How would having weird eye lenses affect a species' ability to understand geometry?

Question

Suppose you have a race of early people similar to humans, except that they have fish eyes to help with peripheral vision. Would this adversely affect their spacial reasoning skills? It seems that AS A HUMAN, my eyes are able to perceive Euclidean space in its "true" form and that this is a pretty big advantage. Is this assumption correct? Would fish-eyed people evolve the ability to intuit the shape of objects in their environment, or would they be at a disadvantage?

For example, would they have a harder time discovering the wheel or fabricating smooth surfaces? Would their artists have a harder time understanding perspective or other aspects of technical drawing?

Answer 1

Euclidean space is the 'true' form of space

There are three spatial dimensions in our universe. This will be equally true for all species existing in the universe. Just to be clear, there are no other spatial dimensions that an alien eye could perceive in this universe (as far as we understand).

'Fish-eyes' may have poor or excellent binocular vision

Binocular vision is the ability to focus two eyes on the same target. Because the angle to the target is slightly different in each eye, this gives the brain the ability to compute a distance to target.

Almost all creatures have some range of binocular vision. Horses have eyes on opposite sides of a large head, and yet they still have about a 65 degree range of binocular vision. Most fish are probably similar.

The separation distance between the eyes can actually increase the acuity of binocular vision, even as it decreases the field of vision. There is some evidence that hammerhead sharks have excellent binocular vision.

Binocular vision is not an absolute requirement

It is important to note that even with a small range of binocular vision, horses don't have trouble running into objects; neither do one-eyed animals of many species. I've seen plenty of one-eyes people and dogs that get through life just find.

The brain has other ways to determine distance to objects, most notably using the size of the object relative to other objects in the field of view. While this is not a effective as the rangefinding of binocular vision, and certainly makes archery more difficult, this shouldn't have much effect on closer up objects.

As a thought experiment, cover one eye and try to do common manipulation of objects. I have no trouble picking up my coffee mug, tying my shoes, or writing on a pad of paper. Even with monocular vision, a species should be able to manipulate their environment and develop crafts just fine.

Answer 2

Your eyes don't perceive Euclidean space very well, actually. Consider this picture. You're lying on your back, looking up at power lines above you. You know there's a vanishing point in one direction, and a vanishing point in the other. At infinity, the power lines appear to merge. You know that the powerlines must slowly separate, reach a maximum separation overhead, and then merge back together, but there's no point where this happens. Your brain sees "straight line" the whole way.

What's actually happened is that the image of the power lines on your retina is curved, but your brain "straightens it out" because it knows its straight.

In fact, our "knowledge" of Euclidean spaces gets broken all the time.

In fact, we aren't actually all that good at perceiving at all. Consider this famous one where black dots appear and disappear as artifacts of how our brain and retina process the image:

In reality, our brain does some mighty find tweaking to create a proper Euclidean space around us. Consider your blind spot. There's a direction you literally cannot see, one in each eye. Your brain literally stitches together an image of what it thinks should be there. It works great, until it doesn't. (hopefully you're looking at this on a large enough screen for this to work:)

The truth is, we experience the world around us through imperfect lenses and dented and distorted hearing horns. Our sense of touch is constantly being fooled. And yet our mind stitches things together to make an amazing sort of sense.

And who says Euclidean is "right" anyways?