Horn-shaped spiral tiling
By Sándor Kabai

Initially a unit of four spirals of trapeziums are prepared, then such units are placed side by side. The image exhibit a number of rotational and translation symmetries. Also a square lattice can be seen, which indicates how a single square tile could be used for establishing the tiling of any size.

In[]:=

ManipulatevA=1

2

,1

2

,t;vB=0,1

2

,t;vC=0,

2

4,t;vD=

2

4,

2

4,t;vE=1

2

,0,t;vA1=1

2

,1

2

,-t;vB1=0,1

2

,-t;vC1=0,

2

4,-t;vD1=

2

4,

2

4,-t;top={Specularity[0.6],Polygon[{vA,vB,vC,vD}]};bot=Polygon[{vA1,vB1,vC1,vD1}];side=Polygon[{vA,vB,vB1,vA1}];face=Polygon[{vA,vD,vD1,vA1}];tile={face,side,top,bot};mult=TableScaleRotate[{tile},i*Pi/4,{0,0,1}],1

2

^i,1

2

^i,1,{0,0,0},{i,n};multc=Table[{Hue[0.25i],Rotate[{mult},i2Pi/4+Pi/4,{0,0,1}]},{i,4}];mx=Table[Translate[Rotate[multc,iPi,{0,1,0}],{i,0,0}],{i,m}];my=Table[Translate[Rotate[mx,iPi,{1,0,0}],{0,i,0}],{i,o}];Graphics3D[my,ImageSize300,BoxedFalse,SphericalRegionTrue,ViewPoint{0,0,100},ViewAngle->0.01],{{n,4,"trapezoid tiles in a spiral"},3,7,1,RadioButton},{{m,4,"spiral tiles in a row"},1,6,1,RadioButton},{{o,4,"spiral tiles in a column"},1,6,1,RadioButton},{{t,0.2,"thickness"},0.1,1,ImageSizeTiny,ControlPlacementLeft},TrackedSymbolsManipulate,SaveDefinitions->True

In[]:=

Graphics3D[{Glow[White],EdgeForm[Thick],my},ViewPoint{0,0,100},ViewAngle->0.01]

Out[]=