Edit: epic fail, and yet not. The rectangle within a hexagon is based on the square route of 3. Looks enough like a golden rectangle to fool me. However the following article is actually not entirely disproven, just not for the reasons I initially thought — Golden Ratios do exist in hexagons; see the comments section for the articles linking to the proof, also here and here.
The ideas about the nature of the hexagon as the compromise between angle and curve, producing the golden section/ratio, still hold true — and then the golden ratio of course produces the golden rectangle — just not the hexagon’s own inner rectangles. Oops. Learn something every day. I’ll leave my mistakes up that the masses may view my imperfection.
The Golden Ratio (1.618…), or “Phi” in Greek, and the Golden Rectangle derived from using that ratio, are considered one of the core bases of all aesthetic beauty. That both the Golden Ratio and Golden Rectangle are derived from the equilateral hexagon and its properties as a universe-cohering shape is a huge revelation and explained below.
We start from the notion that the equilateral hexagon is a sort of central shape of the universe, at least in the two-dimensional geometry. It is the ultimate compromise between circularity and polygonality, or unity between angle and curve.
If you have a simpler polygon than the hexagon, with fewer sides and angles (etc square, triangle, pentagon), it is more angular, and loses similarity to a circle, covering less area. If you add more sides to a hexagon, it is more circular, rounder, but less angular, less polygonal. It tiles together less effectively.
The hexagon is, after all, the solution to the tiling problem, covering area better than any other shape, and the best shape for providing torque (for wrenches/bolts/nuts etc.).
The universe tends towards producing hexagons because this shape, and the compromise between angle and curve they embody, are a sort of spontaneous resting or optimal state of matter. Matter consists of specific particles which gravitate toward and combine with each other, but they need to be arranged at specific points, therefore at angles rather than everything being a curve. However, because particles orient towards centers of gravity, or towards each other equally, they form circles, spheres, and rings (shapes in which all points are equally distant from the center/the opposite point on the circle), and therefore curves. The specificity and particularity of matter being arranged into definite, distinct, separate particles forces it to be angular, but its general equal attraction to itself and each other of its particles causes it to be curved.
Thus the hexagon is the expression of matter’s simultaneous specificity and generality, its angularity and curvature, in a geometric form, in ordinary interactions. It expresses itself this way whether we mean in bee hives, rock formations, the omnipresent carbon-based molecules and sugars of life on Earth, or Saturn’s hexagonal hurricane on its north pole.
But then we notice that the hexagon is in actuality an overlap and intersection between three Golden Rectangles.
The Golden Ratio emerges from the proportion of the sides of the rectangle underpinning the hexagon, the hexagon created by the relationship between angle and curve inherent in the equilateral hexagon.
The Golden Ratio is therefore the constant which represents the universe compromising between angularity and curvature, or uniting the two.
Hence the nautilus spiral.
Edit: apparently the nautilus spiral does not perfectly correlate to the golden ratio, and is instead a logarithmic spiral. The golden/Fibonacci rectangle spiral is instead a better representation of this concept.
Angle and curve are the two fundamentals of geometry and artistic existence. It makes sense that finding the unity between them would be the basis of all aesthetic excellence. Therefore it makes sense that the Golden Ratio, the mathematical-geometric expression of the compromise between them, is the also the mathematical-geometric constant of aesthetic beauty itself.
Therefore, the basis of the Golden Ratio as the constant of aesthetic beauty is based in the origin of the Golden Rectangle in the hexagon, due to the hexagon’s role as the shape which balances angle and curves, because of material particles’ tendency to express themselves as both specific and general, being both distinct separate particles arranging themselves at angles but also generally attracted to each other or to centers of gravity, creating circles, rings, and spheres. This is the ratio which emerges from the rectangle created from the hexagon, the hexagon in turn being created by these behaviors and tendencies of matter itself.