How can I draw isometric dot paper with a scale size of 1cm with TikZ?

Question

I would like to put 1cm isometric dot paper into an exercise of the exam for my class, so my students can draw on. My apologies for no MWE. I haven't got a clue where to start. See image

Answer 1

Isometric dots can easily be created by modifying the x and y coordinates and then putting dots on a grid. However we would need to clip the result, because we basically rotated the coordinate system. The scale has to be adopted to your liking, as I asked how the 1cm shall be defined and now used the assumption of @AlexG.

\documentclass[tikz]{standalone}
\begin{document}
\begin{tikzpicture}[x={(0.86cm,0.5cm)},y={(-0.86cm,0.5cm)}]
\clip (0,12.5) rectangle (25,12.5);
\foreach \x in {0,...,25}
    \foreach \y in {0,...,25}
    {
    \fill (\x,\y) circle (2pt);
    }
\end{tikzpicture}
\end{document}

Edit after the comments of Jon

I took some measurements with Adobe Acrobat in the PDF created from the code above and this is the result:

So which measurement needs to be 1cm, if you take this image as a reference?

The isometric A4-paper

\documentclass[tikz,border={0.23cm 0.25cm}]{standalone}
\begin{document}
\begin{tikzpicture}[x={(0.86cm,0.5cm)},y={(-0.86cm,0.5cm)}]
\clip (0,25.5) rectangle (37.5,29);
\foreach \x in {0,...,50}
\foreach \y in {0,...,50}
{
    \fill (\x,\y) circle (2pt);
}
\end{tikzpicture}
\end{document}

Answer 2

Update 2: new coordinate system defined with pgfkeys

Coordinates are given in the same style as the native implicit coordinates of tikz, i.e. 3 numbers separated by commas. They are prefixed by iso cs: as for example: (iso cs:0,1,7)

\documentclass[tikz,border=5mm]{standalone}
\usetikzlibrary{arrows.meta}
\pgfkeys{/isometrique/.cd,
      coordonnee/.code args={#1,#2,#3}
        {
            \def\myx{#1}
            \def\myy{#2}
            \def\myz{#3}
        }
}
\tikzdeclarecoordinatesystem{isometric}
{
    \pgfkeys{/isometrique/.cd,
              coordonnee={#1}}
    \pgfpointadd{\pgfpointxyz{0}{\myz}{0}}{\pgfpointadd{\pgfpointpolarxy{-30}{\myx}}{\pgfpointpolarxy{30}{\myy}}}
}
\tikzaliascoordinatesystem{iso}{isometric}
\begin{document}
\begin{tikzpicture}[>={Triangle[angle=45:4pt 3]}]
\newcommand{\nbx}{11}%<--number of point on one row
\newcommand{\nby}{9}%<-- number of point on one column
\foreach \j in {0,...,\the\numexpr\nby-1} {
  \foreach \i in {0,...,\the\numexpr\nbx-1} 
  {\fillblack++(0:{2\icos(30)})circle[radius=1pt]+(30:1)circle[radius=1pt];
}}
\drawvery thick,red,->--node[sloped,below]{$y=6$}(iso cs:0,4,0);
\draw[very thick,blue,->](iso cs:0,4,0)-- node[sloped,above]{$x=2$}++(iso cs:2,0,0);
\draw[very thick,red,->](iso cs:2,4,0)-- node[sloped,below]{$z=3$}++(iso cs:0,0,3);
% Arrows showing the newest coordinate system "iso"
\draw blue,thick,->--node[below]{x}++(iso cs:1,0,0);
\draw red,thick,->--node[left]{y}++(iso cs:0,1,0);
\draw violet,thick,->--node[left]{z}++(iso cs:0,0,1);
\node[below,align=center,draw,fill=white] at (iso cs:0,1,2.7){New \textbf{iso} \ coordinate system};
\begin{scope}[shift={(iso cs:2,4,3)}]
\draw[blue,thick] (iso cs:0,0,0)--++ (iso cs:3,0,0)
--++ (iso cs:0,3,0)
--++ (iso cs:0,0,3)
--++ (iso cs:-3,0,0)
--++ (iso cs:0,-3,0)
--++(iso cs:0,0,-3)
(0,3)--++(iso cs:3,0,0)--+(0,-3)
(iso cs:0,3,0)--+(iso cs:0,3,0);
\end{scope}
\end{tikzpicture}
\end{document}

Update Addition of another coordinate system with a vertical key z (at Tobi's request)

Its disadvantage is to be more verbose since you have to write 3 coordinates instead of 2.

With keyvals since here the keys are defined with keyval package, we can define default values and write for example (trio cs:x,y=2,z) instead of (trio cs:x=0,y=2,z=0). Here, the keys have default values, that is to say that if no value is given, they are worth the default value.

\documentclass[tikz,border=5mm]{standalone}
%\usepackage{tikz}
\usetikzlibrary{arrows.meta}
\makeatletter
\define@key{triangularokeys}{x}[0]{\def\myx{#1}}
\define@key{triangularokeys}{y}[0]{\def\myy{#1}}
\define@key{triangularokeys}{z}[0]{\def\myz{#1}}
\tikzdeclarecoordinatesystem{triangularo}%
{%
\setkeys{triangularokeys}{#1}%
\pgfpointadd{\pgfpointxyz{0}{\myz}{0}}{\pgfpointadd{\pgfpointpolarxy{-30}{\myx}}{\pgfpointpolarxy{30}{\myy}}
}
}
\makeatother
\tikzaliascoordinatesystem{trio}{triangularo}
\begin{document}
\begin{tikzpicture}[>={Stealth[]}]
\newcommand{\nbx}{11}%<--number of point on one row
\newcommand{\nby}{9}%<-- number of point on one column
\foreach \j in {0,...,\the\numexpr\nby-1} {
  \foreach \i in {0,...,\the\numexpr\nbx-1} 
  {\fillblack++(0:{2\icos(30)})circle[radius=1pt]+(30:1)circle[radius=1pt];
}}
\drawvery thick,red,->--node[sloped,below]{$y=6$}(trio cs:x=0,y=4,z=0);
\draw[very thick,red,->](trio cs:x,y=4,z)-- node[sloped,above]{$x=2$}++(trio cs:x=2,y,z);
\draw[very thick,red,->](trio cs:x=2,y=4,z)-- node[sloped,below]{$z=3$}++(trio cs:x,y,z=3);
% Arrows showing the newest coordinate system "trio"
\draw blue,thick,->--node[below]{x}++(trio cs:x=1,y,z);
\draw red,thick,->--node[left]{y}++(trio cs:x,y=1,z);
\draw violet,thick,->--node[left]{z}++(trio cs:x,y,z=1);
\node[below,align=center] at (trio cs:x,y=1,z=3){New trio \ coordinate system};
\begin{scope}[shift={(trio cs:x=2,y=4,z=3)}]
\draw[blue,thick] (trio cs:x,y,z)--++ (trio cs:x=3,y,z)
--++ (trio cs:x,y=3,z)
--++ (trio cs:x,y,z=3)
--++ (trio cs:x=-3,y,z)
--++ (trio cs:x,y=-3,z)
--++(trio cs:x,y,z=-3)
(0,3)--++(trio cs:x=3,y,z)--+(0,-3)
(trio cs:x,y=3,z)--+(trio cs:x,y=3,z);
\end{scope}
\end{tikzpicture}
\end{document}

First answer With a coordinate system called tri with the x and y keys.

In addition to the Cartesian coordinates, I have defined a new coordinate system that makes it "simpler" to draw figures on this grid. It is called triangular and its alias is tri.

For example, the first red arrow is drawn like this:

\draw[very thick,red,->](0,0)--(tri cs:x=0,y=7);

The second arrow is defined as follows:

\draw[very thick,red,->](tri cs:x=0,y=7)--++(tri cs:x=2,y=0);

You'll notice that you can mix the two coordinate systems in the same path and use the relative coordinate.

Code

\documentclass[tikz,border=5mm]{standalone}
%\usepackage{tikz}
\usetikzlibrary{arrows.meta}
% new coordinate system called triangular 
\makeatletter
\define@key{triangularkeys}{x}{\def\myx{#1}}
\define@key{triangularkeys}{y}{\def\myy{#1}}
\tikzdeclarecoordinatesystem{triangular}%
{%
\setkeys{triangularkeys}{#1}%
\pgfpointadd{\pgfpointpolarxy{-30}{\myx}}{\pgfpointpolarxy{30}{\myy}}
}
\makeatother
% end of new coordinate system
\tikzaliascoordinatesystem{tri}{triangular}%<-- define the alias tri for triangular
\begin{document}
\begin{tikzpicture}[>={Stealth[]}]
\newcommand{\nbx}{11}%<--number of dots in a single row
\newcommand{\nby}{9}%<-- number of dots in a single column
% Drawing of the isometric grid
\foreach \j in {0,...,\the\numexpr\nby-1} {
  \foreach \i in {0,...,\the\numexpr\nbx-1} 
  {\fillblack++(0:{2\icos(30)})circle[radius=1pt]+(30:1)circle[radius=1pt];
}}
% The following code below shows how to draw on this grid
% Arrows showing the new coordinate system
\draw blue,thick,->--node[below]{x}++(tri cs:x=1,y=0);
\draw red,thick,->--node[left]{y}++(tri cs:x=0,y=1);
% Big red arrow going from the bottom left to the perspective cube
\drawvery thick,red,->--node[sloped,below]{$y=7$}(tri cs:x=0,y=7);
\draw[very thick,red,->](tri cs:x=0,y=7)-- node[sloped,above]{$x=2$}++(tri cs:x=2,y=0);
% Cube perspective drawing
\begin{scope}[shift={(tri cs:x=2,y=7)}]
\draw (tri cs:x=0,y=0)circle(3pt)--++ (tri cs:x=3,y=0)
--++ (tri cs:x=0,y=3)
--++ (0,3)
--++ (tri cs:x=-3,y=0)
--++ (tri cs:x=0,y=-3)
--++(0,-3)
(0,3)--++(tri cs:x=3,y=0)--+(0,-3)
(tri cs:x=0,y=3)--+(tri cs:x=0,y=3);
\end{scope}
\end{tikzpicture}
\end{document}

Answer 3

Just for fun, a pure PostScript solution for making 1-cm-scale isometric dotted paper. Can be directly sent to a PostScript printer.

Use ps2pdf if you need a PDF; but it is much bigger [38 kB] than the PS [242 B]. (The PS code was somewhat optimized for size, though not too aggressively in order not to sacrifice legibility.)

isometricdottedA4.ps:

%! 
<</PageSize [595 842]>> setpagedevice 
/cm {28.346457 mul} def 
[.866 .5 -.866 .5 595 2 div 842 41 cm sub 2 div] concat 
0 1 41 { cm 
 0 1 41 { cm 1 index exch moveto 
  gsave initmatrix currentpoint 2 0 360 arc fill grestore 
 } for pop 
} for 

The following version can be used for both, A4 and Letter, formats. Just replace false with true in the 2nd line for Letter paper. Dots still spaced by 1 cm.

isometricdottedA4orLetter.ps:

%! 
/letter false def % replace with `true' for Letter paper 
letter {/width 612 def /height 792 def} {/width 595 def /height 842 def} ifelse 
<</PageSize [width height]>> setpagedevice 
/cm2bp {28.346457 mul} def % conversion 
/dots height width 60 sin 60 cos div div add 1 cm2bp div cvi def % # dots filled rhombus height [cm] 
[30 cos 30 sin 60 sin neg 60 cos width 2 div height dots cm2bp sub 2 div] concat % axes rotated (30°, 60°) & translated 
0 1 dots { cm2bp 
  0 1 dots { cm2bp 1 index exch moveto 
    gsave initmatrix currentpoint 2 0 360 arc fill grestore 
  } for pop 
} for

Answer 4

A PSTricks solution only for either fun or comparison purposes.

\documentclass[pstricks,border=12pt]{standalone}
\begin{document}
\begin{pspicture}(10,10)
    \multips(0,0)(0,1){11}{%
        \multips(0,0)(1,0){11}{%
            \qdisk(0,0){2pt}\qdisk(.5,.5){2pt}}}
\end{pspicture}
\end{document}

Answer 5

Like this? I see here.

\documentclass[border=3.14mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{3d,perspective}
%https://tex.stackexchange.com/questions/690566/how-to-create-a-mapping-like-the-following-figure/690569#690569
\begin{document}
\begin{tikzpicture}[isometric view ,declare function={a=3;b=3;h=3;
        }]
        \path
        foreach \X in {-1,...,5}
        {foreach \Y in {-1,...,5}
            {foreach \Z in {-1,...,3}
                {(\X,\Y,\Z)node[circle,inner sep=1pt,fill]{}}}}
        (0,0,0) coordinate (A)
        (a,0,0) coordinate (B)
        (a,b,0) coordinate (C)
        (0,b,0) coordinate (D)
        (0,0,h) coordinate (E)
        (a,0,h) coordinate (F)
        (a,b,h) coordinate (G)
        (0,b,h) coordinate (H)
        ;
        \draw  (E)-- (F) -- (G) -- (H) --cycle 
        (E) -- (A) -- (D) -- (H)
        (A) -- (B) -- (F)
        ;
    \end{tikzpicture}
    \end{document}