Tessellation Projects
Introduction
You are going on a
cyberspace adventure to discover what a tessellation is and where you
can find tessellations in art, in nature, and in the rest of the world
around us.
You
will also have the opportunity to create your own unique tessellations!

People
have always been
interested in patterns, both planar patterns
and spatial patterns. Classification of patterns started two and a half
millennia ago with the Pythagorean discovery that there are five
regular solids: the tetrahedron, the cube, the octahedron, the
dodecahedron, and the icosahedron.
Archimedes generalized these to some
nearly regular solids, now called Archimedean solids, such as the solid
made out of pentagons and hexagons that is used for soccer balls.
Kepler found other nearly regular solids and noted the regular
tessellations (tilings) of the plane.
Tessellations (or
tilings, as they are sometimes called) have a
broad appeal, maybe because they exemplify how mathematics
can unify the aesthetic, natural and rational worlds. From the art of
M.C. Escher to crystal growth to the mathematics of Penrose tilings,
tessellations fascinate everyone, from mathematician to artist to
interior decorator to mathematics student.
Task
You will work on your own for 3 of the 4 parts of
this WebQuest. You may choose to do the presentation part alone, with
another student,
or in a group of up to three individuals.

This WebQuest involves the completion of 4 different assignments.
You will complete 3 of the 4 Tessellation Projects as an individual.
You are to develop a PowerPoint about Tessellations with your group or
as an individual.
Your assignments will lead you to:
 Discover and explain what a
tessellation is.
 Compare and contrast
different types of tessellations.
 Find out how symmetry
relates to tessellations.
 Create a unique
tessellation using computer software.
 Locate, photograph, and
describe examples of tessellations in the real world (three different
tessellations per person).
 Uncover the history of
tessellations.
 Create your own unique
tessellations.
 Report your discoveries and
present your information to the class via PowerPoint with your group.
Process
 Use online or printed sources (books) to help you define
and describe what a tessellation is using your own words.
 Compare and contrast different
types of tessellations. You also need to discuss the various
mathematical methods that can be used to create them. (Use online or
printed sources to assist you.)
 Find out how symmetry relates to tessellations
and explain how the different types of symmetry are used to create
tessellations.
 Use The
Geometer's Sketchpad^{(R)} or some other software or
online program to create a unique tessellation (at least one per
person). You
may utilize the
classroom computer or a computer in the school computer lab.
 Find three or more examples (per person) of
tessellations in the world around you. Look for unique examples.
Photograph the tessellations and include them in your presentation
along with a description.
 In your own words present what you have found on
the history of tessellations. Relate what you have found out about some
of the individuals famous for their work with tessellations?
 Follow the directions outlined in the Tessellation Projects and complete this part
of the project
as an individual.
 Combine the knowledge you have acquired, the
products you have created, and the pictures you have taken or found
into a
PowerPoint presentation. Present your project to the class.
Resources:
The following links are provided to help you complete
your assignments.
Feel free to use printed media in the library if you care to do so.

Tessellation
Links
What is a
Tessellation? 

What is
Tiling?


What will
tessellate?


How do I
create my own tessellations?

Online
Software
The Geometer's
Sketchpad
Other Sources

Where can
I find information about the history of tessellations?


Where can I
go to find out more about
people famous for their work with tilings and tessellations? 

Where can
I find examples of real world tessellations and patterns? 

What is
symmetry? 

Where can
I view examples of tessellations that other people, artists and
students, have created? 

Where can I
go to find out how to use PowerPoint? 

Where can I
find some miscellaneous websites on tessellations?


More
Tessellation
Links
Grotesque
Geometry: Andrew Compton's Tessellations: http://www.cromp.com/tess/home.html
Tessellating
Alphabet: Each letter of this tessellating alphabet fits
together with copies
of itself to tile the plane.
Click a letter to see how it tiles. Some letters
tile in more than one way.
http://www.scottkim.com/inversions/gallery/tessellatingalphabet.html
History of
Crystallographic Groups and Related Topics by David E. Joyce Department of
Mathematics and Computer
Science
Clark University Worcester, Massachusetts: http://www.clarku.edu/~djoyce/wallpaper/history.html
Computer Art inspired
by M.C. Escher and Victor Vasarely by Hans Kuiper with
many wallpaper patterns:
http://web.inter.nl.net/hcc/Hans.Kuiper/
Tilings and
Tessellations by U Science at the Geometry Center: http://www.scienceu.com/geometry/articles/tiling/index.html
 Kali, by Jeff
Weeks, Geometry Center, Minneapolis, MN, 1995
(get free from http://www.geometrygames.org/Kali/index.html).
 Kali is interactive program for the Macintosh or Windows that
lets the user draw pictures
under the action of wallpaper,
frieze or rosette groups. As the user freely
draws line segments with the mouse, the program draws several copies
simultaneously, under the action of the selected symmetry group.
Curved segments can be created with a "smoothing" option.
A Java version of Kali by
Mark Phillips is
also available from the Geometry Center. This program will run on any
computer with a Java capable web
browser, such as Netscape 2.0 and
higher or Internet Explorer 3.0.
Evaluation
Grades are an evaluation of learning.
Therefore, everyone who displays a certain level of competency will
receive a matching grade.

 If you have chosen to work with a partner or in
a group of three, combine your information and collaborate for your
PowerPoint presentation. Each group member will receive the same
PowerPoint presentation grade.
 You will be provided with the rubrics that will
be used to score your work.
 Everyone will receive individual grades on their
Tessellation Projects.
 Your scores for the PowerPoint and Tessellation
Projects will be averaged together and the averaged score will count as
a two test grades for each student.
Conclusion
At
the beginning of this webquest, you were told that people have always
been interested in patterns. Maybe it has something to do with the
natural patterns that are all around us. Have you begun to find yourself
constantly noticing where
patterns are evident in nature and the world surrounding you? Because of its beauty and
applicability in ancient as well as modern
times, the art of tiling and tessellations has interested mankind for
centuries. Roman
mosaics and Moslem religious buildings are among some of the oldest
examples of tiling. Famous and more modem examples of tiling in art
form are seen in the works of the M. C. Escher. The author, Senechal
(1990) said that the study of tilings is important in mathematics
education
because the study of shape "draws on and contributes to not only
mathematics but also the sciences and the arts."
You deserve a huge pat on the
back for all of the work that you have done to complete this project.
You gained background information and from there you developed
expertise in the area of tessellations. At times, you may have felt
confused with thoughts and ideas spinning every which way in your
brain. That's perfectly normal when you're working on building new
mental connections.
By the time you completed Our Tessellation Webquest
you learned what
tessellations are and you began to notice that tessellations truly are
found all around us in the real world. By now you have discovered
how tessellations can be created and you have created your own,
following in the legacy of the famous artist, Escher. As
you showcase your final product, share your thoughts on the process of
working on this project with the class. Was this assignment
difficult or easy for you? Do you think you will ever use
tessellations in areas
other than math? What discoveries have you made by
completing this assignment (self, abilities/talent, math, etc.)? You now
have firsthand experience in developing and using tessellations.
Think about how someday you might apply this knowledge and experience
to
other
situations? Science? Art? Interior design?
Landscaping? Architecture?
You might be interested in applying your new found talents right away.
Each year, the World of Escher Tessellation Contest is held. Students
compete with other students from all over the world. Take a look at
previous winners on the web site listed below. If you decide to
pit your artistic abilities against that of other students, note that
entries are due by the 31st of December.
Standards Addressed
NCTM
Standards
According to the NCTM
Standards geometry "provides an opportunity for students to experience
the creative interplay between mathematics and art."
 #2 Mathematics as Communication (model situations
using oral, written, concrete and pictorial methods)
 #8 Patterns and Functions
 #12 Geometry (explore transformations of
geometric figures)
South Carolina 7th Grade Mathematics Process
Standards
71.1 Generate and solve complex abstract
problems that involve modeling physical, social, or mathematical phenomena.
71.2 Evaluate conjectures and pose followup
questions to prove or disprove conjectures.
71.3 Use inductive and deductive reasoning to
formulate mathematical arguments.
71.4 Understand equivalent symbolic expressions
as distinct symbolic forms that represent the same relationship.
71.5 Generalize mathematical statements based
on inductive and deductive reasoning.
71.6 Use correct and clearly written or spoken
words, variables, and notation to communicate about significant mathematical
tasks.
71.7 Generalize connections among a variety of
representational forms and realworld situations.
71.8 Use standard and nonstandard
representations to convey and support mathematical relationships.
South
Carolina 7th Grade
Mathematics Standards
Geometry
Standard 74: The student will demonstrate through the mathematical
processes an understanding of proportional reasoning, tessellations, the use of
geometric properties to make deductive arguments. the results of the
intersection of geometric shapes in a plane, and the relationships among angles
formed when a transversal intersects two parallel lines.
Indicators
74.1 Analyze geometric properties and the
relationships among the properties of triangles, congruence, similarity, and transformations
to make deductive arguments.
74.2 Explain the results of the intersection
of two or more geometric shapes in a plane.
74.3 Illustrate the cross section of a solid.
74.4 Translate between two and
threedimensional representations of compound figures.
74.5 Analyze the congruent and supplementary
relationships—specifically, alternate interior, alternate exterior,
corresponding, and adjacent—of the angles formed by parallel lines and a
transversal.
74.6 Compare the areas of similar shapes and
the areas of congruent shapes.
74.7 Explain the proportional relationship
among attributes of similar shapes.
74.8 Apply proportional reasoning to find
missing attributes of similar shapes.
74.9 Create tessellations with
transformations.
74.10 Explain the relationship of the angle
measurements among shapes that tessellate.
Gifted
and Talented Goals
 The content, process, and
product standards included in this WebQuest exceed the state adopted
standards for students at the 7^{th} grade level.
 This unit requires students to
demonstrate depth and complexity of knowledge and skills.
 The instructional strategies
used in this unit accommodate the unique needs of gifted learners.
 This WebQuest incorporates
acceleration and enrichment for the gifted student.
 Opportunities exist for
worldwide communication and research via the WWW.