Drawing an imperfect spiral in TikZ

Question

I want to draw a spiral to denote the orientation of a tetrahedron (either left or right handed). I played around with using a helix, but I'm not very happy with the look. Here is my attempt.

\documentclass{article}
\usepackage{pgfplots,tikz}

\begin{document}
\begin{tikzpicture}[scale=.75]
\coordinate (a) at (4,2.5);
\coordinate (b) at (3,.8);
\coordinate (c) at (4.5,0);
\coordinate (d) at (5.3,1.2);
\draw[thick, fill=black!20] (a) -- (b) -- (c) -- (d) -- cycle;
\draw[very thick] (a) -- (c) node[at start, above]{$v_3$} node[at end, below]{$v_1$};
\draw[thick, dashed] (b) -- (d) node[at start, left]{$v_0$} node[at end, right]{$v_2$};
\end{tikzpicture}
\begin{tikzpicture}[scale=.5]
\begin{axis} [
view={5}{50},
axis lines=none,
ymin=-2,
ymax=5,
xmin=-2,
xmax=2]
\addplot3 [very thick, ->, domain=.75*pi:4.25*pi, samples = 100, samples y=0] ({sin(deg(-x))}, {cos(deg(-x))}, {x});
\end{axis}
\end{tikzpicture}
\end{document}

What I would prefer is a less uniform spiral that sticks up at the end, to really give the reader the idea that we are orienting the vertices of this tetrahedron with the ordering $(v_0,v_1,v_2,v_3)$. Here is my attempt to draw what I want using paint.

The idea is to give the helix/spiral variable curvature and torsion. I still want the smooth lines seen in the first picture.

Ideally, I would like some simple tikz code that uses arcs or bends rather than pgfplots. I would also like to be able to reverse the spiral and make it left-handed so that I can give an example of the same tetrahedron but with opposite orientation.

Alternatively, if someone can put the helix in the same tikzpicture as the tetrahedron and bring it a little closer to the right side of the tetrahedron I would be fine with that. I tried moving the helix around on my own and I couldn't figure out how to position it where I want it. The code for the helix was found at How to draw vertical spiral using TiKZ?.

My Solution Here is the plot I finally decided on in case anyone wants to use it.

\documentclass{article}
\usepackage{pgfplots,tikz}

\begin{document}
\begin{tikzpicture}[scale=.75]
\coordinate (a) at (4,2.5);
\coordinate (b) at (3,.8);
\coordinate (c) at (4.5,0);
\coordinate (d) at (5.3,1.2);
\draw[thick, fill=black!20] (a) -- (b) -- (c) -- (d) -- cycle;
\draw[very thick] (a) -- (c) node[at start, above]{$v_3$} node[at end, below]{$v_1$};
\draw[thick, dashed] (b) -- (d) node[at start, left]{$v_0$} node[at end, right]{$v_2$};
\end{tikzpicture}
\begin{tikzpicture}[scale=.5]
\begin{axis} [
view={0}{75},
axis lines=none,
ymin=-2,
ymax=5,
xmin=-2,
xmax=2]
\addplot3 [very thick, ->, domain=2.9*pi:6*pi, samples = 100, samples y=0]
          ({.5*sin(deg(-x))}, {.5*cos(deg(-x))+1}, {2*x*x*x});
\end{axis}
\end{tikzpicture}
\end{document}

Answer 1

Here is how I would do it.

\documentclass{article}
\usepackage{pgfplots,tikz}

\begin{document}

\begin{tikzpicture}[scale=.75]
\coordinate (a) at (4,2.5);
\coordinate (b) at (3,.8);
\coordinate (c) at (4.5,0);
\coordinate (d) at (5.3,1.2);
\draw[thick, fill=black!20] (a) -- (b) -- (c) -- (d) -- cycle;
\draw[very thick] (a) -- (c) node[at start, above](v3){$v_3$} node[at end, below](v1){$v_1$};
\draw[thick, dashed] (b) -- (d) node[at start, left](v0){$v_0$} node[at end, right](v2){$v_2$};
\draw[gray,opacity=0.5,-latex] (v0.south) to[out=-75,in=180] (v1.west);
\draw[gray,opacity=0.5,-latex] (v1.east) to[out=0,in=-90] (v2.south);
\draw[gray,opacity=0.5,-latex] (v2.north) to[out=90,in=0] (v3.east);
\end{tikzpicture}

\end{document}

Now for your question in the comment: You set view={5}{50} which defines the viewing angles as explained in the pgfplots manual. Changing it will also lead to a different picture. And all I did was to change the curve by replacing the constant velocity in z-direction to something in which this velocity has some hills.

As I mentioned, you might benefit from drawing all things in one picture with tikz-3dplot. This makes things more intuitive because you then know at least what the directions mean. In order to see them more clearly, I drew the axes but commented them out, all you need to do is to uncomment them. And you could play with the angles in \tdplotsetmaincoords{60}{110} in order to see what the view does. Try e.g. \tdplotsetmaincoords{60}{70}. Notice, however, that you cannot overdo it since these are no real 3D pictures. That is, if you rotate too much, you'll discover that some faces have not been drawn

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{tikz}   
\usepackage{tikz-3dplot}
\begin{document}
\tdplotsetmaincoords{60}{110}
\begin{tikzpicture}[scale=5,tdplot_main_coords]
\coordinate (a) at (0, 0, {sqrt(3/2)/2});
\coordinate (b) at ({-1/(2*sqrt(3))}, -1/2,  {-1/(2*sqrt(6))});
\coordinate (c) at ({1/sqrt(3)}, 0, {-1/(2*sqrt(6))});
\coordinate (d) at ({-1/(2*sqrt(3))},   1/2, {-1/(2*sqrt(6))}); 
\draw[thick, fill=black!20] (a) -- (b) -- (c) -- (d) -- cycle;
\draw[very thick] (a) -- (c) node[at start, above]{$v_3$} node[at end, below]{$v_1$};
\draw[thick, dashed] (b) -- (d) node[at start, left]{$v_0$} node[at end, right]{$v_2$};
\draw plot[variable=\x,samples=100,domain=.75*pi:4.25*pi]
({0.5*sin(deg(-\x))}, {0.5*cos(deg(-\x))+1.5}, 
{(\x/(4.25*pi))^2});
% % Uncomment these lines if you want to know where x', y' and z' point to
% \draw[-latex,blue] (-1.5,0,0) -- (1.5,0,0)  node[above right]  {$x'$};
% \draw[-latex,blue] (0,-1.5,0) -- (0,1.5,0)  node[below] {$y'$};
% \draw[-latex,blue] (0,0,-1.5) -- (0,0,1.5)  node[above left]  {$z'$};
\end{tikzpicture}
\end{document}

ORIGINAL ANSWER: Apart from the possibility to change the viewing angle, you could play with the parametrization:

\documentclass{article}
\usepackage{pgfplots,tikz}

\begin{document}

\begin{tikzpicture}[scale=.75]
\coordinate (a) at (4,2.5);
\coordinate (b) at (3,.8);
\coordinate (c) at (4.5,0);
\coordinate (d) at (5.3,1.2);
\draw[thick, fill=black!20] (a) -- (b) -- (c) -- (d) -- cycle;
\draw[very thick] (a) -- (c) node[at start, above]{$v_3$} node[at end, below]{$v_1$};
\draw[thick, dashed] (b) -- (d) node[at start, left]{$v_0$} node[at end, right]{$v_2$};
\end{tikzpicture}
\begin{tikzpicture}[scale=.5]
\begin{axis} [
view={5}{50},
axis lines=none,
ymin=-2,
ymax=5,
xmin=-2,
xmax=2]
\addplot3 [very thick, ->, domain=.75*pi:4.25*pi, samples = 100, samples y=0] ({sin(deg(-x))}, {cos(deg(-x))}, 
{x+4*sin(deg(x+1.5))+1*x*x*x*x});
\end{axis}
\end{tikzpicture}

\end{document}

Answer 2

add some random value

\documentclass{article}
\usepackage{pgfplots,tikz}

\begin{document}
\begin{tikzpicture}[scale=.75]
\coordinate (a) at (4,2.5);
\coordinate (b) at (3,.8);
\coordinate (c) at (4.5,0);
\coordinate (d) at (5.3,1.2);
\draw[thick, fill=black!20] (a) -- (b) -- (c) -- (d) -- cycle;
\draw[very thick] (a) -- (c) node[at start, above]{$v_3$} node[at end, below]{$v_1$};
\draw[thick, dashed] (b) -- (d) node[at start, left]{$v_0$} node[at end, right]{$v_2$};
\end{tikzpicture}
\begin{tikzpicture}[scale=.5]
\begin{axis} [
view={5}{50},
axis lines=none,
ymin=-2,
ymax=5,
xmin=-2,
xmax=2]
\addplot3 [very thick, ->, domain=.75*pi:4.25*pi, samples = 100, samples y=0] ({sin(deg(-x))+0.1*rnd}, {cos(deg(-x))+0.1*rnd}, {x});
\end{axis}
\end{tikzpicture}
\end{document}