> Vowels, unlike consonants, fill a continuous space in two dimensions.  If you were
> designing a language from first principles, you might try to distribute vowels to maximize
> their distinctiveness, as if you were going to modulate a carrier wave for digital
> transmission.

I would say there are three vowel dimensions: frontness, backness, and
rounding (though rounding is more binary than the other two). There
are two things missing in your analysis: distinctiveness is not
uniform throughout the vowel space, and ease of articulation is not
uniform throughout the vowel space. Languages have to balance these
constraints in their vowel systems. And these explain certain
characteristics of all natlang vowel systems: height distinctions are
more salient on front vowels than on back vowels, back vowels have a
strong preference to be rounded and front vowels have a strong
preference to be unrounded, etc.

>  However, this tends to produce unrealistic systems where neighboring
> vowels differ in two or more features:
> Diagonal vowel positions
> i u  (front, back)
> @\ (center)
> E O (front, back)
> A  (center)

This doesn't strike me as terrible unrealistic.

> Complementary rounded/unrounded three-vowel triangles
> i u\ M
> 9 O
> A

This is pretty bizarre. Definitily not a naturalistic set.

> Perhaps this is an indication that we naturally analyze sounds by first decomposing them
> into discrete features, rather than nearness to an ideal point in a continuous space.  Why
> do natlangs work this way?

Again, I think you're missing out on the competing constraints of
acoustic distinctiveness and ease of articulation. The core vowel
triangle /i u a/ is found in something like 95% of all languages,
because those are all easy to say and maximally distinct. The triangle
/y M Q/ is theoretically just as distinct, but involves much more
difficult articulations. The set /@ @: @::/ is very easy to pronounce,
but not very distinctive.

JS Bangs
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"Have you not seen the heaps of bones piled on each other, skulls stripped of
flesh, staring fearsome and horrible from empty eye-sockets? Have you seen the
grinning mouths and the rest of the limbs lying casually about? If you have seen
those things, then in them you have observed yourself." -St. Gregory of Nyssa