An aperiodic monotile, sometimes called an "einstein", is a shape that tiles the plane, but never periodically.[br][br]Einstein [br]P[sub]0[/sub], P[sub]1[/sub], ... P[sub]13[/sub] - P[sub]0[/sub] move point, P[sub]1[/sub] rotation point[br][br]SuperTiles[br]SuperTile[sub]b[/sub] = poly[sub]A'[/sub], poly[sub]B'[/sub], poly[sub]C'[/sub] poly[sub]D'[/sub] [br]Connect monotile to SuperTiles[br]A'[sub]0[/sub] connect B'[sub]0[/sub], (A[math]\mapsto[/math]B'[sub]0[/sub]) , A'[sub]0[/sub] connect C'[sub]0[/sub], A'[sub]0[/sub] connect D'[sub]0[/sub] [br][br]Grid: Isometric, n=13, L1...L13 Master Kachel[br]L[sub]i[/sub] : Flip at Segment-Axis, M': Connect C_{0}, base M'_{0} - Move C_{0} by M'_{0}[br][br][size=150][url=https://www.geogebra.org/m/ffvqpxws]Working with Isometric grid[math]\nearrow[/math][/url][/size][br][br]I - Isometric Grid, X EinStein[br]I=Surface(O + u (cos(30°), sin(30°)) + v (cos(-30°), sin(-30°)), u, -10, 10, v, -10, 10)[br]X={O, I(1,-1), I(1,-1.5), I(2,-2), I(2.5,-3), I(2,-3), I(1.5,-2.5), I(1,-3), I(0,-2.5), I(0,-2), I(-1/2,-1.5), I(0,-1), I(-1/2, 0)}[br][br]J - Isometric Grid rotate α[br]α=0..360°[br]ROT={{cos(α), sin(α)}, {-sin(α), cos(α)}}[br]J=Surface((ROT (O + u (cos(30°), sin(30°)) + v (cos(-30°), sin(-30°)) - O)) + O, u, -10, 10, v, -10, 10)[br]Y={O, J(1,-1), J(1,-1.5), J(2,-2), J(2.5,-3), J(2,-3), J(1.5,-2.5), J(1,-3), J(0,-2.5), J(0,-2), J(-1/2,-1.5), J(0,-1), J(-1/2, 0)}[br][br]Isometric Grid XFig - Ein-Stein[br]XFig= {(0, 0), (1,-1), (1,-1.5), (2,-2), (2.5,-3), (2,-3), (1.5,-2.5), (1,-3), (0,-2.5), (0,-2), (-0.5,-1.5), (0,-1), (-0.5, 0)}[br]A anchor point: Fig_A=Polygon(A + Zip(I(pp), pp, XFig)) [br]B anchor point: Fig_B=Polygon(B + Rotate(Zip(I(pp), pp, XFig), 30°))[br][br][br][size=150]Gruppierung von Kacheln in Clustern[/size] [br][br]Zusätzlich zu Vier-Kachel-Clustern, die aus einem Dreieck [color=#ff7700]EinStein [/color]und seiner [color=#ff0000]Drei-EinStein-Hülle[/color], ein Sechseck, bestehen, identifizieren wir[color=#0000ff] parallelogrammförmige Cluster, die aus Kachelpaaren[/color] bestehen. [br]Die Parallelogramme gibt es in zwei Varianten: [br]Die eine trennt zwei benachbarte Cluster und die andere verbindet sich mit zwei gedrehten Kopien zu einer [color=#9900ff]dreiarmigen Propellerform[/color], die Fylfot genannt wird. [br][br][i]A grouping of tiles into clusters in the example patch. In addition to four-tile clusters consisting of a reflected hat and its three-hat shell, we identify clusters consisting of a single tile, and parallelogram-shaped clusters consisting of pairs of tiles. The parallelograms come in two varieties: one separates two nearby shells, and the other joins up with two rotated copies to make a three-armed propeller shape called a fylfot. [/i] [br][br][url=https://www.derstandard.de/story/2000144902031/hobbyforscher-entdeckt-ersten-einstein-der-aperiodische-muster-liefert]Die Kachel zur Unendlichkeit[math]\nearrow[/math][/url][br][br][url=https://www.ingenieur.de/technik/fachbereiche/rekorde/raetsel-geloest-eine-fliese-unendlich-viele-muster/]Rätsel Einstein-Kachel gelöst[math]\nearrow[/math][/url][br][br][url=https://cs.uwaterloo.ca/~csk/hat/]An aperiodic monotile_{David Smith}[math]\nearrow[/math][/url][br][br][url=https://www.chiark.greenend.org.uk/~sgtatham/quasiblog/aperiodic-tilings/]Two algorithms for randomly generating aperiodic tilings[math]\nearrow[/math][/url]