Calculating f-stop progression . . .how are the numbers derived?

[Drew Bedo]

I have noodled around online, but can't quite find out what I would like to know:

How are the f-stop numbers derived? i\I understand that it is a ratio of focal length to aperture . . .but can't quite get the math right. I have inklings of the natural log of 2 being involved somehow.



I do not have any pressing reason to do this, but its still early and the coffee has kicked in.

[Len Middleton]

Drew,

I believe it is a progression of the square root of 2.

But would be interested in hearing what the right answer might be as well.

Len

[Dan Fromm]

Drew,

I believe it is a progression of the square root of 2.

But would be interested in hearing what the right answer might be as well.

Len

That's right. Each full stop step halves (going up from 1.0 to 1.4, 1.4 to 2.0, ...) the amount of light the film receives given subject illumination and shutter speed.

[Havoc]

Aperture controles light intensity. So it controls the SURFACE of the opening of the diaphragm. But it is expressed as a ratio of linear dimensions. Doubling the surface is done by making the diameter square root of 2 larger. So you get a progression by square root of 2: 1-1.4-2-2.8-4...

[Bob Wagner]

A pretty much complete explanation:
https://www.slrlounge.com/a-thorough-explanation-of-the-math-behind-f-stops/
For me the key was remembering that doubling the area of a lens opening, and thereby the amount of light, is accomplished by changing the diameter by the square root of 2 (1.4 rounded)

[Nodda Duma]

As Havoc notes, f/# values change by the area size. As such, the numbers change by the square root if 2.

[Bill Kumpf]

On a side note, the longer the lens focal length, the larger the glass surface area must be to give the same f/stop. The diameter of the aperture on a 360mm lens is considerably larger than on a 50 mm lens.

[Emmanuel BIGLER]

Hello from France!
The present f-number system was not standardized when manufacturers started to sell photographic optics.

See this comparative chart of old f-stop systems (http://bigler.blog.free.fr/public/docs-en-pdf/2021-01-19-old-f-stops-4.pdf)

And after the f-number was adopted, it took again some time before all lenses would be engraved (except for max aperture) according the modern standard series 1.4 - 2 - 2.8 - 4 - 5.6 - 8 - 11 - 16 - 22 - 32 - 45 - 64

[Jim Michael]

Another way of looking at it is the progression of the area of a sphere with change in radius, where radius is the f number in any units (inches, meters, furlongs). The area doubles with each f number. The area of a sphere is given by 4 pi r^2, so taking the ratio of 2 radii r1 and r2 yields r1^2/r2^2.

[Dan Fromm]

Another way of looking at it is the progression of the area of a sphere with change in radius, where radius is the f number in any units (inches, meters, furlongs). The area doubles with each f number. The area of a sphere is given by 4 pi r^2, so taking the ratio of 2 radii r1 and r2 yields r1^2/r2^2.

surely the area of a circle is all that's needed

[ridax]

I understand that it is a ratio of focal length to aperture . . .but can't quite get the math right.

All said above on the math is just right.... But one more thing though: the ratio is not of focal length to aperture but of the focal legth to the entrance pupil diameter. In most of the 'normal' lenses, the entrance pupil is slightly larger than the iris opening. It is considerably larger than the iris in telephoto lenses, and it is much smaller in retrofocus lenses. That presents some difficulties in finding the actual apertures in an unknown lens.