Deck Transformation

Author

Eric W. Weisstein
January 27, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Topology/DeckTransformation.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/DeckTransformation.html.

Additional Authors

Todd Rowland
Ed Pegg, Jr.

Diagram for ℂ/Γ or Δ/Γ

The animation shows the action of some deck transformations.  The spaces in question are the torus and its universal cover, the plane.  An element of the fundamental group, shown as the path in blue, defines a deck transformation of the universal cover.  It moves around the points in the universal cover.  The points moved to have the same projection in the torus.  The blue path is a loop in the torus, and all of its preimages are shown. (Works fine with AnimationDirection->Forward.)

Animation

intlat1[n_Integer]:=Flatten[Table[{i,j},{i,-n,n},{j,-n,n}],1]

intlatpg[n_Integer]:=Point/@intlat1[n]

liftline[n_Integer,t_,p0_List,v_List]:=Line[{p0+#1,p0+#1+t v}]&/@intlat1[n]

Clear[cen,bck,hartog];

hartog[n_][v_List]:=
Module[{bck2=Join[bck,{Red,Disk[cen,.3],Purple,Disk[cen+{-1,0},.3],Green,Disk[cen+{0,-1},.3],Orange,Disk[cen+{-1,-1},.3]}],t},
t=Table[
Show[Graphics[{bck2,Blue,liftline[n,t,{.5,.5},v]}],AspectRatio->Automatic,PlotRange->{{-2.1,2.1},{-2.1,2.1}}],
{t,0,1,.1}];
cen+=v;
t
]
cen={.5,.5};
bck=intlatpg[3];

movie=Flatten[hartog[3]/@{{0,1},{1,0},{0,-1},{0,-1},{-1,0},{-1,0},{0,1},{1,0}}];

Show[GraphicsArray[Partition[Take[movie,8],4]]]

Export[gifs<>"decktran.gif",movie]

Graphic

kpoints={{1,1},{1,3},{3,3},{3,5},{5,5},{5,3},{5,1},{3,1},{1,1},{1,3}};

GraphicsGrid[Partition[Table[Graphics[{Black,Point/@Flatten[Table[{a,b},{a,0,8,2},{b,0,8,2}],1],{Orange,Disk[kpoints[[n]],0.6],Cyan,Disk[kpoints[[n]]+{0,2},0.6],Green,Disk[kpoints[[n]]+{2,0},0.6],Red,Disk[kpoints[[n]]+{2,2},0.6]},Blue,Table[Arrow[{kpoints[[n]]+{a,b},(kpoints[[n]]+kpoints[[n+1]])/2+{a,b}}],{a,0,2,2},{b,0,2,2}]}],{n,1,8}],4],Spacings->{20,20},Frame->All,ImageSize->400]