Because hexagon is bestagon!

Based on what I recall of the explanation by the person who figured it out: spinning makes fluid near the edge spin faster than fluid near the middle. The difference in speed creates a wave. Since it's finite and moving, the wave interferes with itself and because of math, makes a hexagon. Something about how the wave pattern changes density and brings different glasses to the surface on the planets.
Then they showed an example by spinning a bucket, and it kinda fell flat because they had to explain that a bucket isn't a sphere so you have to spin it just right to get it to work, but it did work in the end.

To better understand how nature doesn't always make smooth circles out of circular patterns, this Minute Physics video does a banger job using the Earth's moon as an example.

https://www.youtube.com/watch?v=KBcxuM-qXec

For Saturn, you're talking about storm patterns that aggregate near the poles, but the concept is somewhat similar, which is that forces acting on objects (storms) can arrange circles into wave-like shapes.

All that said, I believe that Saturn's hexagon is still not fully understood, and still may be signs of a deeper alien death-star hiding in the clouds and we should probably like... I dunno, stock up on canned beans and toilet paper.

It's less weird when you realize it's not a hexagon, it's a sine wave in cylindrical coordinates. There are a lot of negative feedback loops such that a sine wave can turn into a standing wave. You just have to get a little lucky with a couple important things like your rossby number et voila, hexagon.

Hexagons are just nature's way of making arrays of triangles.

Because hexagons are the bestagons.

Lazy rendering

If only there was a Wikipedia article - https://en.wikipedia.org/wiki/Saturn's_hexagon

A hypothesis developed at Oxford University is the hexagon forms where there is a steep latitudinal gradient in the speed of the atmospheric winds in Saturn's atmosphere.[22] Similar regular shapes were created in a laboratory when a circular tank of liquid was rotated at different speeds at its centre and periphery. The most common shape was six sided, but shapes with three to eight sides were also produced. The shapes form in an area of turbulent flowbetween the two different rotating fluid bodies with dissimilar speeds.[22][23]A number of stable vortices of similar size form on the slower (south) side of the fluid boundary, and these interact with each other to space themselves out evenly around the perimeter. The presence of the vortices influences the boundary to move northward where each is present, and this gives rise to the polygon effect.[23] Polygons do not form at wind boundaries unless the speed differential and viscosity parameters are within certain margins and thus absent at other likely places, such as Saturn's south pole or the poles of Jupiter.

Other researchers claim that lab studies exhibit vortex streets, a series of spiraling vortices not observed in Saturn's hexagon. Simulations show a shallow, slow, localized meandering jetstream in the same direction as Saturn's prevailing clouds are able to match the observed behaviors of Saturn's hexagon with the same boundary stability.[24]

Developing barotropic instability of Saturn's North Polar hexagonal circumpolar jet (Jet) plus North Polar vortex (NPV) system produces a long-living structure akin to the observed hexagon, which is not the case of the Jet-only system, which was studied in this context in a number of papers in literature. The NPV, thus, plays a decisive dynamical role to stabilize hexagon jets. The influence of moist convection, which was recently suggested to be at the origin of Saturn's NPV system in the literature, is investigated in the framework of the barotropic rotating shallow water model and does not alter the conclusions.[25]

A 2020 mathematical study at the California Institute of Technology found that a stable geometric arrangement of the polygons can occur on any planet when a storm is surrounded by a ring of winds turning in the opposite direction to the storms itself, called an anticyclonic ring, or anticyclonic shielding.[26][27]Such shielding creates a vorticity gradient in the background of a neighbor cyclone, causing mutual rejection between the cyclones (similar to the effect of beta-drift). Although apparently shielded, the polar cyclone on Saturn cannot hold a polygonal pattern of circumpolar cyclones such as Jupiter's due to the bigger size and slower wind speed of Saturn's polar cyclone, so the side-adjacent vortices and deep barotropic instability (Cassini's wind speed measurements preclude shallower barotropic instability at least at the time of the Cassini encounter), or possibly baroclinic instabilities remain as the most viable explanations for Saturn's sustained hexagon.[28]

TLDR That's what happens when circles get squished together.

Because Saturn is Catan.

It's bees.

TIL all the Civilization maps are on Saturn

Because storms want to be circles but any given gas giants atmosphere is basically a series of nothing but storms and when you tile circles you get a hexagonal grid due to the spaces in between them?

That's where all the 10mm sockets end up

Standing wave. Earth kind of has one in the jet stream (3 peaks and troughs though, usually), but you can't see it with visible light.

What, you want to tighten the axis with a torx?

Things like Hexagons and the golden spiral occurring in nature are interesting - but very well-travelled.

Aguiar, Ana C. Barbosa, et al. "A laboratory model of Saturn’s North Polar Hexagon." Icarus 206.2 (2010): 755-763.