As you know, cameras and other optical applications frequently use the f-stop system. The focal ratio (f/#) set in common usage is:
1 – 1.4 – 2 – 2.8 – 4 – 5.6 – 8 – 11 – 16 -22 -32 -45 -64
The f-number set is grounded on the geometry of circles. The f/# is obtained by dividing the focal length by the working diameter of the aperture. The resulting ratio is universally used to compare one lens to another with regards to its light
transmission potential. The above set has a delta of 2X. Each increment going right decreases light transmission 50%. Each increment going left increases light transmission 100%. The basis is; Multiply or divide the diameter of any circle by the square root of 2 computes a revised circle diameter with an area that is twice or half, in other words such adjustment of the diameter of the aperture stop of a camera, doubles or halves the exposing light energy. For this set, each number going right is its neighbor on the left multiplied by 1.414. Each number going left is its neighbor on the right divided by 1.414.
Now modern cameras often need to adjust using a finer increment. Industry practice is to use a number set in 1/2 f-stop increments. The number set is: 1 – 1.2 – 1.4 – 1.7 – 2 – 2.4 – 3.5 – 4 – 4.5 – 5.6 – 6.7 – 8 etc.
The multiplier/divisor is the 4 root of 2 = 1.1892
Often camera lenses require 1/3 f-stop changes. The number set is:
1 – 1.1 – 1.3 – 1.4 – 1.6 – 1.8 – 2 – 2.2 – 2.5 – 2.8 – 3.2 -3.6 – 4 etc.
The multiplying/dividing factor is the 6th root of 2 = 1.1225
One would expect that if you constructed a series of circles the delta for the 1/2 f-stop set would be a 50% area change. It is not, it is 41%.
One would expect that if you constructed a series of circles the delta for the 1/3 f-stop set would be a 33% area change. It is not, it is 26%.
I computed revised circle areas using the multiplying/dividing factors, then a percent change. I am surprised by fact that what I thought was 1/2 f-stop increments only yield a delta of 41% and the 1/3 f-stop set yields a delta of only 26%.
Why is this? Asked by me on Stack Exchange Mathematics
One of the answeres:
You want the area to double when you change 11 full f-stop. If you want to make two steps with the same fractional increase, they should each multiply the area by 2–√2. In that case, each one is 2–√−1≈41%2−1≈41% larger than the previous one. Similarly if you want to make three steps, each one should multiply the area by 2–√3≈1.2623≈1.26 so each should be 26%26% larger than the previous one.