Could an eye that is triangular-looking from the outside have any sort of advantage over a round eye? I'm interested in any nonconventional eye form discourse so anything on the subject is appreciated.
Eye shape only determines eye mobility, not eye function
There are two questions here, I will answer the title question: Would a non-round eye be possible or functional?
From the real world, there are many advantages to eyes of different shapes. While I think the question mostly concerns the outside “eyeball” shape, we can’t really talk about that until we talk about the most important parts of the eye, which actually let light in and focus it on the “film”, or retina. Those parts are the pupil and the lens.
The pupil is what allows light to enter the eye, and pupils definitely do NOT have to be round. When we manufacture camera lenses, they are usually round, but not always. This is because we have very simple film manufacturing processes that make an even, flat, and uniform surface for the light to be recorded on. Our film (or digital recording chip, as the case may be), is not intended for one specialized purpose. We want it to be versatile, and allow the customer to simply reproduce whatever comes into the camera as accurately as possible.
An eye doesn’t have this limitation. An animal is not creating fine art, or making a sales brochure. An animal is surviving, and its eye is a functional tool that will process the light around it in the way that help it survive best. An animal will have an eye that gives it an advantage in the environment it needs to live and find food in.
“Advantage” is a relative word, there is no single feature that makes something an advantage. Let’s start with something simple.
Eyelids are an advantage to animals living in a dry desert where dust and arid conditions dry eyes out, but in a pond they serve no purpose at all and only get in the way. So fish don’t have eyelids, and that is an advantage. Camels do have eyelids, and that is an advantage. This is why the question is a bit of a puzzle. How can anyone answer if a square or triangle eye is an “advantage” without knowing the world the creature lives in? You see, it is the same as asking if an eyelid is an advantage. Yes it is, and no it isn’t. It depends on what the creature’s world looks like.
On earth, there are very many different environments and animals have come up with many different pupil schemes to create an advantage in their worlds. The most extreme example I can think of is the W-shaped pupil of the cuttlefish.
We often make the mistake of thinking an eye is a camera that reproduces the outside world for our brains. That is one function, of course, but not all of it. The pupil shape of this eye allows the cuttlefish to see its environment clearly nearly 360 degrees around itself, so it can swim and hunt what is in front of it without loosing track of what is behind it (something is always hunting you on the ocean!) The dip in the pupil is a sort of light hood which prevents too much surface light from getting in.
But what sort of picture would this pupil and lens make? How can the cuttlefish make sense of the warped image it gets? The answer is two fold: First, it doesn’t need to “make sense” of the light that comes in because the cuttlefish eye is linked directly to its most valuable weapon: its skin. The cuttlefish has the most amazing camouflage on earth as it can change its colors, skin texture, and body shape to resemble spiny coral, flat sand, a mottled rock, or almost anything it is near. Even though the cuttlefish itself is color blind, its body will match the colors of a rusty pipe, deep violet coral, or green sea grass. New research has found that the cuttlefish doesn’t need to “understand” that the grass below it is green, it simply needs to stop thinking about what it is seeing altogether and its skin will mimic the spectrum of light coming into its weird light-bending pupil. The picture that gets to its brain is probably more simple. It sees the food, knows where the food it, and eats the food. It sees the shark, knows where the shark is, and hides from the shark.
So an eye is not only designed to “see” and create realistic pictures for the brain. Cameras do this, but eyes do much more specialized functions.
So why are so many eyeballs shaped like a ball? Because a ball can rotate and spin in a socket. This way, the area of the eye which has the best focus can be pointed at the thing it wants to watch without moving its whole head or body.
The lens needs to deliver the light where it belongs. If the animal wants an accurate picture, then the lens needs to focus it clearly like an eagle. If the eye needs to filter out light spectra to copy its environment quickly and accurately, like the cuttlefish, then the lens should act like a prism and provide the spectrum information needed. Why would a cuttlefish care about fine details in a coral bed, when all it needs is the basic pattern and color information? The eye doesn’t give the cuttlefish useless information to process. It gives the skin texture and spectrum information, which helps it stay alive by quickly and unconsciously copying its surroundings.
There are situations where a simple round eye is less advantageous than a specialized eye.
If an eye can focus without moving around, like a Google Earth 360 camera, then it only needs to be “round” if the retina is a simple surface. A retina can be shaped in any way needed to make a triangle eye work. Your story needs to create the world that makes it an advantage.
I'm not quite sure what you mean about "from the outside," but many avian eyes are not spherical. How dramatically non-spherical they are varies with species, but owl eyes are shaped like cocktail shakers, with a large domed cornea transitioning to a tapered cylinder. This maximizes the amount of light gathering capacity while still fitting into the physical constraints of the owl's skull size.
This may not answer your question if by "from the outside" you mean outside the creature's body, since owl eyes do appear round to an outside observer.
To the limit of my understanding, the ideal eye is always round because the lens effect which allows eyes to focus on objects requires that incoming light be evenly dispersed in all directions within the eye - so a triangular eye would needlessly warp incoming images which would then require additional brain power to correct - but then I'm told our eyes interpret the world upside-down and our brains switch it around for us, so so much for efficiency.
If you willing to go for compound eyes (where you don't have one eye with all the light receptors inside but rather thousands of mini-eyes with a single light receptor each) then I guess you could arrange your compound eye receptors in triangles. As long as you have two compound eyes you should still get depth perception and whatnot. /shrug
If you looking for personal sensors which interpret the world and don't specifically need to be eyes however, you can go with any shaped plate you want. You can have light, sound or even electromagnetic sensors which appear as scales, plates or dots on the skin - and those can be whatever shape and size you want.
I am an example of a being with a non round eye.
My eyeball is roughly spherical, but the question asked about the eye, which mightnot be the same thing as the eyeball.
Fom the outside the visible part of my eye looks sort of like the space between two interesting segments of arc. With my eyes open the upper arc is usually much more curved that the lower arc.
I suggest that you go look at your face in a mirror. I expect that the part of your eye visible to other people will have a similar shape to what I have described as the shape of my visible eye. In fact, if you claim that the shape of your eye that is visible on the outside doesn't look like the shape I described for my eye, you should enclose a photograph if you want me to believe you.
I note that animals on earth usually have spheroidal eyeballs, but the shapes of their visible eye parts differs a bit. Thus I can imagine an alian animal or person with a spheroidal eyeball and a rectangular or triangular or star shaped visible part of the eye.
In fact, a rather small modification of the upper eyelids of humans could make the visible parts of their eyes seem like flattened triangles, with the upper eyelid having two straight lines meeting at a point in the center.
A similar modification to the bottom eyelid would make the shape of the visible part of the eye a lozenge.
Eye shape can be an indicator if an animal is predator or prey. There is a huge amount of variety in how animals see. Cats having vertical pupils sheep having horizontal slits. Compound eyes. Even scallops have an set of photosensitive cells that lined up in row. The ways the different retinas work can be different too, for example being more sensitive to motion than stationary objects for some animals like some frogs. Some animals are sensitive to different types of polarization or can see more colors than humans.
So I think for a story you could have eyes be pretty divergent from a human eye, but the question might be what is the evolutionary advantage.