Three different kinds of rhombuses are supplemented with a heptagon to produce various tiling.
The three rhombuses (t1, t2, t3).
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r=0.5/Sin[Pi/7];d=rCos[Pi/7];h=d/Sin[Pi/14];j=(h+2d);y1=Cos[Pi/7];x1=Sin[Pi/7];vA={x1,0,0};vB={0,y1,0};vC={-x1,0,0};vD={0,-y1,0};t1=Polygon[{vA,vB,vC,vD}];t17=Table[Rotate[Translate[{t1},{0,y1,0}],i2Pi/7+Pi/2,{0,0,1}],{i,7}];y2=2y1Sin[Pi/7];x2=Sqrt[1-y2^2];vE={x2,0,0};vF={0,y2,0};vG={-x2,0,0};vH={0,-y2,0};t2=Polygon[{vE,vF,vG,vH}];t27=Table[Rotate[Translate[t2,{1+x2,0,0}],i2Pi/7,{0,0,1}],{i,7}];y3=(1+2x2)Sin[Pi/7];x3=Sqrt[(1+2x2)^2-y3^2]-2y1;vI={x3,0,0};vJ={0,y3,0};vK={-x3,0,0};vL={0,-y3,0};t3=Polygon[{vI,vJ,vK,vL}];t37=Table[Rotate[Translate[t3,{2y1+x3,0,0}],i2Pi/7+Pi/7,{0,0,1}],{i,7}];t127={Green,t17,Pink,t27};t1237={Cyan,t17,LightRed,t27,Yellow,t37};Graphics3D[{t1237},Boxed->False,ViewPoint->{0,0,100},ImageSize->200]
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If 14 pieces of t3 rhombuses are places at the centre, then 6 rings of 14 rhombuses can be assembled.
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t114=Table[Rotate[Translate[Rotate[t1,Pi/2,{0,0,1}],{y1+1,0,0}],i2Pi/14,{0,0,1}],{i,14}];t214=Table[Rotate[Translate[t2,{0,-y2-2y3,0}],i2Pi/14,{0,0,1}],{i,14}];t314=Table[Rotate[Translate[t3,{0,-y3,0}],i2Pi/14,{0,0,1}],{i,14}];y4=(2y3+2y2)Sin[Pi/14];x4=(y4/Tan[Pi/14])-(1+2y1);vM={x4,0,0};vN={0,y4,0};vO={-x4,0,0};vP={0,-y4,0};t4=Polygon[{vM,vN,vO,vP}];t414=Table[Rotate[Translate[t4,{1+2y1+x4,0,0}],i2Pi/14,{0,0,1}],{i,14}];t514=Table[Rotate[Translate[t1,{2y3+2y2+x1,0,0}],i2Pi/14+Pi/14,{0,0,1}],{i,14}];t614=Table[Rotate[Translate[t3,{1+2y1+2x2+x3,0,0}],i2Pi/14,{0,0,1}],{i,14}];form={Yellow,t314,Cyan,t114,Green,t214,LightRed,t414};Graphics3D[{Cyan,t114,LightRed,t214,Yellow,t314,Green,t414,Cyan,t514,Yellow,t614},ImageSize200,Boxed->False,ViewPoint->{0,0,100}]
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v7=Table[r{Sin[i2Pi/7],Cos[i2Pi/7],0},{i,7}];p7=Polygon[v7];Graphics3D[{p7},ImageSize100,Boxed->False,ViewPoint->{0,0,100}]
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Five heptagons can be arranges in two different rings with tip to tip connections.
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p7up=Translate[p7,{0,2.59,0}];u1a={Table[Rotate[p7up,i2Pi/7,{0,0,1}],{i,7}]};t3d7=Table[Rotate[Translate[t3,{0,-1.61,0}],i2Pi/7,{0,0,1}],{i,7}];t2d7=Table[Rotate[Translate[t2,{0,-3.37,0}],i2Pi/7,{0,0,1}],{i,7}];t1da=Rotate[Translate[t1,{0,y1,0}],Pi/2,{0,0,1}];t1db=Rotate[Translate[t1,{0,y1,0}],-Pi/2,{0,0,1}];t1ab=Table[Rotate[Translate[{t1da,t1db},{0,3.745,0}],i2Pi/7,{0,0,1}],{i,7}];t1k=Table[Rotate[Translate[Rotate[t1,Pi/2,{0,0,1}],{0,4.17,0}],i2Pi/7,{0,0,1}],{i,7}];sum1={t1k,Green,t1ab,LightRed,t2d7,Cyan,t3d7,Yellow,u1a};Row[Graphics3D[#,Boxed->False,ImageSize->200,ViewPoint->{0,0,100}]&/@{u1a,sum1}]
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p7up=Translate[p7,{0,-2.59,0}];u1b=Table[Rotate[p7up,i2Pi/7,{0,0,1}],{i,7}];t2u=Table[Rotate[Translate[t2,{0,1.3,0}],i2Pi/7,{0,0,1}],{i,7}];t3u=Table[Rotate[Translate[t3,{0,3.05,0}],i2Pi/7,{0,0,1}],{i,7}];t2sa=Rotate[Translate[t2,{x2,0,0}],-Pi/7,{0,0,1}];t2sb=Rotate[Translate[t2,{-x2,0,0}],Pi/7,{0,0,1}];t2s=Table[Rotate[Translate[{t2sa,t2sb},{0,4.03,0}],i2Pi/7,{0,0,1}],{i,7}];p7u=Table[Rotate[Translate[{p7},{0,4.66,0}],i2Pi/7+Pi/7,{0,0,1}],{i,7}];t1uu=Table[Rotate[Translate[t1,{4.45,0,0}],i2Pi/7+Pi/2,{0,0,1}],{i,7}];t2ss=Table[Rotate[Translate[{t2sa,t2sb},{0,-4.89,0}],i2Pi/7+Pi/7,{0,0,1}],{i,7}];sum2={Green,t2ss,LightRed,t1uu,Cyan,p7u,Green,t2s,Pink,t3u,Cyan,t2u,Yellow,u1b};Row[Graphics3D[#,Boxed->False,ImageSize->200,ViewPoint->{0,0,100}]&/@{u1b,sum2}]
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Fourteen heptagons arranged in two different rings with tip to tip connections and with side to side connections.
Four rings of rhombuses fill the space within the ring of 14 heptagons.
An additional ring of 14 heptagons is added to the purpose of making connections.
The rings of 14 heptagons are shown together with rings of rhombuses. This indicates how the patterns could be printed on rhombuses.
Rings of 14 heptagons are fitted into the rhombuses.
Inside and outside the rings of 14 heptagons.
Examples for other arrangements
See the predecessor submissions:
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Seven-fold rhombus tiling using three rhombuses and a heptagon
by Sándor Kabai
Wolfram Community, STAFF PICKS, September 28, 2023
https://community.wolfram.com/groups/-/m/t/3023665
by Sándor Kabai
Wolfram Community, STAFF PICKS, September 28, 2023
https://community.wolfram.com/groups/-/m/t/3023665