Something Must Be Going On

The heart of OCD is a feeling of "not being right" or repeating a ritual until it "feels right." A creative mathematician experiences intuition as a feeling there is "something going on here but I don't know what it is" according to William Byers in How Mathematicians Think. You were probably taught in high school that mathematics is a rigorous and logical endeavor and that for every mathematical principle there is a proof. It was implied to you that mathematicians seek out new principles by following threads of logic from an existing proof to a new proof. You were taught a myth. Most mathematical breakthroughs began with an intuition. Only later, after the instruction was explored well enough to believe it was true, to believe it was worth proving, perhaps even after it was proved to the satisfaction of the mathematician was an "official" proof created for the record. Proof comes after the fact, not before it. An interesting relationship between obsessive compulsive disorder and mathematics.

Moreover, instruction plays a vital role in creative mathematics. Just as in other creative arts, a shift of frame is required to turn the ordinary into the novel. The author relates the story of how he along with fellow mathematician John McKay noticed something curious about a single number. If you express adding one to 196884 as an equation you get 196884 = 196884 + 1. On the surface, it hardly seems worth the interest of a mathematician. You can add one to any integer on to infinity, something obvious to even non-mathematicians. What is so fascinatingly curious about this instance? As Byers writes, "...these are not just any two numbers. They are significant mathematical constants that are found in two different areas of mathematics." The relationship of the constants could not be a coincidence, thought McKay, who began a line of inquiry leading to a series of conjectures, which went under the fanciful but telling name of "monstrous moonshine." I want to linger a moment on this point. Here we have a mathematician who sees something curious, which prompts a "gut feeling" something systematic must be going on, a suggestion there may be a relationship between two systems of mathematics, who starts inquiring into the possibility, and as he finds more support for the reality of the intuition, he begins to make conjectures about how the two systems might be connected through the curiosity he discovered. At this point, we can hardly blame a mathematician for feeling he was chasing "moonshine." But that is exactly what creative people do. They chase moonshine and rainbows. Yet, somehow they end up driving the process of scientific rational, mathematical and artistic discovery. McKay's conjectures were later proved.

Byers does relate mathematical creativity to artistic creativity, observing good mathematicians (the creative ones) are very sensitive to the feeling of something going on, and ties mathematical intuition to the poet's, quoting the poet Denise Levertov saying "You can smell a poem before you see it."

This is all a blow to anyone raised on the rhetoric of rationalism. The human mind is a reasoning machine. Human beings are rational actors seeking the most efficient path. This ought to be nonsense to any carnival barker or snake oil salesman, but for most educated people it is a conceit they sustain because they enjoy the belief they are rational. Reason has become a virtue and virtues cannot be questioned.

At the bottom of human irrationality may be rational decisions, observations, the machinery of the mind is not metaphysical, but the abstract layers above the fine grain of deterministic reasoning are irrational. The mind is connected to a body. People get "gut feelings" as their mind tries to tell itself something from its emotional, pattern recognizing centers. How else could the pattern recognizing centers of the brain communicate with this supremely rational being, other than by kicking it in the gut?

I take away from this you will not be a creative scientist, mathematician or musician unless you learn to use your intuition. Exercise your curiosity. Keep a childlike sense of astonishment about the world around you or the inner worlds you explore. Experiment. Follow instruction. Don't worry about the result, the path to a Nobel prize in mathematics is not by seeking that which is likely to win a prize, but by following up an intuition, seeing where the thread will lead, without any thought to where it will go, other than to satisfy curiosity and that feeling of something must be going on.

Thoughts on the 4:3 Format and Golden Rectangles

I have a suspicion the near 4:3 ratios of the traditional photographic print sizes (probably based on traditional canvas sizes, but I am unsure of this...it seems likely) emerged due to a concentration of photography on portraiture in its early days and that photography adapted the canvas sizes used in painting, which very likely emerged out of portraiture. I am not entirely sure of this, but it seems reasonable to assume the majority of traditional paintings, as painting emerged in the Renaissance as an important feature of Western art, were portraits at first. The landscape I assume is a later invention as nature began to be seen less of a threat to life and more as an enjoyable extension of human space. We have to remember that nature, i.e. the forest, was a terrifying place for our ancestors and only in the 19th century did the modern conceit of the 'pastoral' emerge. So I hazard that most paintings were portraits. I doubt many of the first patrons wanted paintings of the landscape, they wanted paintings of themselves.

So, I conclude from this the 4:3 ratio may be ideal for portraits. Despite it not being as close to the golden rectangle as 3:2 format. This many explain why photographers who love 4:3 often speak of the difficulty they have with portraits using 3:2 format cameras and why landscape photographers say they prefer the wider 3:2 landscape. It may not be that there is _one_ ideal ratio for all images, but that there are ideal ratios for different _types_ of images. I remember my own struggles using a 35mm camera (3:2 aspect ratio) to take portraits, trying to frame the subject head and shoulders, either getting too much ceiling or too much waist in the finder. The 3:2 ratio frame is just too tall and narrow to comfortably fit the human head and shoulders, which may explain why 8 x 10 and other close to 4:3 ratio forms were favored in painting or early photography.

I would have to learn more about the size frequencies of traditional Western paintings before the 19th century to know for sure.

It is interesting to note 8 x 10, 5 x 7, 11 x 14 all are closer to 4:3 and are also the traditional photographic print sizes in the United States, which emerged in the 19th century with photographic print making and plate (glass negative) sizes. The one exception is 11 x 17, which is 1.54 and very close to the golden rectangle. I do not know how prevalent this size was as an enlargement in the 19th and early 20th century, but it seems to be rare. The 5 x 7 and 8 x 10 were the most common sizes from the mid-19th century to mid-20th century, up to the 35mm camera boom from mid-century to end of century. In the box camera era, many millions of snapshots were small, R3 or R4 I think they call it, 3 1/2 x 5 or 2 1/2 x 3 1/2 inches.

8 x 12 is close to the golden rectangle (12.94 / 8.0 exactly). I always thought the 11 x 14 was an odd size, but seemed to be commonly used in the 35mm days for enlargements, but is very far from 1.6, very distant from 8 x 12 since the nearest golden rectangle is approximately 8 x 14 (9 x 14.56)! An interactive golden rectangle calculator is available at
http://www.mathopenref.com/rectanglegolden.html

I always felt the pull to use the whole 35mm frame. It was just natural. I do not know why, but I loved to frame my compositions using the full viewfinder and hated to crop my images to the traditional sizes. I hated the enlarger frame used to hold the paper down with its fixed print sizes. I wanted to get one with sliding frames so I could choose print sizes like 8 x 12, but there was also the problem of obtaining paper in non-traditional sizes. I had some color prints made later in 8 x 12 after the influence of 35mm point and shoot cameras began to make prints and frames available in the 8 x 12 size for a while in the early 1980s. I used (horribly non-archival) "frameless" frames that sandwiched the 8 x 12 print between a piece of Masonite and glass and appeared to hang magically on the wall. Nothing interfered with the image I had seen through the lens at the moment I chose to trip the shutter. It was only later I learned that Cartier-Bresson had claimed using the whole 35mm frame introduced some special 'magic' to image making. I'm still not convinced he was not pulling our collective leg. There are images that 3:2 butchers and images that it helps.

It was just my style to want to see through the lens and then capture what I saw without thinking about cropping. I still prefer to print my 4:3 images (from an Olympus Four Thirds camera) 9 x 12 inches because I have always found 8 x 10 induces a "claustrophobic" feeling, where slightly upsizing to 9 x 12 gives the image room to breath and a feeling more like the 8 x 12 for some reason. It may have something to do with human visual perception, that as an image gets larger, it encompasses more of the visual field of the eye and "wideness" becomes less important.

An interesting question is when shooting in a 4:3 format, how does the "non-goldenness" of the frame affect compositional elements placed at or the frame divided by a golden rectangle? What happens when a 4:3 ratio rectangle is divided by the 3:2 ratio? We are told that an interesting property of the golden rectangle is if a section whose side is equal to the shortest side is marked off, a new golden rectangle is formed. So the frame is a golden rectangle and at about the third of the longest side is another golden rectangle, which is about equal to photographer's rules of thumb to place the horizon line at thirds vertically. Also, the "rule of thirds" points are about where the golden rectangle would place them. Does the 4:3 disturb the relationship between the outer frame and these golden divisions of the frame?

This is all mostly speculation based on intuition and memory, so don't take it as gospel but as a starting point for thinking about aspect ratio and composition.