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## MAKE IT, TAKE IT, FRAME IT

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**MAKE IT, TAKE IT, FRAME IT**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.**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?**Heptagons?**No, see the overlaps.**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.**Irregular Tessellation**Irregular tessellations are formed from irregular polygons.**Irregular Tessellations areall around us**Look for tessellations in walls, patios and pavements.**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.”**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**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.**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/>.