John's metric as compound lens

Since it reminded me of the classical lens equations in optics (in Gaussian form), I equated a couple of fundamental formulae in optics with John's metric to see if anything useful might be noticed by such an analogy. Here's the one that seemed most interesting to me.

Compound lens equation for 2 thin lenses:

The usual convention is that the object position is positive when to the left of the lens and negative to the right, and the image position is positive to the right and negative to the left of the lens. THESE HAVE BEEN CHANGED, SO THAT THEY ARE NOW ON A CONSISTENT AXIS.

1/f=1/x-1/y-d/xy

f: focal length of combined system
x: position of the object
y: position of the image
d: distance separating thin lenses

The magnification factor is then y/x.

Now let's restate John's metric:

"(2 + 1/x - 1/y - diss(x,y) )/2

The diss(x,y) is a function.
If y/x <= 0.9375 the diss(x,y) is simply y/x
If y/x > 0.9375 the diss(x,y) is (1 - y/x)*15
x should be greater than y and both should be less than 256."

Just to make it a little simpler, let's subtract 1 and then scale it by 2. Notice it will still be useful as a metric, since it is monotonic with John's original and still has unique values everywhere for any x and y. Let's call it 1/f, where f is a measure of dyadic dissonance, and so 1/f is a measure of consonance; then

1/f=1/x-1/y-diss(x,y)

Let's equate the 2-thin-lens formula and this formula, solving for d. For John's 2 cases, corresponding values of d are

d=y^2 y/x <= 15/16
d=15y(x-y) y/x > 15/16

In Newtonian form, this simplification of John's metric could be written
f^2=qd-(x+d-f)(y+f)

This is some interesting stuff, although I'm not sure that John's
metric is all that valid to begin with.

On Sat, Feb 26, 2011 at 2:27 PM, funwithedo19 <nielsed@...> wrote:
>
> Since it reminded me of the classical lens equations in optics (in Gaussian form), I equated a couple of fundamental formulae in optics with John's metric to see if anything useful might be noticed by such an analogy. Here's the one that seemed most interesting to me.
>
> Compound lens equation for 2 thin lenses:
>
> The usual convention is that the object position is positive when to the left of the lens and negative to the right, and the image position is positive to the right and negative to the left of the lens. THESE HAVE BEEN CHANGED, SO THAT THEY ARE NOW ON A CONSISTENT AXIS.
>
> 1/f=1/x-1/y-d/xy
>
> f: focal length of combined system
> x: position of the object
> y: position of the image
> d: distance separating thin lenses
>
> The magnification factor is then y/x.
>
> Now let's restate John's metric:
>
> "(2 + 1/x - 1/y - diss(x,y) )/2
>
> The diss(x,y) is a function.
> If y/x <= 0.9375 the diss(x,y) is simply y/x
> If y/x > 0.9375 the diss(x,y) is (1 - y/x)*15
> x should be greater than y and both should be less than 256."
>
> Just to make it a little simpler, let's subtract 1 and then scale it by 2. Notice it will still be useful as a metric, since it is monotonic with John's original and still has unique values everywhere for any x and y. Let's call it 1/f, where f is a measure of dyadic dissonance, and so 1/f is a measure of consonance; then
>
> 1/f=1/x-1/y-diss(x,y)
>
> Let's equate the 2-thin-lens formula and this formula, solving for d. For John's 2 cases, corresponding values of d are
>
> d=y^2 y/x <= 15/16
> d=15y(x-y) y/x > 15/16
>
> In Newtonian form, this simplification of John's metric could be written
> f^2=qd-(x+d-f)(y+f)

Yeah, I understand, Mike B, but it's a start in one direction at least.
BTW, a correction is needed in the last line of my message:

f^2=qd-(x+d-f)(y+f)

should be

f^2=yd-(x+d-f)(y+f)

Okay, I just got back to this & realized I flipped a sign incorrectly, so
what I said wasn't quantitatively valid.

I'm going out again, but just wanted to attach a quick disclaimer.