TESSELLATIONS AND ESCHER Martha Mitchell Western Illinois University What are tessellations? Basically, a tessellation is a way to tile a floor (that goes on forever) with shapes so that there is no overlapping and no gaps. Tessellations are all around us REGULAR TESSELLATIONS: RULE #1: The tessellation must tile a floor (that goes on forever) with no overlapping or gaps. RULE #2: The tiles must be regular polygons - and all the same. RULE #3: Each vertex must look the same. Squares? Triangles? Hexagons? Will pentagons work? Heptagons? No, see the overlaps. Will octagons tessellate? Nope! They'll overlap too. In fact, all polygons with more than six sides will overlap! In fact, all polygons with more than six sides will overlap! So, the only regular polygons that tessellate are triangles, squares and hexagons! WHY? A regular polygon will tessellate the plane if and only if the measure of its interior angle in degrees divides by 360 ˚ exactly. The 360 ˚ is just not there. Irregular tessellations are formed from irregular polygons. Irregular Tessellations are all around us Look for tessellations in walls, patios and pavements. More irregular tessellating polygons. Tessellations by M.C. Escher Maurits Cornelis Escher was an influential Dutch graphic artist. He is known for his work with tessellations. Interesting because despite lacking mathematical training, his work displays a strong mathematical component. Escher’s understanding of math was largely intuitive & visual. (Self-portrait to the left). •Largely famous for his seemingly impossible art depictions. •Often referred to as the “Father” of modern tessellations. •It is said that he became obsessed with producing art with pictures that did not overlap or leave spaces. •He’s been quoted saying: "Filling the plane has become a real mania to which I have become addicted and from which I sometimes find it hard to tear myself away.” Tessellations by M.C. Escher China Boy, 1936 Tessellation by M. C. Escher Squirrels, 1936 Tessellation by M. C. Escher Fish, 1938 Tessellation by M. C. Escher Horsemen, 1946 Tessellation by M. C. Escher 4 Motifs, 1950 Tessellation by M. C. Escher Scarabs, 1953 Tessellation by M. C. Escher Pegasus, 1959 Tessellation by M. C. Escher Fishes, 1958 Mural Birds, 1967 Tessellation by M. C. Escher Escher’s Last Tessellation His last tessellation was a solution to a puzzle sent to him by Roger Penrose, the mathematician. Escher solved it and, true to form, changed the angular wood blocks into rounded 'ghosts'. Penrose 'Ghosts' - 1971 Make Your Own Escher Tessellations Start with a square and cut segments from two sides and fix them to the opposite sides like this. Make Your Own Escher Tessellations You could start with a hexagon and cut semi-circles from three of the sides and fix them to the opposite sides like this. Step 1: Start with a simple shape that will tessellate e.g. a rectangle. Step 2: Remove a shape or shapes from one side of the rectangle and fix them to the opposite side. A Full Life Escher died on March 27, 1972. He had produced 448 woodcuts, linocuts and lithos and over 2,000 drawings. Self-Portraits Works Cited Britton, Jill. Symmetry and Tessellations. White Plains, NY: Dale Seymour Publications, 2000. Print. "Coolmath4kids.com - Tessellations." Cool Math 4 Kids - Math Games, Math Puzzles, Math Lessons - Designed for Kids and Fun! Coolmath.com,Inc. Web. 22 Mar. 2011. <http://www.coolmath4kids.com/tesspag1.html>. "How To Create Tessellations." Barcodes Inc - Barcode Printer, Barcode Scanner, Point of Sale, Mobile Computing and RFID Experts. Barcodes Inc. Web. 22 Mar. 2011. <http://www.barcodesinc.com/articles/create-tessellations.htm>. Kawas, Terry. "Mathwire.com | Tessellations." Mathwire.com | March 2011. Mathwire.com. Web. 22 Mar. 2011. <http://mathwire.com/geometry/tess.html>. "M.C. Escher The Official Website." The Official M.C. Escher Website. The M.C. Escher Company, B.V. Web. 22 Mar. 2011. <http://www.mcescher.com/>. McClung, Peggy J. "Tessellations." TEC Lesson Plan. Web. 22 Mar. 2011. <http://www.schools.pinellas.k12.fl.us/educators/tec/McClung/index.html>. Lesson plan "PowerPoint Slide Show." Numeracy Software to Support Maths Teaching in Schools. Web. 22 Mar. 2011. <http://www.numeracysoftware.com/freePowerPoint.html>. "Regular Tessellations « Mathematics and Multimedia." Mathematics and Multimedia. Mathematics and Multimedia, 2009. Web. 22 Mar. 2011. <http://mathandmultimedia.com/tag/regular-tessellations/>.