### Tessellations - Brittany Broughton

```TEKS
(7.8) Geometry and spatial reasoning. The student uses geometry to model and describe the
physical world.
The student is expected to:
(A) sketch three-dimensional figures when given the top, side, and front views;
(B) make a net (two-dimensional model) of the surface area of a three-dimensional figure;
and
(C) use geometric concepts and properties to solve problems in fields such as art and
architecture.
(7.13) Underlying processes and mathematical tools. The student applies Grade 7 mathematics
to solve problems connected to everyday experiences, investigations in other disciplines, and
activities in and outside of school.
The student is expected to:
(A) identify and apply mathematics to everyday experiences, to activities in and outside of
school, with other disciplines, and with other mathematical topics
TEKS
(8.6) Geometry and spatial reasoning. The student uses transformational geometry to develop spatial sense.
The student is expected to:
(A) generate similar figures using dilations including enlargements and reductions; and
(B) graph dilations, reflections, and translations on a coordinate plane.
(8.7) Geometry and spatial reasoning. The student uses geometry to model and describe the physical world.
The student is expected to:
(A) draw three-dimensional figures from different perspectives;
(B) use geometric concepts and properties to solve problems in fields such as art and architecture;
(8.14) Underlying processes and mathematical tools. The student applies Grade 8 mathematics to solve
problems connected to everyday experiences, investigations in other disciplines, and activities in and
outside of school.
The student is expected to:
(A) identify and apply mathematics to everyday experiences, to activities in and outside of school, with
other disciplines, and with other mathematical topics
NCTM Standards
Use visualization, spatial reasoning, and geometric modeling to solve problems
Draw geometric objects with specified properties, such as side lengths or angle measures;
use geometric models to represent and explain numerical and algebraic relationships;
Recognize and apply geometric ideas and relationships in areas outside the mathematics
classroom, such as art, science, and everyday life.
Instructional programs from prekindergarten through grade 12 should enable all students to—
create and use representations to organize, record, and communicate mathematical ideas;
select, apply, and translate among mathematical representations to solve problems;
use representations to model and interpret physical, social, and mathematical phenomena.
Instructional programs from prekindergarten through grade 12 should enable all students to—
recognize and use connections among mathematical ideas;
understand how mathematical ideas interconnect and build on one another to produce a
coherent whole; recognize and apply mathematics in contexts outside of mathematics.
Tessellations
• A tessellation or tilling of
the plane is a collection of
plane figures that fills the
plane with no overlaps and
no gaps.
• Tessellation comes from the
Latin word “tessera” which
means a small cube.
• These cubes made up
“tessellata” the mosaic
pictures forming floors and
tiling in Roman buildings.
History of Tessellations
• They can be traced all the way back to the Sumerian
civilization (about 4000 B.C.) in which the walls of homes and
temples were decorated by designs of tessellations
constructed from slabs of hardened clay.
• Not only did these tessellations provide decoration but they
also became part of the structure of the buildings.
• Since then, tessellations have been found in many of the
artistic elements of wide-ranging cultures including the
Egyptians, Moors, Romans, Persians, Greek, Byzatine, Arabic,
Japanese, and Chinese.
Examples of tessellations from different cultures
Chinese
Italian
Persian
Islamic
Egyptian
Korean
Tessellations
•The study of tessellations is concerned
with the use of multiple identical, none
overlapping copies of a certain figure to
cover the Euclidean plane (a flat surface
unbounded in all directions).
• The first known attempt of tessellations
was the tiling in the Alhambra in Spain. It
was laid out by the Moors in the 14th
century.
•They were made of colored tiles forming
patterns that were symmetrical,
geometrical, and beautiful.
• Some were not tessellations because they
didn't cover a surface with a repetitive
design without gaps or overlaps.
This is a picture of the
Alhambra tiling.
M.C. Escher
• A Dutch graphic artist
• Regarded as the 'Father' of modern
tessellations.
• Born in Holland on June17th, 1898 and was
enrolled in the “School for Architecture and
Decorative Arts” in Harlem where he studied
until 1922.
• Much of his work is based on the ancient on
the periodic designs of ancient Moorish
mosaics, Moors of Alhambra, Spain.
The Alhambra Palace - Granada, Spain
There are tessellated patterns on the lower portions of the walls. This
room, like in many Islamic buildings, is perfectly symmetrical (exhibiting
reflective symmetry) as its left and right sides are identical. This was the
inspiration of M.C. Escher’s work.
M.C. Escher
Escher produced '8 heads' in 1922 - a hint of things to come. You can immediately
see 4 different heads but the others are not apparent until the picture is turned
upside-down
8-heads
8-heads turned
Famous tessellations
Sky and Water
Relativity
Content that is useful to understanding tessellations
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Polygons: regular and irregular
Vertex
Angle
Degree
Interior angles
Symmetry
Transformation: translation, reflection, rotation, and
glide reflections
• Direction
• Magnitute
Regular & Irregular Tessellations
• A regular tessellation is a highly symmetric tessellation made
up of congruent regular polygons.
– There are only 3 regular tessellations: those made up of equilateral
triangles, squares, and hexagons.
• A semiregular tessellation uses more than one regular
polygon and has the same polygon arrangement at each
vertex.
– There are 8 of these.
• A demiregular tessellation uses more than one regular
polygon and has two or three different polygon
arrangements.
• An edge-to-edge tessellation is even less regular: the only
requirement is that adjacent tiles only share full sides (For
example, no tile shares a partial side with any other tile.)
What tessellates?
• Not all polygons will tessellate
– Qualifications for tessellations:
• Equal side lengths
• Equal internal angles
• The measure of an internal angle must be an exact divisor of 360.
Why is this?
• Based on this definition, the following polygons are the only regular
polygons that will tessellate:
– Triangle
– Square
– Hexagon
Non-regular polygons
• There are some polygons that don’t meet the previous conditions, thus we
call them non-regular polygons.
• The reason these tessellate is when you put these objects back to back
they form either a triangle, square, or hexagon which we know already
tessellates.
• Examples:
– Kites
– Diamonds
– Parallelograms
– Rectangles
Principles of Tessellations
• There are 4 ways of how a diagram can be
“mapped onto” itself
– Translation
– Rotation
– Reflection
– Glide reflection
Translation
• Translation is the moving of a pattern over a
certain distance, such that it coincides and
cover the underlying pattern again.
http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03Escher/translation.swf
Rotation
• Rotation similarly is the rotation of a pattern
at a fixed origin and fixed angle such that it
covers the underlying pattern also.
http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03Escher/rotation.swf
Reflection
• Reflection is the mapping of a pattern by
“mirroring” the image with respect to an axis
of reflection. The image is therefore a mirror
image of the original pattern.
http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/reflection.swf
Glide Reflection
• Glide reflection is basically a combination of
translation and reflection of the diagram
http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03Escher/glidereflection.swf
Creative Tessellations from Polygons
http://fc23.deviantart.com/fs5/f/2004/340/5/e/bunnytess.swf
Real World Examples
Activity
• Using the regular polygons, create your own
tessellation.
• You can create a regular, semiregular, or
demiregular tessellation.
Activity Examples
Integrating Tessellations
Works Cited
•
Chapin, S. H. Math Matters: Understanding the Math You Teach Grades K-6. California: Math Solutions Publications,
2000.
•
“Chapter 111. Texas Essential Knowledge and Skills for Mathematics Subchapter B. Middle School” Texas Education
Agency. 1 September 2006. Retrieved: 2 November 2008.
<http://www.tea.state.tx.us/rules/tac/chapter111/ch111b.html>.
•
Flournoy, V. The Patchwork Quilt. New York: Dial Books for Young Readers, 1985.
•
Hope, Martin. Integrating Mathematics Across the Curriculum. Arlington Heights, Ill: IRI/Skylight Training and
Publisher, 1996.
•
“Maurits Cornelius Escher.” Artinthepictures.com: An introduction to art history. 2008. Retrieved 4 November 2008.
<http://www.artinthepicture.com/artists/MC_Escher/biography.html>.
•
“Overview: Standards for Grades 6–8.” Principles and Standards for School Mathematics. NCTM: National Council of
Teachers of Mathematics. 2004. Retrieved: 2 November 2008.
<http://standards.nctm.org/document/chapter6/index.htm>.
•
“Principles of Tessellations.” The Mathematics behind the art of M.C. Escher. Retrieved 31 October 2008.
<http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/main2.html#Principals>.
Works Cited
•
Taschen. Escher, M.C. London: Taschen, 2006.
•
Tessellations: Escher and how to make your own. October 2003. Annal, David and Bareiss,
Seth. Retrieved: 2 November 2008. <http://tessellations.org/index.htm>.
•
Totally Tessellated. 1998. Bhushan, A., Kay, K., & Williams, E. Retrieved: 2 November 2008.
<http://library.thinkquest.org/16661/history.html>.
•
“What is a Tessellation?” The Math Forum. Drexel University. 2002. Retrieved 2 November
2008. <http://mathforum.org/sum95/suzanne/whattess.html>.
•
Ziring, Neal. “M.C. Escher Brief Biography.” Escher Pages. 27 January 2003. Retrieved 4
November 2008. <http://users.erols.com/ziring/escher_bio.htm>.
Pictures Cited
Slide 1 – Background:
Artist: Carlos Gershenson <http://hawmkoonstormbringer.deviantart.com/art/Fractal-Tessellation-89159878>
Slide 5 – Roman Catholic Bull tiling
http://tessellations.org/tess-what.htm
Slide 7 – Various Culture Tessellations
http://library.thinkquest.org/16661/gallery/index.html
Slide 8 – Alhambra Tiling
Taschen. Escher, M.C. London: Taschen, 2006. Page 49.
Slide 10 – Alhambra Palace
http://library.thinkquest.org/16661/gallery/26.html
Slide 11 – Escher’s “8 Heads”
http://www.tessellations.org/tess-escher4.htm
Slide 12 – Escher Drawings
Escher’s “Drawing Hands.” Taschen. Escher, M.C. London: Taschen, 2006. Page 174.
Escher’s “Relativity.” Taschen. Escher, M.C. London: Taschen, 2006. Page 168.
Escher’s “Sky and Water.” Taschen. Escher, M.C. London: Taschen, 2006. Page 56.
Slide 14 – Types of Tessellations
<http://library.thinkquest.org/16661/of.regular.polygons/index.html?tqskip1=1&tqtime=0709>.
Slide 15 – Regular Tessellations
<http://mathforum.org/sum95/suzanne/whattess.html>.
Slide 16 – Non-Regular Tessellations
<http://mathforum.org/sum95/suzanne/whattess.html>.
Pictures Cited
Slide 18 – Translation
Translation Flash Movie – <http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/main2.html#Principals>.
Escher’s “Flying Horses.” - Taschen. Escher, M.C. London: Taschen, 2006. Page 113.
Slide 19 – Rotation
Rotation Flash Movie – <http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/main2.html#Principals>.
Escher’s “Humans.” – <http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/main3.html>.
Slide 20 – Reflection
Reflection Flash Movie – <http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/main2.html#Principals>.
Slide 21 - Glide Reflection
Glide Reflection Flash Movie – <http://www.math.nus.edu.sg/aslaksen/gem-projects/maa/0203-2-03-Escher/main2.html#Principals>.
Escher’s “Horseman.” - - Taschen. Escher, M.C. London: Taschen, 2006. Page 103.
Slide 22 – Rabbit Tessellation
Rabbit Tessellation Flash Movie - <http://bardaux.deviantart.com/art/Rabbit-Tesselation-12930458>.
Slide 23 – Real World Examples
Honeycomb - <http://www.prsd.k12.pa.us/hs/tessellations%20revised.ppt>.
Fish Scales - <theartofnature.org/id20.html>.
Chain Link Fence - <houstonchainlink.com/>.
Soccer ball - <www.barrowga.org/rec/soccer/u6.html>.
Slide 25 – Activity Examples
SemiRegular Pictures – <http://commons.wikimedia.org/wiki/Special:Search?search=Tiling+Semiregular&go=Go>.
DemiRegular Pictures - <http://www.prsd.k12.pa.us/hs/tessellations%20revised.ppt>.
Slide 26 – Integrating Tessellations
Venn Diagram - Hope, Martin. Integrating Mathematics Across the Curriculum. Arlington Heights, Ill: IRI/Skylight Training and
Publisher, 1996. Chapter 5.
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