rot60[xy_]:=(({{Cos[((1π)/3)], -Sin[((1π)/3)]}, {Sin[((1π)/3)], Cos[((1π)/3)]}})).xy; (*rotation matrix for 60 degrees translation of points*)
Rot60[xylist_]:=rot60/@xylist; (* only rotation of 60 degrees is used in this example*)
Rot120[xylist_]:=Rot60[Rot60[xylist]];
Rot180[xylist_]:=Rot60[Rot60[Rot60[xylist]]];
Rot240[xylist_]:=Rot120[Rot120[xylist]];
Rot300[xylist_]:=Rot120[Rot180[xylist]];
ROT60[xylistlist_]:=Rot60/@xylistlist; (* only rotation of 60 degrees is used in this example*)
ROT120[xylistlist_]:=Rot120/@xylistlist;
ROT180[xylistlist_]:=Rot180/@xylistlist;
ROT240[xylistlist_]:=Rot240/@xylistlist;
ROT300[xylistlist_]:=Rot300/@xylistlist;

pntA={2*(Sqrt[3]),8};
pntB={0,6};
pntC={4*(Sqrt[3]),6};
pntD={2*(Sqrt[3]),0};
pntO={0,0};

transDA[xy_]:=xy+(pntA-pntD);
TransDA[xylist_]:=transDA/@xylist;
TRANSDA[xylistlist_]:=TransDA/@xylistlist;

transOD[xy_]:=xy+{2(Sqrt[3]),0}; 
TransOD[xylist_]:=transOD/@xylist;
TRANSOD[xylistlist_]:=TransOD/@xylistlist;
 
transDO[xy_]:=xy-{2(Sqrt[3]),0}; 
TransDO[xylist_]:=transDO/@xylist;
TRANSDO[xylistlist_]:=TransDO/@xylistlist;
(*
Slider[Dynamic[XXX],{0,10,0.1}]
Slider[Dynamic[YYY],{0,10,0.1}]
Slider[Dynamic[ZZZ],{0,10,0.1}]
Slider[Dynamic[QQQ],{0,10,0.1}]

{Dynamic[XXX],Dynamic[YYY]}
{Dynamic[ZZZ],Dynamic[QQQ]}

I have used these sliders to control the outcome, however this slowed the process down in such a significant way that it was not really useful.
*)

testRuitAB={{6/(Sqrt[3]),8},{4/(Sqrt[3]),8},{0,6}}; (* draw the points of the figure*)
testRuitBD={{0.000,6.000},{0.010,5.800},{0.200,5.300},{0.360,3.500},{0.240,1.900},{0.200,1.700},{0.300,1.600},{0.400,1.550},{0.600,1.500},{0.700,1.502},{0.900,1.570},{1.000,1.630},{1.100,1.670},{1.300,1.840},{1.400,2.220},{1.900,1.550},{2.800,0.630},{2*(Sqrt[3 ]),0}};(* points of the figure *)
Graveer={{1.4,2.4},{1.6,2.9},{2.0,3.1},{2.5,3.0}}; (* separate engraving figures/points*)
Graveer2={{1.4,2.7},{1.6,3.2},{2.0,3.4},{2.5,3.4}};
Graveer3={{1.4,3.0},{1.6,3.5},{2.0,3.7},{2.5,3.7}};
Graveer4={{0,6},{0.21,5.8},{0.4,5.3},{0.56,3.5},{0.44,1.9}};
Graveer5={{0,6},{0.41,5.8},{0.6,5.3},{0.76,3.5},{0.64,2.1}};
Graveer6={{0,6},{0.61,5.8},{0.8,5.3},{0.96,3.5},{0.84,2.1}};

totaleRuit={testRuitAB,(*, testRuitBD , testRuitAC ,*)testRuitBD}; (* make a list of all the point-lists created above and leave out the double lines*)
totaleGraveer={Graveer, Graveer2, Graveer3, Graveer4, Graveer5, Graveer6}; (* separate engraving list, in order to assign different colours*)

transBDBD[xy_]:=xy+{2*4*(Sqrt[3]),0}; (* translating of points to satisfy the C3C3C6C6 figure*)
TransBDBD[xylist_]:=transBDBD/@xylist; (* translating of lists of points to satisfy the C3C3C6C6 figure*)
TRANSBDBD[xylistlist_]:=TransBDBD/@xylistlist; (* translating of lists of lists of points to satisfy the C3C3C6C6 figure*)

transADAB[xy_]:=xy+{4*(Sqrt[3]),8+4}; (*different translation*)
TransADAB[xylist_]:=transADAB/@xylist;
TRANSADAB[xylistlist_]:=TransADAB/@xylistlist;

T2[xylistlist_]:=TRANSBDBD[xylistlist]; 

ROT60B[xylistlist_]:=TRANSOD[ROT60[TRANSDO[xylistlist]]]; (*rotate one tile 60 degrees*)
ArneRuit:=Table[Line[Nest[TRANSADAB, Nest[TRANSBDBD,Nest[ROT60B, totaleRuit,i],j],k]],{i,0,5},{j,0,3},{k,0,3}]; (* to greate a table of the desired figure, first rotate the start-figure (two lines) 6 times by 60 degrees, then translate that figure by length BDBD and by length ADAB in the other direction to create the field of figures*)
ArneGraveer:=Table[Line[Nest[TRANSADAB, Nest[TRANSBDBD,Nest[ROT60B, totaleGraveer,i],j],k]],{i,0,5},{j,0,3},{k,0,3}]; (*same for the engravings*)
Laseren=Show[Graphics[{Red,ArneRuit,Black,ArneGraveer}]] (*to show the result the two tesselations are shown in red and black*)
Export["Laseren.eps", Laseren]; (* exported as an eps file*)