"Redundant" coordinate system for triangular grid

You could use sth. like this:

€dit: Triangles.

%documentclass[]{article}
documentclass[margin=5pt, tikz]{standalone}
usepackage{tikz}
usetikzlibrary{shapes.geometric}
begin{document}
pgfmathsetmacroxMax{8}
pgfmathsetmacroyMax{7}

pgfmathsetmacros{2*sqrt(3)/3}
begin{tikzpicture}[font=footnotesize,
x={(1,0)}, y={(60:1)}, 
>=latex, 
Point/.style={draw=red, fill=white},
shape triangle/.style = {draw, thick, regular polygon, regular polygon sides=3, minimum height=s cm, % anchor=center, 
inner sep=0pt, 
},
triangle/.style={shape triangle, shift={(0.5,0.25*s cm)}},
triangle*/.style={shape triangle, shape border rotate=180, , shift={(0.5,-0.25*s cm)}},
]
% Contents
fill[purple] (0,3) -- (0,4) -- (4,0) -- (3,0) --cycle;

node[triangle*, fill=green] at (4,4)  {$downarrow$};
node[triangle, fill=green!77!red] at (4,4)  {$uparrow$};

node[triangle, fill=red] at (2,5)  {a};
node[triangle, fill=cyan] at (2,3)  {};
node[triangle*, fill=yellow] at (0,7)  {};

% CoSy
begin{scope}[very thin]
clip[] (0,0) -- (xMax,0) -- (xMax cm+pgflinewidth, yMax) -- (pgflinewidth,yMax)  --cycle; 

% Contents - cliped
fill[blue!33, opacity=0.7] (0,1) rectangle (xMax,2);

fill[purple!33, opacity=0.7] (6,0) -- (7,0)--(7,8)--(6,8)--cycle;

foreach y in {0,...,yMax}{
tikzset{  temp/.style={fill=pink, draw=none,opacity=0.1}   }
node[triangle*, temp] at (6,y)  {};
node[triangle, temp] at (6,y)  {};
}

% CoSy 1/2 - "grid"
foreach x in {0,...,xMax}{
draw[] (x,0) -- (x,yMax); 
}
%
pgfmathsetmacroyMAX{2*yMax}
foreach y in {0,...,yMAX}{
draw[] (0,y) -- (xMax,y); 
draw[] (0,y) -- (y,0); 
}
end{scope}
% CoSy 2/2 - "axes"
draw[thick,->] (0,0) -- (xMax+0.5,0) node[below]{$x$};
draw[thick,->] (0,0) -- (0,yMax+0.5) node[left]{$y$};
foreach x in {0,...,xMax}{
draw[] (x,2pt) -- (x,-2pt) node[below] {x}; 
}
foreach y in {0,...,yMax}{
draw[] (2pt,y) -- (-2pt,y) node[left] {ify 0 else yfi}; 
}

% Contents - Points
draw[Point, draw=green] (4,4) circle(2pt);

draw[Point] (0,1) circle(2pt);
draw[Point] (3,3.5) circle(2pt);
draw[Point] (4,6) circle(2pt);
end{tikzpicture}
end{document}