Thursday, February 14, 2008

The Fibonnaci sequence, In Our Time and why digital cameras should adopt the 35 mm aspect ratio forever

Not to mention my old neglected web pages. All intimately connected through the 1/Golden Ratio of "1.6180339887". That's a smidgen more than 8/5.

How connected?

To begin with, here's a (slightly revised) discussion from an ancient web page of mine about photographic aspect ratios, long the bane of the art:

John's Digital Photography Page

I thought printing was a peculiar curse of digital photography, but "Socrates" corrected me:

Printing is the curse of ALL photography and has been for a hundred years. 35 mm. has been the most popular format for probably fifty years. Until very recently (with 4X6 paper), there was NO paper available that matched the 3X2 aspect ratio. We had a choice of 3.5X5 or 5X7 or 8X10. The 8X10 was perfect for a 4X5 view camera used by professional portrait photographers. The smaller sizes were "almost" the 4X5 aspect ratio ...

In the case of digital photography, consumer camera sensors follow the convention of monitors: 640:480 = 4:3; on the other hand 35 mm film has a ratio of 3:2. Digital SLRs most often go with the 35 mm ratio of 3:2 (width to height), and some oddball Panasonic cameras use the video ratio of 16:9.

Not quite the same. The most popular print size, 4x6, is a good match to 35mm film, but not to most digital sensors. (BUT, Apple's photo books expect the 4:3 ratio.) If you print to a 4x6 paper; either there's cropping (most common) or dark bands are seen (better really, but unsightly).

This list of aspect ratios helps clarify the problem (here I use height/width as it makes the ratio easier to compare):

4x5: 0.80 (view camera and 8x10 prints)
3x4: 0.75 (most digital cameras and Apple's PhotoBook)
5x7: 0.71 (print size)
3.5x5: 0.7 (print size)
2x3: 0.67 (35mm and 4x6 prints - 35 mm is the diagonal for the old netatives) 
5x8: 0.63 (Golden Ratio, more or less)
9x16: 0.56 (video ratio, widescreen monitors)..

So, we clearly see that the so-called 4x6 aspect ratio of 35 mm film and dSLRs is closest to the Golden Ratio. Which makes that the best choice for digital photography sensor size.

Huh? How did I make that leap? Well, it turns out the "Golden Ratio" is baked into the mathematics of the universe, and it has some odd aesthetic appeal to the human brain (Parthenon, art, architecture, music, etc).

The Golden Ratio is derived by looking at the convergence of the ratio of number pairs in the Fibonacci sequence. That sequence was introduced to Northern Europe in the early 13 century by Fibonacci (In Our Time: 11/29/2007), who might have come across them while learning about Arabic (Indian actually*) numbers when adventuring in Northern Africa. The sequence goes like this:

1, 1, 2, 3, 5, 8, 13, 21, 34 and so on (each digit is the sum of the preceding pair).

the ratios look like this (note the 3/2 -- the 35 mm negative size:

1/1, 2/1, 3/2, 8/5, 13/8, 21/13, 34/21 and so on ..

So we see that our familiar 4x6 photographs are the 3rd item in the ratio sequence, and not too far from the covergent value of the Golden Ratio: 1.6180339887.

There's no arguing with the fundamental structure of the universe, adjusting slightly for tradition.

The 4 x 6 printed image rules. Let no pretender emerge.

* I thought our digits were an Arabic invention. They turn out to be Indian, but they came to Europe via Arabia. This is an interesting example of wrong things I learned as a child that I have not since revisited, an increasingly common phenomena.

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