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.

 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.

Process
1. Use online or printed sources (books) to help you define and describe what a tessellation is using your own words.
2. 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.)
3. Find out how symmetry relates to tessellations and explain how the different types of symmetry are used to create tessellations.
4. 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.
5. 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.
6. 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?
7. Follow the directions outlined in the Tessellation Projects and complete this part of the project as an individual.
8. 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.

 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?

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.
• 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 first-hand 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.

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

7-1.1      Generate and solve complex abstract problems that involve modeling physical, social, or mathematical phenomena.

7-1.2      Evaluate conjectures and pose follow-up questions to prove or disprove conjectures.

7-1.3      Use inductive and deductive reasoning to formulate mathematical arguments.

7-1.4      Understand equivalent symbolic expressions as distinct symbolic forms that represent the same relationship.

7-1.5      Generalize mathematical statements based on inductive and deductive reasoning.

7-1.6      Use correct and clearly written or spoken words, variables, and notation to communicate about significant         mathematical tasks.

7-1.7      Generalize connections among a variety of representational forms and real-world situations.

7-1.8      Use standard and nonstandard representations to convey and support mathematical relationships.

South Carolina 7th Grade Mathematics Standards

Geometry

Standard 7-4:     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

7-4.1       Analyze geometric properties and the relationships among the properties of triangles, congruence, similarity, and transformations to make deductive arguments.

7-4.2       Explain the results of the intersection of two or more geometric shapes in a plane.

7-4.3       Illustrate the cross section of a solid.

7-4.4       Translate between two- and three-dimensional representations of compound figures.

7-4.5       Analyze the congruent and supplementary relationships—specifically, alternate interior, alternate exterior, corresponding, and adjacent—of the angles formed by parallel lines and a transversal.

7-4.6       Compare the areas of similar shapes and the areas of congruent shapes.

7-4.7       Explain the proportional relationship among attributes of similar shapes.

7-4.8       Apply proportional reasoning to find missing attributes of similar shapes.

7-4.9       Create tessellations with transformations.

7-4.10     Explain the relationship of the angle measurements among shapes that tessellate.

• The content, process, and product standards included in this WebQuest exceed the state adopted standards for students at the 7th 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.

Our Tessellation WebQuest developed by Cynthia R. Parker