This is a delayed portmanteau reply.  I won't bore you with the sob story
reasons for the delay, but the portmanteau is because in some cases the
answers in the three reply posts I've seen differ markedly, and I want
to address those differences clearly.

In any event, it seems obvious that I'm not going to get references to
published material here, yes?  OK; I just thought asking here was better
(at least, more polite) than buttonholing random optometrists or
ophthalmologists at their offices.  What I wanted was not so much
references as indications of whether it was even still worth *looking*
for certain categories of information.  My thanks for those indications.

Kept for reference some ways below.

In article <1162947993.602611.191590@m73g2000cwd.googlegroups.com>,
"Dr Judy" wrote similarly:

"Smith (1991) pointed out that the relationship between spherical
refractive error and visual acuity becomes highly variable when factors
such as pupil size, illumination, target type, the threshold acuity,
instructions to the patient, and the patient's background are not
controlled.  He reviewed a number of earlier studies and concluded
that a better expression of the relationship was determined by the
following equation:

    VA = square root of [1 + (K * D * S)squared]

where VA = visual acuity in minutes of arc, or the minimum angle of
resolution; K = a constant with mean of 0.83 or 0.85 [?]; D = diameter
of the entrance pupil in millimeters; S = spherical refractive error
in diopters.
    "Smith (1991) found a linear relationship between VA and S in
high errors, for which the equation becomes VA = K * D * S.  The
relationship of visual acuity and refractive error is covered in
more detail in Chapter 7."  (Except that it isn't, actually.)

This is on p. 637 (bottom of first column and top of second) of
<Borish's Clinical Refraction>, ed. William J. Benjamin, OD, MS, PhD;
Philadelphia [et many cetera]:  W. B. Saunders Company, c 1998.  The
context is a discussion of fogging, of all things, in the chapter
"Monocular and Binocular Subjective Refraction" by Irvin M. Borish
and William J. Benjamin, chapter 19, pages 629-723.  Unfortunately,
and oddly, this chapter's references were not printed in the copy
accessible to me, though I think I can find Smith's article from
other references.

Anyway, my point in all this is that Smith apparently did look at
the physical variables you two listed, though he then came up with
a simplistic outcome.  I'll clearly have to look at the articles;
I'd hoped y'all would just say "Oh, don't look at those outdated
things, it's now well known in the field that Murgatroyd 2004 is
the Gospel truth."  Or, flipside, "Snerf 1999 clearly shows that
these guys with equations were full of it."

The psychological variables are obviously trickier.  But since the
equation I'm looking for is one that takes *in* objective refraction
and best corrected visual acuity, and then spits *out* uncorrected
visual acuity for the *same* person, I would *hope* that the
psychological variables would *usually*, more or less, on average,
cancel out.

I recognise that "best corrected visual acuity" is, at best, a
problematic concept with limited relationship to what a person
can actually see.  But on the other hand I also recognise that
my own life experience is clear evidence that the relationship
*exists*, no matter how limited.  And unlike "good eyesight" or
"bad eyesight", visual acuity can be quantified, however
subjectively.

Separately, in article <vm6512t4o2i7g8k0s04fpqhr42ps48ilvn@4ax.com>,
Mike Ruskai wrote some correct comments about patents (but I should
note that I'd seen a legitimate source refer to the article the
patent was based on, before I saw the patent refer to it, so this
isn't completely kook material) as well as:

< Beyond that, you need to remember that 20/20 is *defined* as "normal"
< vision.  There's no coincidence if you end up at that value when you
< eliminate refraction errors.

I don't wish to be rude, but this simply isn't true except in an
unuseful sense.  20/20 is defined as normal in a sense similar to
the sense in which orange is defined as the colour of an orange; it's
good enough for lay use, but it's not what the pros mean.  See on this
the discussion of "visual acuity" in pretty much any book that devotes
a chapter to it, for example the chapter by Ian L. Bailey, chapter 7 and
pp. 179-202 of the book I already cited.  20/20 means you see at 20 feet
what you should be able to see at 20 feet, where "should" is defined in
terms of an angle.  This angle is either "minimum angle of resolution",
in which case it's 1 minute, or "minimum angle of recognition", in which
case it's 5 minutes, depending who you ask.  This is because the letters
eye charts are built out of are built such that the entire letter
occupies, at 20 feet, 5' or some multiple or fraction thereof, while
the individual strokes that make up the letter are 1' wide.

Pretty much *any* set of statistics on how well people see will
explicitly note that 20/20 is subnormal for healthy young people.
The guy Snellen who came up with this system himself acknowledged
this.  In the 1960s, normal American men in their 20s or so could see,
on average, about 20/15, and a few could see 20/10.  See on this
<http://www.cdc.gov/nchs/data/series/sr_11/sr11_003.pdf>, more or less
passim.  It's just one of many examples.

In contrast, Mike Ruskai answered:

< I think so, though I've certainly heard that position (and believed it
< until I grew up and thought better of it).
<
< In a normal eye, the focal plane for an object at infinity falls on
< the retina with relaxed focusing muscles.  In myopes, the focal plane
< with relaxed muscles falls in front of the retina.  That means the
< "infinity" focus is actually set for an object nearer than infinity.
<
< The stronger the myopia, the closer the relaxed focus position.
<
< That allows an uncorrected myope to see clearly without eye strain at,
< say, a comfortable reading distance.  But there's no more acuity, all
< else being equal, versus a person who has to accomodate to reach focus
< at that distance.

So which of you is right?  The one who provides the explanation I already
know for why my naive view was wrong, or the one who provides an
explanation I hadn't thought of for why it could be right?

Fortunately, on this one, elementary optics should help.  I didn't like
optics in high school, and skipped the quarter of college physics that
covered it.  But I can probably relearn enough to pay attention to the
more challenging parts of Brian Curtin's <The Myopias>.  I just didn't
want to bother if there was no reason to.

I wasn't able to parse the reply to this by Dr Judy at all.  What I'm
concerned with is "Does myopia improve near vision in any way other
than bringing the near point closer?  Does hyperopia improve distance
vision in any way at all?"  Definining "normal" is part of my inquiry,
in a sense, but isn't really relevant to this particular question.
Even if I used the *word* "normal" in stating it.

OK, all of this makes sense to me...  So then the question is whether
any studies of this variation exist.

My impression that none do is somewhat strengthened by something I came
across in a detailed guide to perimetric techniques on the Web.  
Apparently one kind of perimetry - ?Goldmann perimetry - is best suited
for extent determinations (<http://www.eyetec.net/group3/M12S1.htm>);
and I can imagine that not everyone has Goldmann perimeters, or some
such.

But it still mystifies me.  Decades, indeed over a century, of studies
of variation in visual acuity.  A single study way back in 1922 of
variation other than age-related in accommodation.  And no studies at
all of variation in visual field extent, except as caused by disease.

Unless I'm just not looking in the right place.  Volume 5 of <Vision
and Visual Dysfunction>, general editor Prof. John Cronly-Dillon,
Boca Raton [et alii]:  CRC Press, probably c 1991, has the intriguing
title <Limits of vision>.  For all I know, this contains a detailed
discussion of the topic; volume 5 is the one volume the library I'm
now typing from doesn't own.

Mike Ruskai answered:

< Were any of those figures justified by testing?

Of course not.  None was even footnoted.  Snarl.

Dr Judy's reply included:

Well, for starters, I'm researching "normal".  In this case, is it
180, 190 or 200 degrees?

In general, I'm researching the range and curve of variation.  I get
the impression that in things medical, or at least things sensory,
bell curves are vanishingly rare.  The only kind of curve most of the
books I've consulted seem to acknowledge is one where the maximum and
the mode are identical, and then there's a tail leading off to the
minimum:  Trichromatism is the norm, dichromatism happens, and
monochromatism is rare; 20/20 is the norm (or 20/15, or whatever), and
worse acuity happens; in hearing, back in the 1960s they didn't even
*test* for hearing significantly better than the standards they went
in with, even though it turned out that those standards were wrong,
and they were jettisoned a few years later.

Repeatedly, I'm finding evidence that this "Normal and bad are the only
options" thing is *wrong*.  Tetrachromats exist.  20/20 is not the norm,
and anyway you can (perhaps) decompose visual acuity into "best corrected
visual acuity", which is bell curve-ish, "spherical refraction error",
which is bell curve-ish, and "cylindrical refraction error", which I
don't know about.  And, well, that standard was jettisoned, but it's
*still* routine for studies of hearing to begin "My group hears *much*
better/worse than the [current] standards suggest..."

I'm aware that there *are* reasons to expect "normal or bad, but not
good" to happen in human biology.  Presumably lots of potential bell
curves get throttled by evolution or some such.  It's just that I'm
becoming decreasingly convinced that there are as *few* bell curves
as I'm finding discussed.

The fact that the only datum I have on super-normal visual field extent
is an anecdote presumably drawn from some sort of individual case
report is, from my POV, a *bad* thing.

Um, I don't understand.  You're saying rods and cones slow down as we
age?  Why would this be more of an issue at night?  (I'd think anything
that was more an issue at night than by day would be something that
differentially affected rods more than cones, no?)

Anyway, though, I'm interested in variation due to aging, but not only
in such variation.  During WWII, the US Navy and Air Force put a lot
of effort into finding out stuff about variation in human (healthy
young male, in fact) night vision, and AIUI what they learned was that
no two variables correlate:  for example, night visual acuity is
unrelated to all of day visual acuity, dark adaptation time, and, oh,
night visual field.  This is obviously not much of an explanation of
what variation is found in, well, any of these variables.  So what I
was asking was, do any of y'all know whether any more order has been
brought into the subject since then?

If what you're saying is "Yes, and that order is that all the other
changes depend on slow photoreceptor response times", OK, but I'm
confused; in that case I'd expect more of them to correlate.

Anyway, thanks for your replies.  Sorry it's taken me so long to say so.

Joe Bernstein

Hello MIKE.

 Title my  grandfather was too  MICHAL/MIKE/ RUSKAI

----------------------------
Milan Ruskai
ruskai@senzor.sk
Ved?ci TPV
SENZOR, s.r.o.
Zajacia 30
04 012, KO?ICE
SLOVAKIA

Tel.: +421 55 6 747 622
Tel.: +421 55 6 747 623
Fax.: +421 55 7 291 801
Mob.: +421 905 404 628

Mike Ruskai nap?sal(a):

Do a Google search something like...

tetrachromatism OR tetrachromat OR tetrachromacy OR ...

well, you get the idea, and, so to speak, you'll find it eye-opening.

Basically, it appears that I wasn't entirely right about the timing;
the websites claim that tetrachromatic vision in humans had been
postulated (but not actually found) as early as 1948.  In the early
1990s, the topic was massively revived by an announcement of an
Actual Tetrachromat Found in England, and although further real
reports seem to be less than numerous, the topic remains mildly hot.
This is probably because it fits in well with that aspect of the
Zeitgeist that emphasises women's health issues (an aspect that, by
the way, I approve of).  'Cause, you see, tetrachromatism in men
would require some rather improbable mutations, but in women it's
not that big a deal.  The only question is how uncommon it actually
is for a woman to jump through all the relevant hoops.

In a nutshell, here's the deal.  The statistics for men are often
distorted; it's not unusual to run into offhand claims that 10% of
men are colourblind, which is bullshit, and relevantly so.  About
2% of European men have either protanopia or deuteranopia, which
are the two forms of what's known as red-green colour blindness.
There are various extremely rare phenomena that lead to blue-yellow
colour blindness or full colour blindness; most of these are equal
opportunity as to which sex they hit.  But most of the BS 10% figure
comes from two things that aren't colour *blindness* at all, just
anomalies:  protanomaly and deuteranomaly.  (The equivalent on the
blue-yellow spectrum would be tritanomaly if it were demonstrated
to exist, but it hasn't been; the going theory is that it's a weak
form of tritanopia, which is the name for full-blown blue-yellow
colour blindness.)

Anyway.  Protanomaly and deuteranomaly both involve mutations to
the genes that code for the pigments in cone cells, which in most
people are the only cells that deal with colour.  (A few true
colour-blind people seem to have made their rods deal with colour
too, which is evidence of how negotiable our neural wiring is; I'll
get back to that.)  Anyway, the deal is that most mammals have only
two kinds of cones, one of which lives on a regular chromosome
(trouble with that causes tritanopia), the other on the X chromosome.
Twice in the primate order, the X chromosome one has been duplicated
in mutated form to produce a third kind; hence Old World monkeys and
apes, including us, have on kind of trichromacy, and howler monkeys
from South America have a different kind.  Well, the part of the
X chromosome in question seems to be remarkably unstable.  For
starters, the majority of people actually have more than one *copy*
of the new gene on the X chromosome, though only one per X chromosome
actually functions.  Furthermore, about 4-5% of men (in Europe,
anyhow) have mutated forms of the new gene (which codes M cones'
pigments), forms which react more or less differently to colours.  
And another 1% or so have mutated forms of the *old* X chromosome
gene (coding L cones' pigments), which do the same thing.

Most of these mutations don't do very much, though.  You get people
with an L cone gene that's like 1% deflected from normal, or an M
cone gene that basically is 1% different from L cones.  The former
person just sees a few colours slightly differently from the rest
of us, while the latter is only a smidgen short of being red-green
colour blind.

But - hypotheses coming! - *sometimes* the mutated gene lands smack
in the middle, and can be considered by the eye as a whole different
colour.  In men, this leads to disagreements on complicated colour
matching tests, but otherwise no big deal:  the guy still has three
colours, they just aren't the same as everyone else's three.

But...  In women, who have *two* X chromosomes, if one and only one
of those has one of those smack-in-the-middle mutations, then that
woman suddenly has *four* functioning pigment colours:  S cones',
M cones', L cones', and what's being called H cones'.  So!  This
suggests that something like 8% or more of women could conceivably
be tetrachromats, which makes it astonishing that it took decades
to find even one, right?

But the plot thickens.  Not only do the odds get greatly reduced
by the tendency for the mutations to be fairly conservative and
not do the H thing, but there's also the fact that the woman with
four functioning *cone* colours doesn't necessarily *see* with them.  
She has to somehow get her neural circuitry wired to expect, instead
of the three contrasts most of us rely on, four.  (I think.)  Now, as
noted way back there, there are people out there whose rods have
learned to contribute to colour vision, so it's hardly rocket science
for an H cone to find a way to do so; it's just a question of how often
it actually happens.

So there you have it:  women with colour vision that puts the rest
of us to shame.  Somewhere between .00001% and 10% of the female
population.  Probably.

Oh, and it gets even better.  A logical implication of all this is
that *pentachromatic* women are also possible:  just find *two*
pigments that are enough-different from the norm, and give her
one of each, along with the normal three.  This is, however, at
least an order of magnitude less likely than tetrachromatism, and
so far, nobody's reported any.

As I said, there's a lot out there.  If you're not interested in
the Google search, you could also try the Wikipedia article on
"tetrachromat", which has half a dozen of the most significant
links listed at the end.

Hope this helps.

Joe Bernstein

Only some women, and only maybe.

The genes that encode for red and green pigments are both on the X
chromosome.  Females, of course, get two copies of this chromosome,
and through a strange bunch of hoops may end up with some cells
expressing one copy, while others express the remaining copy.

The upshot is that these women have four different photoreceptors in
their eyes, with the spare being a slight variation of red or green.
They are also more likely to give birth to colorblind sons (because
the chromosome that provides the extra receptor is defective, in that
only one gene - red or green - actually works, and the son has a 50%
chance of getting that defective copy).

I don't believe it has been definitively demonstrated yet whether or
not this actually results in tetrachromatism.  It's not enough to have
receptors sensitive to slightly different wavelengths of light.  The
brain has to be wired up to make use of that information.  So the
green and greenish (or red and reddish) receptors could trigger the
same response in the brain.

By comparison, we have on the one hand some South American primates,
where all males are dichromatic (i.e. what we would call color blind),
and females trichromatic.  The females can definitely see in
trichromatic color, so having only one sex to work with is clearly not
an impediment to natural selection wiring up the brain appropriately.

In humans, however, it's only a small percentage of the one sex.

Naturally, it's not an easy thing to determine experimentally, though
there are probably some that try to answer the question that I haven't
read about.