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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. Tessellations are all around us. - PowerPoint PPT PresentationTRANSCRIPT

MAKE IT, TAKE IT, FRAME IT

MAKE IT, TAKE IT, FRAME ITTESSELLATIONS AND ESCHER

Martha MitchellWestern 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?

NOPE!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!

11WHY? 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 TessellationIrregular tessellations are formed from irregular polygons.Irregular Tessellations are all around usLook 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.Eschers understanding of math was largely intuitive & visual.(Self-portrait to the left).17Largely 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.

Hes 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. EscherSquirrels, 1936

Tessellation by M. C. EscherFish,1938

Tessellation by M. C. EscherHorsemen, 1946

Tessellation by M. C. Escher4 Motifs, 1950

Tessellation by M. C. EscherScarabs, 1953

Tessellation by M. C. EscherPegasus, 1959

Tessellation by M. C. EscherFishes, 1958 Mural

Birds, 1967

Tessellation by M. C. EscherEschers Last TessellationHis 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 LifeEscher died on March 27, 1972. He had produced 448 woodcuts, linocuts and lithos and over 2,000 drawings.

Self-Portraits

Works CitedBritton, 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. . "How To Create Tessellations." Barcodes Inc - Barcode Printer, Barcode Scanner, Point of Sale, Mobile Computing and RFID Experts. Barcodes Inc. Web. 22 Mar. 2011. . Kawas, Terry. "Mathwire.com | Tessellations." Mathwire.com | March 2011. Mathwire.com. Web. 22 Mar. 2011. . "M.C. Escher The Official Website." The Official M.C. Escher Website. The M.C. Escher Company, B.V. Web. 22 Mar. 2011. . McClung, Peggy J. "Tessellations." TEC Lesson Plan. Web. 22 Mar. 2011. . Lesson plan "PowerPoint Slide Show." Numeracy Software to Support Maths Teaching in Schools. Web. 22 Mar. 2011. . "Regular Tessellations Mathematics and Multimedia." Mathematics and Multimedia. Mathematics and Multimedia, 2009. Web. 22 Mar. 2011. .