Well, I think the answer is "sort of." Probably no particular thing is universally beautiful. The things I consider beautiful are personal. — T Clark
You can see something physical at the heart of aesthetic judgements - natural properties like symmetry, balance, and economy of effort. So there are objective properties that appeal.
I would guess that is because our brains evolved to be pattern recognisers. We need to zero in on what is the most general condition of our world so as to appreciate what is then the particular or often, the wrong, the blemished, the marred, the imperfect.
I would add straight away that the brain also evolved to make good mate choices, to find good food, etc. So that would also play into our aesthetic responses.
But we can see a circle or square as beautiful in being highly symmetric. Or more naturally, our eye is drawn to geometries that speak to perfectly balanced growth and elaboration.
So why is the golden rectangle perhaps more appealing than the ultra-simplicity of a square. Is it because the eye can see the universal growth ratio inherent in it?
And why the appeal of mountains, surf, trees? Again what we likely appreciate is the perfect balance represented by fractal self-similiarity - a scale symmetry.
That's not the whole story of aesthetic judgements, but it seems the most objective part of our evolved responses. We have an eye for natural perfection as that then is what allows us to see its imperfections. The imperfect perceptually pops-out once we have a baseline ability to recognise the perfectly freely growing or elaborating complexity that defines a natural balance.
Then there is a more subjective slant on the same general neurocognitive imperative - to make the world easy to read at a glance.
We like compositions that are balanced in their familiarity and surprise. Again, it is about the Gestalt need to balance figure with ground, event with context, so that the world is felt to be intelligible and yet not dull. We want a work of art to reward us both in being deeply familiar and deeply unfamiliar. And the ratio might be that of the golden rectangle or logarithmic growth. So a spicy dash of surprise at every level while also a healthy base of confirmation of what we reasonably expect.
It is not hard to see how this applies in all aesthetic ventures from an oil painting, to music, to great cooking or a nicely furnished room. There is some ratio of the familiar to the surprising that meets the subjective preferences of a mind that wants to be able to safely predict its world and yet still be always learning, or steadily growing and expanding its range of experience itself.
So aesthetics could be explained ecologically by the needs of our brain to see through to the essential baseline structure of the world. The world is objectively what it is due to the nature of growth as a "mathematical" process itself. And then subjectively - to be good at building a model of the world - we need a psychological architecture that is also based on open-ended growth. That is how we can develop a good fit. We seek to find a good balance between habit and attention, between familiarity and surprise, in every moment.
So every aesthetic object or image should "test" our responses in that fashion. They should contain a brain-suitable balance of predictability and surprisal. There will be a ratio of the two that feels the most informational or meaningful.
Phi, or the golden ratio - 1.618... - could be the right number. Here is someone who suggests that:
https://plus.maths.org/content/golden-ratio-and-aesthetics
There's a ton of psychological studies on preferences for complex stimuli, but I can't recall there being some magic number being identified. You tend to get general comments about a U curve, but not some special value.
The relationship between the complexity of a stimulus and its perceived beauty has been a topic of great interest with influential studies since the earlier experimental investigations of aesthetics. For instance, Berlyne showed that complexity is a dominant determinant of interestingness and pleasingness of a stimulus (Berlyne, 1963; Berlyne et al., 1968). Berlyne (1971) suggested that the relationship between complexity and pleasingness could be explained by an inverted U-curve, where the stimuli with intermediate levels of complexity are the most preferable ones. This concept of an optimal amount of stimulus complexity has been supported by numerous studies that found an inverted U-curve when characterizing aesthetic preference as a function of complexity (Vitz, 1966; Berlyne, 1971; Saklofske, 1975; Farley and Weinstock, 1980; Imamoglu, 2000).
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4796011/
This is probably because it is hard to get people exactly matched in their life experiences. We all might have a similar brain design, but also would all find somewhat different things surprising or familiar.
Plus the world itself is varied as an environment. All of nature might be fractal in its growth, but the slope of that line can be very different. A flat sea and a mountainous sea are both fractal, yet also at opposite ends of the spectrum in terms of beautiful monotony and spectacular excitement.