Drawing polyhedra using TikZ with semi-transparent and shading effect

TikZ also has an xyz-coordinate system that is quite useful here. There's a nice answer making use of this feature: Table, i.e. the piece of furniture on which one eats, in Tikz

Here's a way of drawing your octahedron. To rotate it, play around with the x, y and z options of the tikzpicture:

\documentclass{article}
\usepackage{tikz}
\begin{document}

\begin{tikzpicture}[line join=bevel,z=-5.5]
\coordinate (A1) at (0,0,-1);
\coordinate (A2) at (-1,0,0);
\coordinate (A3) at (0,0,1);
\coordinate (A4) at (1,0,0);
\coordinate (B1) at (0,1,0);
\coordinate (C1) at (0,-1,0);

\draw (A1) -- (A2) -- (B1) -- cycle;
\draw (A4) -- (A1) -- (B1) -- cycle;
\draw (A1) -- (A2) -- (C1) -- cycle;
\draw (A4) -- (A1) -- (C1) -- cycle;
\draw [fill opacity=0.7,fill=green!80!blue] (A2) -- (A3) -- (B1) -- cycle;
\draw [fill opacity=0.7,fill=orange!80!black] (A3) -- (A4) -- (B1) -- cycle;
\draw [fill opacity=0.7,fill=green!30!black] (A2) -- (A3) -- (C1) -- cycle;
\draw [fill opacity=0.7,fill=purple!70!black] (A3) -- (A4) -- (C1) -- cycle;
\end{tikzpicture}

\end{document}

You can use the fill and opacity constructs:

\documentclass{article}
\usepackage{tikz}

\definecolor{cof}{RGB}{219,144,71}
\definecolor{pur}{RGB}{186,146,162}
\definecolor{greeo}{RGB}{91,173,69}
\definecolor{greet}{RGB}{52,111,72}

\begin{document}

\begin{tikzpicture}[thick,scale=5]
\coordinate (A1) at (0,0);
\coordinate (A2) at (0.6,0.2);
\coordinate (A3) at (1,0);
\coordinate (A4) at (0.4,-0.2);
\coordinate (B1) at (0.5,0.5);
\coordinate (B2) at (0.5,-0.5);

\begin{scope}[thick,dashed,,opacity=0.6]
\draw (A1) -- (A2) -- (A3);
\draw (B1) -- (A2) -- (B2);
\end{scope}
\draw[fill=cof,opacity=0.6] (A1) -- (A4) -- (B1);
\draw[fill=pur,opacity=0.6] (A1) -- (A4) -- (B2);
\draw[fill=greeo,opacity=0.6] (A3) -- (A4) -- (B1);
\draw[fill=greet,opacity=0.6] (A3) -- (A4) -- (B2);
\draw (B1) -- (A1) -- (B2) -- (A3) --cycle;
\end{tikzpicture}

\end{document}

PSTricks can handle this one. Run it with xelatex or latex->dvips->ps2pdf

\documentclass{article}
\usepackage[dvipsnames]{pstricks}
\usepackage{pst-solides3d}
\begin{document}

\begin{pspicture}(-2.5,-2)(2.5,2.5)
\psset{lightsrc=10 20 30,viewpoint=40 10 30 rtp2xyz,Decran=40}
 \psSolid[object=octahedron,a=3,linecolor=blue,
          opacity=0.6,hollow,hue=0 1,
          action=draw**]
% \axesIIID(3,3,3)(4,4,4)
\end{pspicture}

\end{document}