Let me begin by sayin I am a high school math teacher who has taught and traveled a great part of this world. My camera has always been by my side, Over the years I have begun to love and appreciate the intimate relationship between the worlds of mathematics and photography. The following is about the fundamentals of math that support our love for the camera.
One thing that becomes apparent in the study of photography is that the founders were scientists and mathematicians first. From their studies and research they established the foundation of this innovative device – the camera.
Nicephore Niepce is considered the Father of Photography. He created the heliography technique in 1825. which he used to produce the oldest surviving photograph. Louis-Jacques-Mandé Daguerre later in 1839 produced his one-of-a-kind images on a highly polished, silver-plated sheet of copper. William Henry Fox Talbot is best known for his development of the calotype, a photographic process that was an improvement over the daguerreotype. Talbot process actually produced a negative from which several prints cold be made. These three men of math and science laid the framework for our digital world today.
Regardless of the type of camera one uses, the entire process of capturing a moment in time is based on a solid foundation of mathematical principles. One could possibly argue that a basic understanding of this “photo math” is essential for the serious photographer.
I must admit that this question may initially perhaps appear either scary or trivial, but photography is by far the most mathematical of all the art forms. Within this little box we carry around on our daily outings, there is the arithmetic of things call f-stops, shutter speeds, and ISO settings, We also have the geometry of focal plane, focal point, and even focal distance. We even have the topology of depth of field and the hyperfocal point.
Numbers of Photography
How can one possibly decide what three numbers are the most influential in photography? Whether it is the iconic range finder Leica, a modern DSLR, or an expensive medium format Hasselblad, what numbers are the most fundamental or noteworthy for a photographer to know? Expressed another way, what numbers are indispensable in this artform without the photographer’s knowledge?
Choice Number Three
After careful soul-searching research, I narrowed the choices down to my Big Three. Beginning with my third choice, I decided to take the philosophical road, rather than the 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.
Definitely the most prestigious photography 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 the world of photography that is still felt today. Their goal was to present a “Purist” view of photography that was in direct opposition to the traditional standard of “Pictorialism.” These photographers believed that photography was more about interpretation than documentation. Radical elements of the times such as soft focus and toning became the standard in the art world.
The Purist movement enabled photographers to be considered peers to the painters and sculptures of the time. Prior the camera was simply used to strictly capture a portrait or an event. 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 fundamental tools of Group f/64 were a large format camera and 8×10 contact prints. Their aim was to push photography beyond the mere duplication of an event. They strove to somehow betray the unseen natural beauty of the moment. 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. Some controversial topics in photo clubs today deal with issues such as changing skies, composite images, photo stacking, and HDR. 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.
Choice Number Two
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. Prior only an elite few were able to master the large camera bodies, the chemical process, and the considerable time needed to produce even one image. The Brownie enabled all to at least capture a moment. The next major extension of photography as an art form, however, was the development ASA 400 film. The number 400 is therefore the second most influential number is photography.
The advent of ASA 400 film expanded the frontier for photography. Prior the camera was only functional in bright midday light. With this new film loaded into the Kodak Brownie, the photographer was free to shoot is relatively low light and enter the world of shade. Prior to this little camera and its new film, the photographers was limited in what they could shoot. However, with this relatively inexpensive camera and fast film, mom and dad could now create their own family photo album filled with life’s memories to be passed down through generations. They could shoot in either the front or back yard for these family heirlooms regardless of the sun’s location.
With the further development of 35mm camera in conjunction with ASA 400 Tri-x in the early 1950’s, the platform of quality photography became became available for the world, 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 today. Over this hundred plus years, photography was transformed from a few carrying large boxes on tripods, to basically one photographer per family with either an instamatic or 35mm camera, and now to a quality camera in everybody’s pocket. One could argue that more photos were taken in the whole world just yesterday than in the entire twentieth century all together. It took almost two centuries, but today nearly everybody is a photographer. Photography has truly become the art venue of the masses.
Choice Number One
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 sensor to half as much time as its predecessor. So therefore exposures are cut in half with each speed. The math is simple!
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 by definition means that the diameter of the lens is equal to the focal length regardless to the actual length of the lens. This is quite literally a one to one relationship! If the the focal length of a lens is five inches long the diameter of the lens must be five inches for it to have an f/1 rating. Obviously having quality glass that is five inches wide is close to impossible.
This is why telephoto lenses cannot be made as fast as a normal lens. The next stop is f /1.4, which would require a lens to allow only half the amount of light to pass through. Realize also that light passing through a lens is strictly a function of area of the lens surface. The next stop of f/2 would require an area half that of f/1.4. and so on. In conclusion, for shutter speeds we simply multiply or divide the speed by to to change the setting. For a lens, we much divide or multiply the area of the lens by 2 in order to accomplish the same. So how did we ever get f/1.4?
Some Critical Definitions
1. Focal Plane: This is basically the plane where the film or sensor is located.
2. Focal Point: Point where all light converges after entering the lens.
3. Focal Length: Distance between the focal point of the lens and the focal plane.
Circles and Lens
Review: An f / 1 aperture would mean that the diameter of the lens is equal to the focal length. The key here is that diameter is important in lens theory, and obviously the area of a lens controls the amount of light passing through. Area is a function of radius, not diameter. We have a problem!
Going back to some basic high school math, one hopefully remembers the above formula for area of the circle, and also the fact that the radius is simply half the diameter. Therefore, if we are cutting the light in half with each stop, we must first divide the diameter by two to determine the radius. The number two just became the driving force with f-stops. Without doing a lot of math, realize to finally undo this radius/diameter dilemma, we will need 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.
What is the square root of 2 ? This should look familiar. The diameter of a lens that allows half the light through the lens is 1.4 of the diameter of an f/1 lens. The first major stop of a normal lens is therefor f/1.4. Math review: The square root of two can be written 2 to the 1/2 power – (2^1/2). Remember also when we multiply like bases we add exponents, so 2^1/2 times 2^1/2 equals 2^2/2. This simply equals equals 2^1, which equals 2. Isn’t f/2 the next lens opening?
The above chart shows that basically each f/stop is simply the square root of two times the previous one. What this means is that every time photographers change an f/stop they are 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 mathematicians of photography and appreciate the rigor involved in creating something rather complicated that could be attached to the simple body of a lens.
Another Reason for Selecting the Square Root of Two
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 shown above.
This may seem trivial, yet when one is printing, the aspect ration of the digital image (1.5) has an effect on the paper size you need. The problem is one basically ends up cropping and leaving wide wasted margins on the photographic paper. The issue is to keep this wasted paper to a minimum.
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 the A-list is dominant. The initial 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. Not this paper size is a little larger than an 11/14 with a more “flattened” aspect ration.
Now let’s compare the A3 size to our standard 11×14 paper used in the U.S. Checking the American paper sizes above and remember that root two is about 1.4142, it appears that the 5×7 crop factor you use in Photoshop will enable you to fill an A3 sheet of paper to the limit However, if you print the same image on 11×14 paper, you will have a lot of wasted area. Actually printing any image on A3 paper gives you almost 20% more geography to showcase your work. Note also that A3, as well as 11×14, can be matted easily on a 16×20 mat board.
The point here is not to discuss the desirability of various print paper ratios. Obviously a 4×6 aspect ratio is suitable for many landscapes, 4×5 for some portraits, and 5×7 for much in between. These are all personal and artistic choices the photographer makes to best showcase the image. However, on A3 paper more image will always be shown.
In conclusion, I believe if we were to do a study of all photographers throughout history, we would discover that their minds function very mathematically regardless of their formal math training. We all seem to use both the right and left side of our brains in the creation process. Besides the technical aspects mention above, consider all the other ways that mathematics enters our art. We constantly consider the geometry of composition with points such as the rule of thirds and the the geometry of composition balance. We obviously love the art of photography, but in a strange and beautiful way we also respect the math that defines and supports it.