The definition of a photograph is, “a picture produced by light making contact with light sensitive material in a camera.” But really, it is much more than that. Ansel Adams said, “[a] great photograph is one that fully expresses what one feels, in the deepest sense, about what is being photographed.” Yes, a true photograph is one that cannot be explained with words, but the process which is used to get to that result is crucial, too. From the smallest piece of a camera, to the most complex photographic concept, every step of creating a photograph somehow relates to math.
The Camera: Convex Lenses, Focal Point, Focal Length, and Field of View
Almost every camera has a convex lens. A convex lens is a lens where the middle of the glass is thicker, while the edges are thinner, as opposed to a concave lens where the middle of the glass is thinner, and the edges are thicker. When light travels through a convex lens, the light passes through the lens at an angle and bends slightly, creating an upside down image. To further explain, take a tree for an example. If light from the top of the tree comes through the lens at an angle, that same light will be at the bottom of the image. The same process happens with the light at the bottom of the tree. Both the angle of the light and the convex lens contribute to the making of an upside down image. Digital cameras are programmed to flip the image so it doesn’t appear upside down when we look at it.
The place where all parallel light rays gather into one point is called the focal point. It is also the place where light traveling through the lens will be in focus. The distance from the center of the lens to the focal point is focal length. Focal length is longer or shorter depending on the length of the lens. Short focal lengths give a wide view and are called wide angle lenses. Long focal lengths have a narrow view and make objects seem closer. These lenses are called telephoto lenses. In between the long and short focal lengths are normal focal lengths. Normal focal lengths see similar to the human eye. The range or view that can be seen in a picture (wide or narrow) is called field of view or angle of view. As mentioned before, the focal length controls the field of view. A long focal length creates a small field of view, and a short focal length creates a large field of view.
Exposure: Aperture, Shutter Speeds, and ISO
Exposure is one of the key components to not only taking a “good” picture, but also to taking a mathematically and photographically correct photograph. Exposure is the amount of light that hits the film or sensor in a camera. A bad exposure would be either when there is too much light and the photo looks too light (overexposed), or when there is too little light and the photo looks too dark (underexposed). A good exposure would result in a photo that looks close to what is seen in real life, or, at least, what the photographer would like it to look like. There are three pieces to exposure: aperture, shutter speeds, and ISO.
The first component of exposure to consider is aperture. To understand aperture, a photographer needs to understand a little bit about how a camera is constructed. A camera is made up of a camera body and a lens. The diameter of the lens opening is the aperture. The opening size is controlled by a ring made of six metal blades. A bigger aperture means more light can get into the camera. The aperture/focal length relationship is a ratio. The longer the focal length, the bigger the aperture has to be to let in the correct amount of light, and vice versa. The size of the aperture controls the amount of light that gets into a camera, just like diameter of a water hose controls how much water comes out the end.
The numbers used to change the aperture are referred to as f-stops. An f-stop is focal length divided by aperture. As the f-stop numbers get bigger, the aperture gets smaller, and the light is reduced by 1/2 for every f-stop. This is because f/2 is really 1/2, and 1/2 is larger than f/16 which is really 1/16. In every camera, no matter how big or small it is, the f-stops 1.0, 1.4, 2.0, 2.8, 4.0, etcetera will always let in the same amount of light. This is because an f-stop is a ratio between the aperture and focal length. A ratio is a relationship between one value and another. So, to increase the amount of light, a large aperture would be picked, which has a small f-stop number; to decrease the amount of light, a small aperture would be picked, which has a large f-stop number. Although this seems odd, it is easy to just remember that the larger the aperture, the more light that comes into the camera, and the smaller the f-stop number.
Not only does aperture control light, but it controls how much of a picture is in focus. To explain in further detail, if a picture is taken of a person using a small aperture, the person and the background would be in focus. If a picture is taken of a person using a large aperture, the person would be in focus but the background would be blurred. This is depth of field. Depth of field is how much of a photo is in focus from front to back, as opposed to left to right. The smaller the aperture, the bigger the depth of field, or the more that is in focus, and vice versa. This is an inverse relationsip. An inverse relationship is when one value goes up, the other goes down. So, as an example, picking a large aperture, such as f/2.8, would create a small depth of field, while picking a small aperture, like f/11, would create a large depth of field with much of the picture in focus.
The next important component of exposure is shutter speed. Shutter speed helps to control the amount of time that light makes contact with the film or sensor in a camera. Common shutter speeds are 1/1000, 1/500, 1/250, 1/125, 1/60, and 1/30. On a camera, these times would be expressed as only the denominator, and not the whole fraction. In a film camera, opposite the lens, is film covered with light sensitive chemicals. To produce a good image, a certain amount of light must hit the film. The amount of light that hits the film is controlled by the amount of time the door is open, and how big it is. This door is called the shutter. To freeze action (a car, a runner, an animal, water, etcetera), a picture must be taken with a fast shutter speed such as 1/1000 of a second. To blur action, a picture must be taken with a slow shutter speed such as 1 second. Once again bringing math into the equation, the relationship between aperture and shutter speed is inverse. The faster the shutter speed, the smaller the shutter speed number, and the bigger the aperture has to be to let in the correct amount of light. There are different combinations of shutter speeds and f-stops that give the same exposure. To clarify, if the shutter speed is 1/125 at f/8, and the shutter speed is changed one increment slower to 1/60, the f-stop must be changed one increment smaller (f/11) to keep the exposure the same. Although this might seem confusing, it should be thought of in increments; all it is, is doubling or halving the size, speed, or amount of light.
The final piece to concretely understanding exposure is ISO. An ISO rating is a measurement of how sensitive film or a sensor is to light. ISO stands for International Standards Organization. In a film camera, ISO cannot be changed unless new film is bought with a different ISO. ISO can be changed in a digital camera. The common ISO numbers in a film camera are 50, 100, 200, 400, 800, and 1600. A digital camera can go all the way up to 3200. This number sequence is a geometric ratio. In this case, a geometric ratio is when the numbers are multiplied by two every time. The smaller the ISO number, the less sensitive the film; the larger the ISO number, the more sensitive the film. The more sensitive the ISO, the more noise or grain there is in a photo. It is better to use a less sensitive ISO unless the place where the picture is being taken doesn’t have sufficient light.
It is important to not only understand the different parts of exposure, but to also understand how and why they would be changed. For ISO, shutter speed, and aperture, moving from one number to another in order, either doubles or halves the amount of light that enters the camera. These increments are called stops. To keep exposure the same, but change one of the components – ISO, shutter speed, aperture – at least one other component must be changed as well. For example, the ISO is 800, and the shutter speed is 1/250 at f/8.0. The shutter speed is changed to 1/30, three stops down from 1/250. So, either the aperture or the ISO would need to be changed by three stops to let in less light. The f-stop could be changed to f/22, or the ISO could be changed to 100. Once a photographer understands the relationship among aperture, ISO, and shutter speed, it is much easier to take an attractive photograph.
Simple and beneficial, histograms help to determine whether a photograph is correctly exposed or not. A histogram is a mathematical graph that diagrams the lights and darks in a picture. This graph helps to determine whether or not a photo is exposed correctly. There is no “ideal” histogram. The one thing to try to avoid is clipping on the left or the right side of the graph. Clipping is when a pixel is on the right or left wall, or when a pixel is pure white or pure black. A histogram for a correctly exposed photo should have most of its pixels in the middle of the graph, or at least have no pixels touching either side. If a histogram is clipped on the left side (or the black side) it shows the photo is underexposed. If a histogram is clipped on the right side (or the white side) it shows the photo is overexposed. A histogram is a very useful way to decide if a photo is exposed correctly.
The Zone System
From zone one to zone nine, the zone system for determining the correct exposure not only helps to produce great photographs, but is also greatly related to math. The zone system is a scale of tonality from one to nine, with one being the blackest black and nine being the whitest white.
Zone one is maximum black with no feeling of any substance; it is flat and featureless. Zone nine also has no feeling of any substance, but it is maximum white instead of black. Zone two is a little lighter than zone one. It might even be mistaken for zone one. It is too dark, flat, but there is a slight feeling of depth. Zone eight is a little darker than zone nine. It is too light, but there is a slight feeling of depth. Both zones three and seven are the first zones to show detail and texture. Although they show depth and substance, they still are not gray tones. Zones four, five and six are the mid-tones; they are all shades of medium gray. Zone four is the lightest dark and details are easily seen. Zone six is the darkest light and it is also easy to see detail. Zone five, the middle tone of the zone system, is the mid gray tone, neutral, and marks the change from shadows to highlights. Zones one through four are shadows, and zones six through nine are highlights. Zone five is neither a shadow nor a highlight.
If a person takes a picture with a traditional camera, the camera’s light meter will take an average of all the tones in a picture, and change the average to zone five, or “neutral.” Sometimes the neutral meter reading will result in the correct exposure, or at least close to being correct, but people often have to change the zone to match the tone of the subject, or the main part of the photograph. To change a picture to a darker zone, a person would increase the speed of the shutter, or decrease the size of the aperture. To change a picture to a lighter zone, a person would need to decrease the speed of the shutter, or increase the size of the aperture. Just like aperture, shutter speed, and ISO, one zone is a stop of light. To move from one zone to another, a person would need to change the amount light the same amount of times a move to another zone was made.
Although this may seem complicated, if the concept of “stops” and the doubling and halving of the amount of light is fully understood, moving from one zone to another is actually quite simple. This would be done using aperture, shutter speed, or ISO. For example, a picture was taken in zone five, with a shutter speed of 1/1000, ISO 200 at f/4.0. If a person wanted it in zone seven, the photo would need to be changed by two stops of light. That could be done by either changing the shutter speed to 1/250, the ISO to 800, or the f-stop to f/8.0. The most common reason a photographer would want to change a photo to a different zone would be because the zone didn’t match the tone of the subject in the picture. To further explain, there are different objects that fit in different zones. In zone one, there is non-reflective black cloth, or anything else that wouldn’t show texture, reflections, or substance. Dark water and any very dark substance with slight texture would fit in zone two. Dark shadows, black hair, black fur, and black cloth all go in zone three. Zone four includes dark leaves, brown hair, jeans, and light shadows, and zone five consists of African American skin, wood, and any average gray material. Zone six is comprised of beach sand, concrete, clear dark blue sky, and average white skin. Blond hair, white clothes, white paint, average snow, and cloudy bright skies all fit in zone seven. Zone eight contains smooth painted wood in sunlight, and printing paper. Finally, zone nine includes bright reflections off windows, water, and glass. Even though the zone system might seem complicated, once the basics are fully understood it is a fantastic way to help correctly expose a photograph.
Most people do not think photography is as complicated as it really is. People think that they can take good pictures or be a photographer just because they can own a camera, even a very nice camera. But to be able to take a truly correct picture, a person would need to know, or at least have a basic understanding of the technical and mathematical part of photography. People do not realize how much there is to know and learn about it, and do not understand how much math is related to photography. “You don’t take a photograph, you make it,” said Ansel Adams. If a person does not have an understanding of the mathematical part of photography, a photo that has been visualized cannot be created, or “made.” To take an accurate and inspired photograph, a photographer needs to concretely understand the math in every piece of photography.
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Ms. Chapman Interview
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