run... i still need to tell you about our bus hitting (and killing) a cow on yesterday's drive, and our experience teaching english on this island only accessible by a bamboo bridge! so i have to go. bridget ran to get me lunch (and for dessert we have a tarantula...! we decided we weren't craving it for breakfast so we've saved it for now)!
ok. now i'm late.
love love!