Worksheet 29: Friday November 20 Tessellations: Tiling
Worksheet 29: Friday November 20Tessellations: Tiling The PlaneDefinitionA tiling of the plane or tesselation is a pattern that covers the plane with non-overlappingfigures A periodic tiling is one in which there exists at least two translations in non-paralleldirections in which the tiling is mapped onto itself. A tiling which is not periodic can benonperiodic or aperiodic. A monohedral tiling is one in which the pattern is formed from asingle identical shape that is repeated.QUESTION: Which regular polygons can tile the plane?ANSWER: the interior angles of the figure must evenly divide 360 degrees.Monohedral TilingsThere are only three edge-to-edger regular monohedral tilings of the plane: square, triangle andhexagon.Any Parallelogram Can Tile The Plane!Any parallelogram can tile the plane. You can lay the parallelograms side to side a form a strip.Then you lay the strips one on top of the other one to cover the plane.
Math 105 Fall 2015Worksheet 29Math As A Liberal ArtAny Triangle Can Tile The Plane!Any triangle (whether regular or not) can tile the plane. This is true because any triangle can beformed into a parallelogram by putting two copies of the triangle together. Note that there is alack of reflection symmetry. We require half-turns (i.e. 180-degree rotations) of the figure tocomplete this tiling.Any Quadrilateral Can Tile The Plane!Any quadrilateral (not just squares and parallelograms) tiles the plane. This is true because youcan create a hexagon with opposite sides parallel and congruent, and these can tile the plane.Again, we require half-turns of the figure (i.e. 180-degree rotations) to complete this tiling.There Exist Irregular Polygons That can Tile The Plane!German mathematician Karl Reinhardt (1895-1941) showed in his Ph.D. thesis that there arethree types of irregular hexagons that can tile the plane.It used to be believed that there are 14 known types of irregular pentagons that can tile the plane.Four were found by Marjorie Rice, a San Diego housewife with no formal mathematical trainingbeyond high school. But no one has proven whether this is all of them or not. Reinhardt found 5(thought this was it, but couldn’t prove it). No new ones have been found since 1985. In August2015 a professor announced that she and her undergraduate students had discovered a 15thirregular pentagon tiling! (It’s the one in the bottom right.)No Convex Polygon With More Than Six Sides Can Tile The PlaneReinhardt proved in 1927 that no convex polygon with more than six sides can tile the plane.2
Math 105 Fall 2015Worksheet 29Math As A Liberal ArtSemi-Regular TilingsA semi-regular tiling is a tiling that uses two or more regular shapes and have the exact samebehavior at each vertex. There are exactly eight “semi-regular” tilings.Can you identify which pairs of regular tilings are used in each case?Note that all of the tilings of the plane we have looked at so far are periodic (i.e. they possesstranslational symmetries in two non-parallel directions).3
Math 105 Fall 2015Worksheet 29Math As A Liberal ArtNot All Tilings Are Periodic!Roger Penrose (1931-) is a British mathematician who discovered a kind of aperiodic tiling (i.e.a tiling of the plane which only produces nonperiodic tiling consisting of only two “prototiles.”These kinds of tilings (seen below) are now known as Penrose tilings.These are the “prototiles” of Penrosetilings known as the “thin rhomb” and“thick rhomb,” respectively. They haveinternal angles of 36 and 144 degrees forthin rhomb and 72 and 108 degrees forthick rhomb.Replicating M.C. EscherNow that we know about regular and irregular monohedral tilings, let’s see how we can make“art” out of them as is done by M.C. Escher and others as shown above.Periodic Tiling AlgorithmBegin with a tiling with a parallelogram.Modify the parallelogram tiles but maintain it as a monohedral tiling by ensuring that translationof the tiles in both a “vertical” and a “horizontal” direction hold.So we simply need to:1. Modify a side and translate it to the opposite side.2. Modify one of the other sides and translate it to the opposite side.4
Math 105 Fall 2015Worksheet 29Math As A Liberal ArtMake Your Own Period Tiling!For example, we start with a parallelogram and then whatever change we make to one side wetranslate that change to the side paralle to it. And then we translate our new shape in the twodirections that the parallelogram is symmetric in.First Make Your Own Irregular TileOnce you have made your desired alteration to the parallelogram, translate your new shape inthe two directions that the parallelogram is symmetric an infinite amount of times to create a newperiodic tiling!5
Math 105 Fall 2015Worksheet 29Math As A Liberal ArtThe Voderberg Tiling: A monohedral non-periodic tiling of the plane by a 9-sided irregularpolygon (nonagon).6
Worksheet 29: Friday November 20 Tessellations: Tiling The Plane A tiling of the plane or tesselation is a pattern that covers the plane with non-overlapping figures A periodic tiling is one in which there exists at least two translations in non-parallel directions in which the tiling is
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