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Dear Stephen J. Gould,
For quite a few months I had been trying to write to you about my thoughts
after reading your and Rhonda Roland Shearer's essay "Boats & Deckchairs".
I have greatly enjoyed your column since the early 90's but this essay
was especially meaningful for two reasons. First because on this occasion
I thought of something you apparently did not. I was initially reluctant
to accept that I might have realized something you (or Duchamp!) had
not, but the more I thought about it the less reluctant I became. Now
I dare to share it with you and ask for your opinion. The second reason
is because on announcing your retirement from the column, I realized
with a mix of joy and sadness that I would barely catch the immense
pleasure and honor of sharing an issue of the magazine [Natural History]
with you. As my article, "Touchy Harvestmen," will be featured next
October. I will begin with my reflections on your 4-D essay, and this
will bring me back to my harvestmen's [daddy-long-legs] 4-D perspective.
I haven't read Abbott's Flatland (I certainly will) but from
your digested excerpts I can conclude that A Square didn't have to fly
too high above Flatland to see the shocking and never before imagined
perspective being offered from a 3-D world. Of course the higher the
better, but just standing a bit above the plane and stretching the neck
and peeping would be enough to see Mr. Circle all at once, though somewhat
deformed as an ellipse. (Similarly to when we are lost in the woods
and need to climb a tree or a hill to have a map view of where the heck
we are and where we are trying to go.) The perfect view of Mr. Circle
is at a right angle from above, but any angle larger than zero allows
for seeing him all at once, even though the shape distortion increases
as the angle diminishes. I would put my money down and say that A Squares'
big "WOW!" was just after taking-off and long before reaching a straight
angle above Mr. Circle. An experience much like the very first time
we fly as children and realize that we can see a whole block or field
all at once just after taking-off, long before reaching a complete view.
If I got that right and I properly understood that the analogy should
work when going from 3-D to 4-D as well, then I think we (especially
us primates) do have a chance to have that 4-D perspective of a 3-D
land. In fact, the great majority of us have it all the time, literally
in front of our noses. The genesis of my argument goes back to my childhood
when staying late in bed. Laying on my side, I would amuse myself by
switching between the two different perspectives of the landscape of
blankets in front of my face, shifting as I closed each eye. Then, I
would force both eyes to focus and converge on something just a few
inches from my nose, and close one, and then the other (you see where
I'm going?). Then I remembered a zoology teacher of mine in college
saying what a "convenient idea" it was in primate evolution to have
two frontal eyes, enabling us to judge distances when jumping from branch
to branch. And the last relevant revelation along this line, before
your essay, came when I took the instructions leaflet of my binoculars
and read it (one wanders who on earth would read the directions for
a pair of binoculars!). This only occurred as I was trying to kill time
while waiting in the rain forest for the end of a butterfly copula that
had lasted several hours already. It said that when you see through
your binoculars (if they are the kind that includes mirrors), the objects
not only look closer, but the 3-D view is "deeper." This was because
the two sources of the image coming from the objects to each tube are
wider apart than your eyes; I thought that was pretty cool too and kept
on peeping at "deeper" butterfly sex.
So when I read your article, I first thought it would be possible to
do something like using two periscopes (the kind people use to see parades
above the crowd) oriented sideways (and maybe slightly forward) to look
at an object in front with one eye on each periscope. I wondered if
the brain could still handle and integrate that (as it can when the
two sources of image are slightly separated when looking at binoculars),
and this would look even "deeper", more in 4-D! However, that would
be like A Square trying to see Mr. Circle from almost directly above,
closer to a straight angle, with less shape distortion. But we are always
looking at things from two different points anyway: from each eye. This
difference is negligible with a distant object, but less and less when
the object gets closer to the point where we could see it from opposite
ends: between our eyes. We know since we were kids we can only focus
so close, even crossing our eyes, but I think that is enough to stretch
our necks out of 3-D land. A practical object to do this with is for
instance is a 3.5" floppy disk (which in fact is a solid "square" case
with a real floppy disk inside, but that doesn't matter now). It is
an object with true volume, although conveniently flattened for our
purposes to a couple of mm, a flattened "cube". If you place it vertical
and perpendicular to your face, just between your eyes at the minimum
distance at which you can focus and converge your eyes on a single image
of the edge facing you (10-20 cm), you are looking at the two full sides
of the disk at once. If you close one eye, you only see the opposite
side and nothing of the other.
In other words, my argument is that if we only had one eye, or if we
had them on opposite sides of our head as many birds and mammals, we
would be true prisoners of the 3-D prison. In that case, we would be
unable to see objects from two points at the same time. As long as we
have two (eyes) views of the same object (depth vision), and if I understood
your essay correctly, we are having a 4-D view of the world, or at least
somewhere between 3-D and 4-D. This is as if A Square stood on a chair,
on its toes, stretched its neck and could see a deformed Mr. Circle.
Leaving primates and owls aside, I was trying to think of animals that
had shape-perception with eyes that could really look at an object from
different sides at straight angles, maybe some mollusk? But even if
there is such we would still need to ask it what that's like. We would
be back to where A Square was trying to explain to their friends what
it's like up there, so let's better try it ourselves (September 16 is
independence day in Mexico and they sell those periscopes in the street
to see the parade, I'm getting myself two of them!).
However, visual animals are probably not the most interesting to consider
for the cum-hyperhypho-embraced perspective, but those whose main perception
of the world come through tactile stimuli, and which can wrap objects
to perceive them. It is true that us primates, especially as kids, handle
a lot of objects and get the "4-D perception" of them through our hands
or mouth. This reminds me of a TV program showing how they allowed this
blind-since-birth sculptor to climb on a specially made structure around
Michelangelo's David to touch and embrace ("observe") it... he was delighted.
But the true masters of cum-hyperhypho-embracing must be something like
flatworms, snakes, octopuses (in spite their good view), and one of
my favorite creatures: harvestmen, or daddy longlegs. Many species,
including the one I have studied, see nothing but changes in light intensity
above them, and their hearing and smelling are hopeless. But they sure
have legs, and they do much more than walking with them. As they progress,
they are constantly assessing their very complex 3-D environment through
their 8 "channels", with an accuracy that must exceed our poor tactile
perception, and that depends clearly on touching objects on several
sides at the time. In short, they might not have the resolution primates
or owls have, but their depth perception is clearly better, and it's
the only one they got!
During the the many field hours I was working with harvestmen for my
dissertation, on top of the great fun they provided me, I frequently
read your column lying in my field hammock. It was then that I shared
that View of Life, never imaging that I would someday have an excuse
to share details of mine with you, which is to a great extent yours
anyway. Regardless of your thoughts on my 4-D speculations, I deeply
thank you for all this time.
Truly yours,
Rogelio Macías-Ordóñez
Departamento de Ecología y Comportamiento Animal
Instituto de Ecología, A.C.
México
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