# [GIF] Anything Goes (Quadrilateral tessellations)

Posted 3 years ago
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| Anything GoesIt's an obvious but kind of cool fact that every quadrilateral tiles the plane. Building on some ideas from Cut the Knot, I decided to write a function which would, given the edges of a quadrilateral (as vectors), produce a tessellation. Here's what I ended up with: BasicBlock[edges_] := Module[{n}, n = Length[edges]; Table[Accumulate[(-1)^i RotateRight[edges, i]], {i, 0, n - 1}] ]; QuadTilingBasis[edges_] := {edges[] - edges[], edges[] - edges[]}; EdgesToTessellation[edges_, numx_, numy_] := Module[{basis}, basis = QuadTilingBasis[edges]; Table[Polygon /@ (x*basis[] + y*basis[] + # & /@ # & /@ BasicBlock[edges]), {x, -(numx - 1)/2, (numx - 1)/ 2}, {y, -(numy - 1)/2, (numy - 1)/2}] ]; Given that, I used some general machinery that I've discussed before (1, 2, 3) for generating random polygons and finding geodesics in polygon space to generate various 1-parameter families of quadrilaterals (and, hence, tessellations). Here are the necessary bits of that code: ToReal[z_] := {Re[z], Im[z]}; ToComplex[{x_, y_}] := x + I y; FrameToEdges[frame_] := ToReal[ToComplex[#]^2] & /@ Transpose[frame]; QuadFrameToTessellation[frame_, numx_, numy_] := EdgesToTessellation[FrameToEdges[frame], numx, numy]; ProjectionBasis[{A_, B_}, {C_, D_}] := Normalize[#] & /@ Eigenvectors[ Transpose[Transpose[{A, B}].{A, B}.Transpose[{C, D}].{C, D}], 2]; PlaneGeo[{A_, B_}, {C_, D_}, t_] := Module[{a, b, c, d, cPerp, dPerp, dist1, dist2}, {a, b} = ProjectionBasis[{C, D}, {A, B}]; {c, d} = ProjectionBasis[{A, B}, {C, D}]; {cPerp, dPerp} = {Normalize[c - (c.a)*a], Normalize[d - (d.b)*b]}; dist1 = ArcCos[a.c]; dist2 = ArcCos[b.d]; {Cos[t*dist1]*a + Sin[t*dist1]*cPerp, Cos[t*dist2]*b + Sin[t*dist2]*dPerp} ]; Eventually, I found two elements of the Stiefel manifold that produced an interesting family of tessellations: frame1 = {{-0.21728090898823724, -0.08834582559466465, 0.7450486536045761, 0.6244089408802944}, {-0.7224486073877968, 0.44620730863225605, -0.4229311116667767, 0.3163800281163406}}; frame2 = {{0.42870301121644105, 0.38283874197110207, -0.633804812205533, -0.5176289074665839}, {-0.6547467745123633, 0.7145404849942669, 0.14903954331161623, -0.19627982855932777}}; And then, finally, here's the code that produces the above animation: Module[{cols}, cols = RGBColor /@ {"#00C3FF", "#F46188", "#3B475E"}; Animate[Graphics[{FaceForm[None], EdgeForm[ Directive[JoinForm["Round"], Thickness[.005], Blend[cols[[;; 2]], Haversine[t]]]], QuadFrameToTessellation[ PlaneGeo[frame1, frame2, .96 Haversine[t]], 10, 8]}, PlotRange -> 2, ImageSize -> 540, Background -> Last[cols]], {t, 0, π}, AnimationDirection -> ForwardBackward] ] Answer - another post of yours has been selected for the Staff Picks group, congratulations !We are happy to see you at the tops of the "Featured Contributor" board. Thank you for your wonderful contributions, and please keep them coming! Answer