Tikz | Galley of Tikz drawings

I would like to create a special environment that encloses a math mode inside a tcolorbox. The tcolorbox title assignment must be optional, but in general I am interested in making optional tcolorbox options assignment. I am trying with:

I would like to change the default background color while writing in math mode in latex (but not for the inline math mode). I can achieve the result that I want using, for example, this piece of code:

I would like to illustrate graphically a bandwidth segmentizer. To do so, I would like to represent the following diagram in TikZ

So far, I have the following LaTeX code

I am "including" a .tex file. I get the error "TeX capacity exceeded, sorry [input stack size=5000].@nomath ...e@font@warning etc." The error is located by latex in the line that says

\pic[current plane, draw,fill=orange!50,fill opacity=.5, text opacity=1,"\footnotesize $\phi$", angle eccentricity=2.2]{angle=ketplus--origin--psiProjectedEquat}; %je nachdem, ob der Psi Pfeil links oder rechts von der z-Achse ist, muss man die Reihenfolge bei "angle=" umkehren }

I should point out that the file that is "included" works when compiled on its own (where the preamble isn't in the masterfile but in the file itself. It seems that the \usepackage[ngerman]{babel} in the masterfile causes the problem.

MWE:

"file.tex":

I want to draw an arrow from one tikzpicture to another, to include in my Beamer presentation I am making in Rmarkdown. I tried following the answer to this question: Connecting two tikz pictures with arrow. However, it does not work for me as it connects the two images together instead. This is my initial code (to make it fit with bullet points in the slide):

Edit: I have inlcuded my YAML in Rmarkdown.

I copied an existing project on overleaf but it doesn't display the icons, instead shows the code, for example, the main project is:

but the copied project is for LinkedIn icon:

here is a minimal reproducible example:

I need to draw a shaded area enveloping a graph like in the example below. I am looking for an automated procedure and a more satisfying result. Compared to the example, I need smoother edges and more regular distances between vertices (circles) and the boundaries of the area.

Drawing arbitrary shapes in TikZ are time consuming, I suspect that a smart trick might do the job, like creating large overlapping areas centered at each vertex. Any suggestions?

And here is a minimal working LaTeX code to generate the graph which has to be enveloped:

I try to achieve that different style classes are chosen in a self-defined tikz function depending on the input number.

However, the \ifnum command doesn't seem to work as I expect it. The error message that I get is:

If I place two nodes adjacent to each other the resulting boundary between the nodes is doubled making it more thick than other boundarys. Is it possible to place them on top of each other in order to keep the original thickness?

Here my code and below some example pictures: