(NASA)
(OMCA)
Saturn’s north pole has a hexagonal cloud boundary
Why we must go there
While deciding on our next home planet, let’s examine a
mystery function $\boldsymbol{f(x,y)}$
discovered by a mathematical expedition
to the hexagonal weather pattern on Saturn’s topside.
A positive test, $f(x,y) = {\scriptsize\raise.4ex+}1 \kern1mu$, indicates an evenly triangular array of points above the clouds.
A subsurface probe, $f(x,y) = -{\large\tfrac54} \kern1mu$, reveals a regular hexagonal array of points.   But we need a modulated survey, $ f(x,y) = 1 {+} \, 3 \cos{\tiny\sqrt{\vphantom{\raise3.5ex~}\normalsize3}} \kern2mu r \kern1mu $, in order to account for the satellite view.
As this is part of a polar expedition,
$r = \sqrt{x^2 \! + y^2} \,$
and that last equation is the same as
$ f(x,y) = 1 {+} \, 3 \cos \! \sqrt{3 (x^2 \! + y^2)} \kern1mu
\LARGE\raise-.4ex\strut $.
 
Its $y$- intercepts are difficult to specify
but it crosses
$y = 0 \,$ at
$ \require{begingroup}\begingroup
\def \3 {{ \tiny\sqrt{ \vphantom{\raise3ex~}\large 3 } }}
\def \X#1{ {\pm}{ \large\tfrac{#1\pi}{3\3} \normalsize\,\raise.5ex{,} \, } }
x = \X{ 2} \X{ 4} \X{ 8} \X{10} \X{14} \X{16} ~ \cdots
\endgroup $
Laboratory measurements have determined that $f(x,y)$ contains 10 secret ingredients, each of which may be a variable, an operator, a trigonometric function, a decimal number, a symbolic constant, or a single bracket / fence such as left parenthesis, right brace, or absolute value bar. For example, $ 1 {+} \, 3 \cos \! \sqrt{3 (x^2 \! + y^2)} \, $ contains 13 ingredients.
So . . . What is a 10 - ingredient formula for $~\boldsymbol{ f(x,y) }\,$?
In the spirit of scientific methodology,
feel free to request a plot based on
$f(x,y)$.
 
Another modulation,
$ f(x,y) = -\sin^2 \!{ \tfrac{\large 2}
{ \tfrac1{\large r} - \tfrac{1}{6\large\pi} } }\, $
for instance, can uncover some nuts and bolts.
Notes
Although $r$, $\theta$ and non-trigonometric functions are not available as ingredients within $f(x,y)$, any solutions that use them anyway would be more than welcome for the sake of interest and education.
The cartoonish “points” in plots of $f(x,y) = {\scriptsize\raise.4ex+}1 \,$ and $f(x,y) = -{\large\tfrac54} \,$ represent true points.
Complex numbers are allowed but not needed.
These implicit plots were made by
EquationExplorer
at KevinMehall.net.
Also good for implicit function plots is
MathGrapher at eMathHelp.