### 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
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
Tessellations
and fix them to the opposite sides like this.
Tessellations
from three of the sides and fix them to the opposite
sides like this.
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.
 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/>.
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