## The Three Most Influential NumbersIn the World of Photography

As a high school math teacher, it is no wonder why I became attracted to this adventure we call photography. No other art genre is more mathematical than that which involves a camera. It is not surprising that William Henry Fox Talbot, the Father of Photography and inventor of the metal-based daguerreotype process, had a strong background in mathematics and engineering. He also published the first book in 1844 with actual photographs featured.

William Henry Fox Talbot (1800 – 1877 )

Regardless of the type of camera one uses, it and the entire process of capturing a moment in time is based on a solid foundation of mathematical principles. One could argue that a basic understanding of this "photo math" is essential for the serious photographer.

I must admit that the thesis question I pose here is slightly trivial, yet I hope interesting. How can one possibly decide what three numbers are the most influential in photography? Whether it is a range finder Leica, a DSLR, or a medium format Hasselblad, what numbers are the most fundamental or noteworthy for a photographer to know?

So many choices

After careful soul searching, I narrowed down my three numbers. Beginning with my third choice, I decided to go philosophical, rather than technical. This number and its place in photographic history is still relevant today. For third place I decided on the number 64. This number may appear as a strange choice, but as important as it was in the last century, it is perhaps more important now.

Definitley the most prestigious photo club of all time was called “Group f/64.” In the early 1930’s a distinguished group of eleven icons of photography including Ansel Adams, Imogen Cunningham, and Edward Weston, formed a union that produced a lasting impact on photography that is still felt today. Their goal was to present a "Purist" view of photography that was in direct opposition to the academic west coast standard of “Pictorialism.” These photographers believed that photography was more about interpretation than documentation. Soft focus and toning were the standard in the art world.

Historical wok of Ansel Adams and Edward Weston

In countering this view and establishing the foundation of the Purist movement, Edward Weston stated, “The camera should be used for a recording of life, for rendering the very substance and quintessence of the thing itself, whether it be polished steel or palpitating flesh.” The basic tools of Group f/64 were 8x10 contact prints and glossy paper. The goal was the sharpest image possible, and the large format camera and glossy paper enabled that end. Their aim was to capture the natural beauty of the moment, not to interpret it. The title “f/64” was derived from the smallest aperture of their large format cameras which obviously produced the maximum in depth of field.

So why is this number important today? The rival two theories of photography still are alive and well, yet they have learned to coexist in a world of mutual respect. Today photography has grown to accept each views. Both tripods and plug-ins are the norms where all photographers strive to either “capture or create” reality as individually seen through their own eyes.

The Kodak Brownie

Before I give my second number, here is a little photo history. The introduction of the Kodak Brownie in the early 1900’s brought photography to the masses, since only an elite few were able to master the large camera bodies and involved chemical process required for development. In conjunction with that, my second choice number is 400 as in ASA 400. The advent of ASA 400 film expanded the frontier that Kodak began even farther as it enabled the photographer to now get out of the sun and venture into the world of shade. Prior to the Brownie, there were no photographs know as snapshots, but with this small and inexpensive camera, mom and dad could have their own portfolio of family photos. With the availability of ASA 400, they had the further option of stepping into either the front or back yard for their snapshots regardless of the sun's location.

With the development of 35mm ASA 400 Tri-x in the early 1950's, the platform of photography became the world, regardless of time of day or position of the sun. The camera was on the verge of becoming a universal fixture in the home, but this transformation took another fifty plus years with the universal growth of the smart phone camera. This competed the photographic revolution as we know it. The world was transformed from an elite few possessing large bulky cameras to a place where a camera is in everybody's pocket. One could argue that more photos were taken in the whole world yesterday than in the entire twentieth century. It took almost two centuries, but today nearly everybody is a photographer. Photography has become the art venue of the masses.

However, for my number one choice, I had to go technical for two reasons. Many photographers may not realize it, but by far the most important and utilized number in the photographic process is the “square root of 2.” The average photographers are probably asking themselves now, what??? One will never see this number on a camera, lens, or even film, but a properly exposed photograph could never have been taken without it. This shy reclusive number remains hidden behind the curtain, but its presence is felt every time a lens is adjusted for an exposure.

Now is the time for a little math lesson without too much detail - I promise. Two things control the exposure of an image. One is the shutter speed with numbers such as 1 sec, ½ sec, ¼ sec, 1/125 of a sec, 1/500 of a sec and so on. The math here is rather simple. Each shutter speed merely exposes the image to half as much time as its predecessor. so therefore exposures are cut in half with each number.

Now let’s look at those crazy numbers on your lens. They are basically as follows: 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, and 22. Could there be a more random group of unrelated numbers anywhere? If there is a reason or pattern, it is not at all obvious – or is it???

This is one fact of which all photographers should be aware. An f/1 aperture would mean that the diameter of the lens is equal to the focal length. They are quite literally one to one! This is why long lenses cannot be made as fast as a short lens. An eight inch f/1 lens would require an eight inch wide diameter, and this would be a monster piece of glass to produce. The next stop, f/1.4, would require a lens with only half the area, so it is becoming more feasible. So with f-stops, the area of the lens is the key, yet with shutter speeds, it is only the time film is exposed.

Area of a Circle

So diameter is important in lens theory, and obviously the area of a lens controls the amount of light passing through. Going back to some basic high school math, one hopefully remembers that area of a circle is a function of the radius, not the diameter, and you may also recall that the radius is half the diameter. Therefore, if we are cutting the light in half with each stop, we must first divide the diameter by two to get the radius. The number two just became the driving force wit f-stops. Also remember that the area of a circle is “PI r squared.” Without doing a lot of math, realize to undo the radius we have to undo its square, which means we have to take its square root. This is a very abbreviated look at how we come up with the square root of 2.

Now let’s suppose we could engineer an acceptable f/1 lens. The next step would be to generate an f-stop allowing half as much light to pass. We therefore would simply multiply the 1 by the square root two. Well, what is one times the square root of two – about 1.4. Does this look familiar? F/1.4!!! If we continue the process of establishing the remaining f-stops, all we need to do is multiply each by "root 2 " F/1.4 multiplied by "root 2" is simply 2; F/2 multiplied by "root 2" is 2.8; F/2.8 multiplied by "root 2" is 4; F/4 multiplied by "root 2" is 5.6; and so on. We have thus generated the f-stops on our cameras with each allowing half as much light to pass through as its previous, and this is all based on this irrational number - the square root of 2.

What this means is that every time a photographer changes an f/stop he or she is simply multiplying by the square root of two, which has the effect of cutting the amount of light passing through the lens in half. It is perhaps a good idea now to think fondly of the early engineers of photography and appreciate the rigor involved in creating something rather complicated that could be attached to the simple body of a camera.

However, there is a second reason for choosing the square root of two as my most influential number in photography, and it does not relate to the camera exactly. It more accurately is associated with the final product – the print. In America our basic paper sizes are all over the place in terms of a unified aspect ratios, which is simply the long side of photographic paper divided by its short side. The basic paper sizes and their aspect ratios are as follows:

This may seem trivial, yet if you are printing, the crop factor you choose for your digital image could have a limiting effect on the paper size you want unless you basically end up cropping again or leaving wide margins on the photographic paper.

Now let’s consider how the rest of the world handles the varying paper sizes and aspect ratios. For the most part they use a system of four major categories defined by the letters A,B, C and D. They are all related, but in photography we probably only use the A list papers. The basic sheet of A paper is called A0, which is the beginning point. One sheet of A0 has an area of one square meter and an aspect ratio of, guess what, the square root of two. Now this is a rather large piece of paper, so if we fold it in half we have an A1 size, yet the aspect ratio remains constant, root 2. If we fold it in half again, we have the A2 size, again with the same aspect ratio. One more fold gives us the popular A3 size, which is 11.7 by 16.5 inches, again with an aspect ratio of root two.

The A0 sheet of paper subdivided into the various A sizes

Now let’s do a little comparison. If we look at the American paper sizes and remember from above that root two is about 1.4, it appears that the 5x7 crop factor you use in Photoshop will enable you to fill an A3 sheet of paper to the limit. The point here is not to discuss the desirability of various print paper ratios. Obviously a 4x6 aspect ratio is suitable for many landscapes, 4x5 for some portraits, and 5x7 for much in between. These are all personal and artistic choices the photographer makes to best showcase the image.

In conclusion, I believe if we were to do a study of all photographers, we would discover that their minds function very mathematically regardless of their formal math training. Think of all the aspects of photography that involve the use of math: Aperture, Shutter Speed, ISO, Depth of Field, Inverse Square Law, Focal Length, Focal Point, Hyperfocal Distance, Sensor Size, Sensor Resolution, and finally the Geometry of Composition. We obviously love the art of photography, but in a strange and beautiful way we also respect the math that defines and supports it.

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