Torsten Sillke, June 1997 Topologically equivalent tilings: --------------------------------- Example: Cairo tiling (A) The Cairo tiling is the dual tessellation of the Archimedean "snub square" tessellation of the plane, as John H. Conway calls it. (B) One way to the Cairo tiling is to tile the plane with X-pentominoes, then cutting each pentomino into four "house-shaped" pentagons. (C) "sailor basketweave" Equilateral tiling (A) Square tiling (B) | / | / / / /`-./ / /`-. `-./ `-./ `-. `-./ / /`-./ / /`-. /`-. / /`-. `-./ / ___/ | ___/ | _ / /`-./ / /`-. `- / | / | / `-./ / /`-./ / / `-./ `-./ /`-. `-./ / /`-./ `-. /`-. /`-. -./ / /`-. `-./ | ___/ | ___/ / /`-./ / /`-. | / | / `-./ / /`-./ / `-./ `-./ /`-. `-./ / / /`-. /`-. /`-./ / /`-. `-./ ___/ | ___/ | _ ./ / /`-./ / /`- / | / | / /`-. `-./ / /`-./ / `-./ `-./ / /`-. `-./ / `-. /`-. /`-. _/ /`-./ / /`-. ` | ___/ | ___/ ' `-./ | _/ /`-./ / | / \ | / \ /`-._.+' `-./ | _/ /`-. .-' `-./ \.-' `-./ \.-' `-./ / | /`-._.+' `-./ \ /`-. .-'\ /`-. .-'\ | _/ /`-./ / | /`-._. \___/ | \___/ | \_ ._.+' `-./ | _/ /`-./ / / \ | / \ | / / | /`-._.+' `-./ | _/ / \.-' `-./ \.-' `-./ /`-./ / | /`-._.+' `-./ `-. .-'\ /`-. .-'\ /`-. -./ | _/ /`-./ / | /` | \___/ | \___/ /`-._.+' `-./ | _/ /`-./ | / \ | / \ / / | /`-._.+' `-./ | _ .-' `-./ \.-' `-./ \.-' | _/ /`-./ / | /`-._.+' \ /`-. .-'\ /`-. .-'\ +' `-./ | _/ /`-./ / | \___/ | \___/ | \_ | /`-._.+' `-./ | _/ /`- / \ | / \ | / /`-./ / | /`-._.+' `-./ / \.-' `-./ \.-' `-./ | _/ `-./ / | /`-._ `-. .-'\ /`-. .-'\ /`-. _.+' | _/ `-./ / | \___/ | \___/ / | _.+' | _/ | \ | \ / `-. / | _.+' .-' \.-' \.-' | _/ `-. / | \ .-'\ .-'\ _.+' | _/ `-. \___ | \___ | \_ . / | _.+' | \ | \ | | _/ `-. / | _.+' \.-' \.-' .+' | _/ `-. / | .-'\ .-'\ | _.+' | _/ ` | \___ | \___ `-. / | _.+' | \ | \ | _/ `-. / | .-' \.-' \.-' _.+' | _/ `-. / \ .-'\ .-'\ / | _.+' | _/ \___ | \___ | \_ _/ `-. / | _.+' -Dan Hoey The Cairo tiling is also topologically equivalent to the following rectangular tiling in 2x1 rectangles: (C) +--+--+-----+--+--+-----+ | | | | | | | | | +-----+ | +-----+ | | | | | | | +--+--+--+--+--+--+--+--+ | | | | | | | +-----+ | +-----+ | | | | | | | | | +--+--+--+--+--+--+--+--+ | | | | | | | | | +-----+ | +-----+ | | | | | | | +--+--+--+--+--+--+--+--+ | | | | | | | +-----+ | +-----+ | | | | | | | | | +-----+--+--+-----+--+--+ -David Wilson This is known to bricklayers as the "basketweave" bond, more precisely as "sailor basketweave", a "sailor" brick being one whose largest face is outermost. There is also "soldier basketweave", in which each component square is formed by three bricks (on edge). -John H. Conway Aesthetics of the Cairo tiling: You also get the effect that degree-four vertices sort of look like they aren't really there. So it looks like two pieces of distorted chickenwire overlaid on each other: ___ ___ ___ . . / \ . / \ / \ .-' `-. .-' `-./ \.-' `-./ \ / \ .-' `-. .-' /`-. .-'\ /`-. \ / \ : : ___/ : \___/ : \___/ \ : : / \ : / \ : / \ / `-. .-' `-./ \.-' `-./ \.-' `-./ \ / `-. .-' /`-. .-'\ /`-. .-'\ /`-. \ / : / : \___/ : \___/ : \___/ : \ : / \ : / \ : / \ .-' `-. \.-' `-./ \.-' `-./ \.-' / \ .-' `-. .-'\ /`-. .-'\ /`-. .-'\ / \ : : \___/ : \___/ : \___/ \ : : / \ : / \ : / \ / `-. .-' `-./ \.-' `-./ \.-' `-./ \ / `-. .-' /`-. .-'\ /`-. .-'\ /`-. \ / : / : \___/ : \___/ : \___/ : \ : / \ : / \ : / \ .-' `-. \.-' `-./ \.-' `-./ \.-' / \ .-' `-. .-'\ /`-. .-'\ /`-. .-'\ / \ : : \___/ : \___/ : \___/ \ : : \ : / \ : / \ / `-. .-' `-. \.-' `-./ \.-' / \ / `-. .-' `-. .-'\ /`-. .-'\ / \ / ' ' \___/ ' \___/ \___/ -Dan Hoey These notes on Cairo tilings are from the mailing list math-fun from may 1997. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Problem 1: Torsten Sillke, June 1997 Show that the following two tilings of the plane are topologically equivalent. _________________________________ _________________________________ | | | | |_______| | | | |_______| | | | |_______| | | |_______| | | | | |_______| | | | |_______| | |___| | | | |___| | | | | | | | |_______| | | | |_______| | | | |___|___| | | |___|___| |_|_|_|_|_______|_|_|_|_|_______| |_|___|_|_______|_|___|_|_______| |_______| | | | |_______| | | | | |_______| | | |_______| | | | |_______| | | | |_______| | | | | | | | |___| | | | |___| | |_______| | | | |_______| | | | | |___|___| | | |___|___| | | | |_______|_|_|_|_|_______|_|_|_|_| |_______|_|___|_|_______|_|___|_| | | | | |_______| | | | |_______| | | | |_______| | | |_______| | | | | |_______| | | | |_______| | |___| | | | |___| | | | | | | | |_______| | | | |_______| | | | |___|___| | | |___|___| |_|_|_|_|_______|_|_|_|_|_______| |_|___|_|_______|_|___|_|_______| |_______| | | | |_______| | | | | |_______| | | |_______| | | | |_______| | | | |_______| | | | | | | | |___| | | | |___| | |_______| | | | |_______| | | | | |___|___| | | |___|___| | | | |_______|_|_|_|_|_______|_|_|_|_| |_______|_|___|_|_______|_|___|_| SPOILER: Matthew Cook segs[z_] := Block[{a={(1-z)/4,z/4},b={.5,z/4},c={(3+z)/4,z/4},d,e,f}, {d,e,f}={0,1}+{1,-1}#&/@{a,b,c}; {{{0,0},a,b,c,{1,0}},{{0,1},d,e,f,{1,1}},{a,d},{b,e},{c,f}}] Do[Show[Graphics[Table[ Line/@Map[If[EvenQ[x+y],#+{x,y}&,Reverse[#]+{x,y}&],segs[z],{2}], {x,5},{y,5}],AspectRatio->Automatic]],{z,0,1,.05}] -Matthew Cook - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - But, here's the real challenge: Problem 2: Collatz (Hamburg), 199? Show that the following two tilings of the plane are topologically equivalent. ________________________. . . . . ________________________. .___. . | | | | |_______________| | | | | | | | |_______________| | | | | | | | |_______________| | | | | | | | | | | | | | | | | | |_______________| | | | | | | | |_______|_______| | | | | | | | |_______________|_|_|_|_| | |___| |_______________|_|___|_| | | | | | | | | |_______________| | | | | | | |_______________| | | | | | | | | |_______________| | | | | | | | | | | | | | | | | | |_______________| | | | | | | |_______|_______| |_|_|_|_| | | | |_______________| |_|___|_| |___| |_______________| ________| | | | | | | | |________ ________| | | | | | |________ ________| | | | | | | | |________ | | | | | | | | | ________| | | | | | | | |________ |_______| | | | | | |_______| ________|_|_|_|_| | | | |________ ________|_|___|_| |___| |________ |_______________| | | | | | | | | |_______________| | | | | | | |_______________| | | | | | | | | | | | | | | | | | |_______________| | | | | | | | | |_______|_______| | | | | | | |_______________|_|_|_|_| | | | | |_______________|_|___|_| |___| | SPOILER: Matthew Cook Yes, you are right, this is much trickier! If you ever run into a machine that's running Mathematica, the following code should make a nice animation of this transformation. (Paste in the code, press shift-enter, and double click on one of the pictures to see the animation.) segs[z_] := Block[{a2=2+z(4z-2),dc2=4-2z,d2=4-z(6-4z),ac2=2+2z, a3=1+z(2z+1),d3=4-z(5-2z)}, {{{-dc2,-a2},{-d3,-a3},{-d3,a3},{-d2,ac2}}, {{d2,-ac2},{d3,-a3},{d3,a3},{dc2,a2},{6z,6-d2}}, {{-d3,-a3},{d3,-a3}},{{-d3,0},{d3,0}},{{-d3,a3},{d3,a3}}}] Do[Show[Graphics[Table[Line/@ Map[If[EvenQ[x+y],#,Reverse[#]]+{6x+(4z-2)y,(4z-2)x+6y}&,segs[z],{2}], {x,-5,5},{y,-5,5}], AspectRatio->Automatic,PlotRange->{{-20,20},{-20,20}}]],{z,0,1,.05}] -Matthew Cook - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Hint for problem 1: An intermediate position is _____ _____ _____ _____ / \/ \/ \/ \ /\.___./\.___./\.___./\.___./\ | |___| | | | |___| | | | | | |___| |_|_| |___| |_|_| | \/ \/ \/ \/ \/ /\.___./\.___./\.___./\.___./\ | | | | |___| | | | |___| | | |_|_| |___| |_|_| |___| | \/ \/ \/ \/ \/ /\.___./\.___./\.___./\.___./\ | |___| | | | |___| | | | | | |___| |_|_| |___| |_|_| | \/ \/ \/ \/ \/ /\.___./\.___./\.___./\.___./\ | | | | |___| | | | |___| | | |_|_| |___| |_|_| |___| | \/ \/ \/ \/ \/ \_____/\_____/\_____/\_____/ -- mailto:Torsten.Sillke@uni-bielefeld.de http://www.mathematik.uni-bielefeld.de/~sillke/