Photography measures the light used by either doubling it or halving it in a odd series of “f-stops”. How the numbers 1.4, 2, 2.8, 4, 5.6, 8, 11, 16 and 22 double or halve anything is what I’ll try to answer. If light were water then the aperture would be the diameter of pipe to let the water flow in onto the sensor. We cannot measure the light transmitting capacity absolutely by the diameter of the aperture since wider lenses gather up more light from a larger field of view than the narrow slice taken by telephoto lenses. The f-stop numbers refer to how wide the diameter is compared to the length of lens. A lens with a focal length of 50mm at f/2 has an aperture that is 25mm: its focal length, indicated by the letter “f”, divided by 2; or 50mm divided by 2; or 25mm. This is why the number seem backwards with small numbers indicating wider apertures and bigger number indicating narrower apertures.
You would think than an aperture of f/2 would let in twice the light of f/4, but really f/2 lets in four times of the light of f/4. One thirtieth of a second is obviously twice the exposure as on sixtieth of a second, but what f stop is twice f/4? The answer is f/2.8. “What is this weird number ’2.8′?”, you might ask. The answer is to be found with that number “1.4″.
“1.4″ is the rounding-up of the square root of 2 to one decimal point. We have seen the square root of 2 when looking at lens lengths. each multiplication of lens length by the square root of 2 gives us view that is half as wide in angle of view and dividing the length by the square root of 2 gives us a view that is twice as wide. Which is why wide angle lenses have shorter lengths than telephotos.
The origin of the square root of two can be explained by the simple slave Ion in the Socratic dialogue named after him by Plato. At first Ion makes the naive mistake of dividing the sides of a square in half in an effort to make a square that is half the area.
Plato encourages him to try again and Ion eventually works out the solution. This is a visual interpretation of the solution.
While squares are not aperture, or should I say, few apertures are squares; all aperture shapes, be they pentagons or circles, follow the same rule. The square root of 2, or its approximation, 1.4, is the key. Dividing the aperture by 1.4 will halve the aperture and multiplying the diameter by 1.4 will double it.
The multiples of 1.4 are approximately: 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, and 22. That series is exactly what you see on an aperture ring. On a 50mm lens f/1.4 will give you an aperture of 35mm, f/2 is an aperture of 25mm, f/2.8 is 18mm and so on.