tikz-3d shifting rectangle

Question

Consider the following code:

\documentclass{article}
\usepackage{tikz}
\usepackage{tikz-3dplot}
\usepackage{amsmath}

\newcommand{\ptsize}{1.5pt}

\usepackage{xcolor}
\definecolor{MyColor1}{rgb}{0.24, 0.59, 0.32}
\definecolor{MyColor2}{rgb}{0.85, 0.49, 0.19}
\definecolor{MyColor3}{rgb}{0.22, 0.42, 0.69}

\begin{document}
\tdplotsetmaincoords{75}{130}
\begin{tikzpicture}[tdplot_main_coords]

% Define all the nodes necessary for the drawing
\coordinate (alpha1) at (1,-1,0);
\coordinate (alpha2) at (0,1,-1);
\coordinate (alpha3) at (0,0,2);

\coordinate (12) at ($ (alpha1) + (alpha2) $);
\coordinate (23) at ($ (alpha2) + (alpha3) $);
\coordinate (123) at ($ (alpha1) + (alpha2) + (alpha3) $);
\coordinate (223) at ($ (alpha2) + (alpha2) + (alpha3) $);
\coordinate (1223) at ($ (alpha1) + (alpha2) + (alpha2) + (alpha3) $);
\coordinate (11223) at ($ (alpha1) + (alpha1) + (alpha2) + (alpha2) + (alpha3) $);

\coordinate (-alpha1) at (-1,1,0);
\coordinate (-alpha2) at (0,-1,1);
\coordinate (-alpha3) at (0,0,-2);

\coordinate (-12) at ($ (-alpha1) + (-alpha2) $);
\coordinate (-23) at ($ (-alpha2) + (-alpha3) $);
\coordinate (-123) at ($ (-alpha1) + (-alpha2) + (-alpha3) $);
\coordinate (-223) at ($ (-alpha2) + (-alpha2) + (-alpha3) $);
\coordinate (-1223) at ($ (-alpha1) + (-alpha2) + (-alpha2) + (-alpha3) $);
\coordinate (-11223) at ($ (-alpha1) + (-alpha1) + (-alpha2) + (-alpha2) + (-alpha3) $);

% Draw the black polygon
\begin{scope}[opacity=0.5]
    \draw (alpha3) -- (11223) -- (-alpha3) -- (-11223) -- (alpha3);
    \draw (alpha3) -- (223) -- (-alpha3) -- (-223) -- (alpha3);
\end{scope}

% Draw the nodes for the roots in appropriate colors
\begin{scope}[MyColor3]
    \draw[fill] (alpha3)  circle (\ptsize);
    \draw[fill] (-alpha3) circle (\ptsize);
\end{scope}

\begin{scope}[MyColor2]
    \draw[fill] (alpha1)  circle (\ptsize);
    \draw[fill] (223)     circle (\ptsize);
    \draw[fill] (1223)    circle (\ptsize);
    \draw[fill] (11223)   circle (\ptsize);
    \draw[fill] (-alpha1) circle (\ptsize);
    \draw[fill] (-223)    circle (\ptsize);
    \draw[fill] (-1223)   circle (\ptsize);
    \draw[fill] (-11223)  circle (\ptsize);
\end{scope}

\begin{scope}[MyColor1]
    \draw[fill] (alpha2) circle (\ptsize);
    \draw[fill] (12)      circle (\ptsize);
    \draw[fill] (23)      circle (\ptsize);
    \draw[fill] (123)     circle (\ptsize);
    \draw[fill] (-alpha2) circle (\ptsize);
    \draw[fill] (-12)     circle (\ptsize);
    \draw[fill] (-23)     circle (\ptsize);
    \draw[fill] (-123)    circle (\ptsize);
\end{scope}

% Draw the orange rectangle
\draw[MyColor2] (11223) -- (223) -- (-11223) -- (-223) -- (11223);

% Draw the upper green rectangle
\draw[MyColor1] (-alpha2) -- (123) -- (23) -- (-12) -- (-alpha2);

% Draw the upper green rectangle
\draw[MyColor1] (12) -- (-23) -- (-123) -- (alpha2) -- (12);

% Nodes for the names
\node (alpha1-name) at (1.4,-1,0) {$\alpha_1$};
\node (alpha2-name) at (0,1.4,-1) {$\alpha_2$};
\node (alpha3-name) at (0,0,2.3) {$\alpha_3$};

\begin{scope}[shift={(0,0,15)}]
    \draw[dashed] (alpha1) -- (1223) -- (-alpha1) -- (-1223) -- (alpha1);    
\end{scope}

\end{tikzpicture}
\end{document}

which produces the picture

I'm currently trying to shift the dashed rectangle by a certain amount in the z-direction of the graph. Following this my attempt can be found in the above code

\begin{scope}[shift={(0,0,15)}]
    \draw[dashed] (alpha1) -- (1223) -- (-alpha1) -- (-1223) -- (alpha1);    
\end{scope}

but this doesn't produce any changes. So presumably I'm misunderstanding how this shift command is supposed to work... Ideally I'd like to define new coordinates at the edges of the shifted rectangle, but I'm not really sure if this can be done..

Answer 1

Symbolic coordinates do not get transformed that way. You can transform them with transform canvas, e.g.

\begin{scope}[transform canvas={shift={(0,0,2)}}]
    \draw[dashed] (alpha1) -- (1223) -- (-alpha1) -- (-1223) -- (alpha1);    
\end{scope}

However, this is a bit dangerous, for instance the bounding box does not get computed correctly. On the other hand, tikz-3dplot loads the calc library, so you might do

\documentclass{article}
\usepackage{tikz}
\usepackage{tikz-3dplot}
\usepackage{amsmath}

\newcommand{\ptsize}{1.5pt}

\usepackage{xcolor}
\definecolor{MyColor1}{rgb}{0.24, 0.59, 0.32}
\definecolor{MyColor2}{rgb}{0.85, 0.49, 0.19}
\definecolor{MyColor3}{rgb}{0.22, 0.42, 0.69}

\begin{document}
\tdplotsetmaincoords{75}{130}
\begin{tikzpicture}[tdplot_main_coords]

% Define all the nodes necessary for the drawing
\coordinate (alpha1) at (1,-1,0);
\coordinate (alpha2) at (0,1,-1);
\coordinate (alpha3) at (0,0,2);

\coordinate (12) at ($ (alpha1) + (alpha2) $);
\coordinate (23) at ($ (alpha2) + (alpha3) $);
\coordinate (123) at ($ (alpha1) + (alpha2) + (alpha3) $);
\coordinate (223) at ($ (alpha2) + (alpha2) + (alpha3) $);
\coordinate (1223) at ($ (alpha1) + (alpha2) + (alpha2) + (alpha3) $);
\coordinate (11223) at ($ (alpha1) + (alpha1) + (alpha2) + (alpha2) + (alpha3) $);

\coordinate (-alpha1) at (-1,1,0);
\coordinate (-alpha2) at (0,-1,1);
\coordinate (-alpha3) at (0,0,-2);

\coordinate (-12) at ($ (-alpha1) + (-alpha2) $);
\coordinate (-23) at ($ (-alpha2) + (-alpha3) $);
\coordinate (-123) at ($ (-alpha1) + (-alpha2) + (-alpha3) $);
\coordinate (-223) at ($ (-alpha2) + (-alpha2) + (-alpha3) $);
\coordinate (-1223) at ($ (-alpha1) + (-alpha2) + (-alpha2) + (-alpha3) $);
\coordinate (-11223) at ($ (-alpha1) + (-alpha1) + (-alpha2) + (-alpha2) + (-alpha3) $);

% Draw the black polygon
\begin{scope}[opacity=0.5]
    \draw (alpha3) -- (11223) -- (-alpha3) -- (-11223) -- (alpha3);
    \draw (alpha3) -- (223) -- (-alpha3) -- (-223) -- (alpha3);
\end{scope}

% Draw the nodes for the roots in appropriate colors
\begin{scope}[MyColor3]
    \draw[fill] (alpha3)  circle (\ptsize);
    \draw[fill] (-alpha3) circle (\ptsize);
\end{scope}

\begin{scope}[MyColor2]
    \draw[fill] (alpha1)  circle (\ptsize);
    \draw[fill] (223)     circle (\ptsize);
    \draw[fill] (1223)    circle (\ptsize);
    \draw[fill] (11223)   circle (\ptsize);
    \draw[fill] (-alpha1) circle (\ptsize);
    \draw[fill] (-223)    circle (\ptsize);
    \draw[fill] (-1223)   circle (\ptsize);
    \draw[fill] (-11223)  circle (\ptsize);
\end{scope}

\begin{scope}[MyColor1]
    \draw[fill] (alpha2) circle[radius=\ptsize];
    \draw[fill] (12)      circle[radius=\ptsize];
    \draw[fill] (23)      circle[radius=\ptsize];
    \draw[fill] (123)     circle[radius=\ptsize];
    \draw[fill] (-alpha2) circle[radius=\ptsize];
    \draw[fill] (-12)     circle[radius=\ptsize];
    \draw[fill] (-23)     circle[radius=\ptsize];
    \draw[fill] (-123)    circle[radius=\ptsize];
\end{scope}

% Draw the orange rectangle
\draw[MyColor2] (11223) -- (223) -- (-11223) -- (-223) -- (11223);

% Draw the upper green rectangle
\draw[MyColor1] (-alpha2) -- (123) -- (23) -- (-12) -- (-alpha2);

% Draw the upper green rectangle
\draw[MyColor1] (12) -- (-23) -- (-123) -- (alpha2) -- (12);

% Nodes for the names
\node (alpha1-name) at (1.4,-1,0) {$\alpha_1$};
\node (alpha2-name) at (0,1.4,-1) {$\alpha_2$};
\node (alpha3-name) at (0,0,2.3) {$\alpha_3$};

%\begin{scope}[transform canvas={shift={(0,0,2)}}]
\draw[dashed] ($(alpha1)+(0,0,2)$) -- ($(1223)+(0,0,2)$) 
    -- ($(-alpha1)+(0,0,2)$) --($(-1223)+(0,0,2)$) -- cycle;    
%\end{scope}

\end{tikzpicture}
\end{document}

I shifted by 2 instead of 15 because 15 is really far away, but you can do that, too.