Mathematicians discover new way for spheres to 'kiss'

> In two dimensions, the answer is clearly six: Put a penny on a table, and you’ll find that when you arrange another six pennies around it, they fit snugly into a daisylike pattern.

Is there an intuitive reason for why 6 fits so perfectly? Like, it could be a small gap somewhere, like in 3d when it's 12, but it isn't. Something to do with tessellation and hexagons, perhaps?

> They look for ways to arrange spheres as symmetrically as possible. But there’s still a possibility that the best arrangements might look a lot weirder.

Like square packing for 11 looks just crazy (not same problem, but similar): https://en.wikipedia.org/wiki/Square_packing

It would be fun to make that square packing for 11 from wood and give it to puzzle enthusiasts with this task: Rearrange the squares so you can add an additional 12th square. And then watch them struggle putting even those 11 squares back in.

Three pennies form an equilateral triangle with (of course) 60 degree angles.

Six of those equilateral triangles will perfectly add to 360 degrees. Intuitive enough? (I'm being a little hand-wavey by skipping over the part where each penny triangle shares two pennies with a neighbor — why the answer is not 18 for example.)

For my mind though, the intuitiveness ends in dimension 2 though. ;-)