F-Stop in the Name of Love

I love photography. When I was younger, my dream was to shoot for National Geographic: to become the next Steve McCurry, the next Mike Yamashita, the next Ami Vitale, Nick Nichols, or Josh Haner.

One afternoon in my mid-20s, though, I visited the Newseum in Washington, DC, which included a photojournalism gallery with every Pulitzer Prize-winning image since 1942. There’s an almost other-worldly quality to great photography, an almost superhuman ability of great photographers to tell a story — to reveal an entire narrative — in the click of a single shutter. I spent hours staring at these images, marveling at the lighting, the layering, the angles and depth. And when I left the museum I thought, “I’ll never be that good.” It wasn’t an angsty realization. I wasn’t frustrated. I just knew, “These photographers literally see the world differently.” So I got into math education instead.

Indeed, photography is a way of looking at the world; it is a way of seeing. The difference with the naked eye, of course, is that this form of seeing involves a camera, an artificial device with its own virtues, but also its own limitations. Taking fully advantage of the artistic potential of photography, then, means understanding how a camera works.

“Photography” derives from Greek words phōtos and graphé, and means drawing with light. When you take a picture, there are two* main settings that control how much light hits the film (or these days, digital sensor): aperture and shutter speed. Aperture refers to the diameter of the lens opening**, and is measured in f-stops. Once you focus on your subject, the aperture determines the depth of field: how much in front of and behind the subject is also in focus. As you can see below, the smaller the aperture, the greater the depth of field.

Though the lens is focused on the wooden mask, a small aperture of f/22 means the chess pieces in the foreground and the bicycle helmet in the background are also in focus. When the aperture is f/2.8, on the other hand, they’re blurry. Here, the large aperture results in shallow depth of field, which is common for portraits.

This is the artistic side of aperture. Of course, there’s also a mathematical side. A small aperture (i.e. a large f-stop number) corresponds to a small diameter, which means that not much light is hitting the film. At f/22, the area of the lens opening is 12 mm2. Contrast this with an aperture of f/2.8, where the area is 726 mm2. With each additional stop, you’re more or less doubling the area.

Why does this matter?

For any given scene — inside by candlelight, outside on the beach — there’s a certain amount of light that’s needed for the image to turn out correctly. If there’s too much light, the image will appear too bright, or overexposed. If there’s not enough light, the image will appear too dark, or underexposed. Compared to f/2.8, an aperture of f/22 will contribute less than 1/60th as much light!

But what if you want to use a smaller aperture? If you’re taking a landscape picture and need more depth of field, how do you ensure that enough light reaches the sensor for the photo to turn out correctly?

This is where shutter speed comes in. While aperture refers to the size of the lens opening, shutter speed refers to how long the film is exposed to light. Inside every camera is a shutter — a guardian door — that prevents light from exposing the film. When you take a picture, the door opens to allow light to pass. The shutter speed refers to how long this door is open. (Actually, since speed is a rate, e.g. miles per hour, a more appropriate term would be shutter time. But anyway.)

If you use a small aperture, you have to compensate with a long shutter speed. Large aperture? Compensate with a short shutter speed. To better understand this balance between aperture and shutter speed, play around with the settings below. The original image was correctly exposed with an aperture of f/8 and a shutter speed of 1/60th second. As you can see, it has a moderate depth of field; the objects in the foreground and background are somewhat in focus, but also somewhat blurry.

But what if you wanted more of a portrait look — i.e. a shallower depth of field — and shot at f/2.8? In this case, the lens opening would be letting in around 8 times as much light. To maintain the correct exposure, you’d need to use a shutter that was 1/8th as long, or 1/480th second. (Since the closest standard shutter speed is 1/500th second, you’d use that.) Meanwhile, if you wanted more of a landscape look and shot at f/22, you’d lose light through the lens opening, and would need to compensate with a much longer shutter speed.

Portrait and low depth of field? Landscape and high depth of field? When you care mostly about aperture, shutter speed will follow. Sometimes, though, you may care more about shutter speed, in which case aperture will follow. Technically, a fast shutter speed doesn’t let in much light. Artistically, it freezes motion. Technically, a slow shutter speed lets in a lot of light. Artistically, it creates the effect of motion blur.

The original image above was correctly exposed at f/5.6 and 1/125th second. The relatively large aperture results in a fairly shallow depth of field (notice how blurry the objects are in the background), while the relatively fast shutter speed freezes the bicycle rider’s shoes. It’s not enough to freeze the wheel, though, which is spinning much faster. To accomplish this, you’d use an even faster shutter speed; at 1/500th second, the spokes are suddenly visible, and the rider’s feet appear almost stationary. Of course, since the shutter speed is contributing so little light, you have to compensate with a much larger aperture, which reduces the depth of field even further.

At the other end of the spectrum***, a shutter speed of 1/8th second (an eternity in photo-time!) means you can almost feel the rider’s legs moving around in a circle, while the corresponding aperture of f/22 decreases the diameter, balances the light, and results in greater depth of field.

And it’s this word — balance — that characterizes the technical side of photography. Every scene requires a certain amount of light, and there are a number of aperture and shutter speed combinations that will result in the correct exposure. However, they’ll all yield different images, and it’s up to you as a photographer to decide which one offers the best, most beautiful, most authentic view of the world.

The slideshow below includes some of the photos I’ve taken in my own travels. As you browse them, try to get a sense of the aperture and shutter speed, and how they contribute to the final image. I’ve included notes to help. Enjoy, and happy shooting. Who knows: maybe you'll be the next National Geographic photographer!

Can you see your students having this conversation in math class? Check out the supporting lesson materials.


*On modern digital cameras, there's actually a third key setting: ISO. The ISO refers to how sensitive the digital sensor is to light. At ISO 50, the sensor is not very sensitive, i.e. it needs a lot of light to correctly expose the image. At ISO 6400, the sensor is much more sensitive to light, which means you can get away with using a smaller aperture or faster shutter speed (or both). However, the higher the ISO number, the grainer the image. The same was true for analog film, which also had ISO (or ASA, depending on the manufacturer); ISO 50 was recommended for sunny days, while ISO 400 was recommended for indoors. The main difference with digital cameras is that, unlike with film cameras, you can adjust the ISO on the fly.

**Every lens has a focal length: the distance from the lens to the film plane. A lens with a focal length of 24mm is a wide-angle lens, while a 200mm lens is a telephoto lens. The diameter of the lens opening is found by dividing the focal length by the f-stop number. In this example, we used an 85mm lens. At f/22, the diameter is 85mm ÷ 22, which rounds to 3.9mm. Had we used a different lens, the diameter would have been different.

***In reality, the spectrum for shutter speed extends far beyond 1/500th second and 1/8th second. Unlike with apertures, the number of which is limited by the mechanics of the lens, there's really no shortage of possible shutter speeds. Cameras can shoot anywhere from 1/8,000th second to "bulb," where a photographer can start the exposure and then come back hours later to stop it. (This is used for stellar photography.) Incidentally, this is why standard shutter speeds don't follow a perfect pattern. From 1/30th second to 1/60th second to 1/120th second...you'd expect the next shutter speed to be 1/240th second. However, it's 1/250th second. This allows the subsequent speeds to be 1/500th second, 1/1000th second, etc., which is a lot "cleaner" than 1/480th second, 1/960th second, etc.


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