2003 December
 Xah
 xah@xahlee.org
 http://xahlee.org/PageTwo_dir/more.html

gp = 
        Table[{GrayLevel[If[EvenQ@(x + y), 0, 1]], 
            Rectangle[{x, y}, {x + 1, y + 1}]}, {x, 0, 7}, {y, 0, 
            7}];

Show[Graphics@gp, AspectRatio -> 1]

e

We want to apply a slant transformation to the checker pattern. First change the Rectangle to Polygon, then apply the slant transform to each coordinate of the Polygon.

gp2 = gp /. 
            Rectangle[{x_, y_}, {a_, b_}] :> 
              Polygon@{{x, y}, {a, y}, {a, b}, {x, b}} /. 
          Polygon[ptsList_] :> 
            Polygon[({First@#, Last@# + (First@#)/8}) & /@ 
                ptsList];

Show[Graphics@gp2, AspectRatio -> 1, 
      Frame -> True]