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LESSON TWO

Concept Introduction: What do two-dimensional tessellations look like? Where in art can they be found?

Lesson Introduction for Unit on Tessellations

This exploration of the world of art and culture, including the works of M.C. Escher, provides students with an opportunity to identify and create original tessellations. Using a wealth of interactive multimedia applications available on the Web, students will explore the artistic representations of tessellations and transformations and enhance their understanding of the underlying mathematical principles.

Bookmarks

Works by and in the style of M.C. Escher

The World of Escher - Artwork Gallery
http://www.WorldOfEscher.com/gallery/

NGA - M.C. Escher -- Life and Work
http://www.nga.gov/collection/gallery/ggescher/ggescher-main1.html

Mathematical Art of M.C. Escher -- a Platonic Realms Mini-Text

Computer Art Inspired by M.C. Escher and Victor Vasarely
http://web.inter.nl.net/hcc/Hans.Kuiper/

Hop's Escher Tessellation Tiles
http://www.tabletoptelephone.com/~hopspage/HopsTiles.html

The Seventeen Wallpaper Tilings
http://www.ScienceU.com/geometry/articles/tiling/wallpaper.html

Boxer Math Toolbox
http://www.boxermath.com/plp/modules/online/workshop/toolbox/mosaictool3

Triangle Tilings and Polyhedra
http://www.ScienceU.com/geometry/articles/tritile/

Interactive Triangle Tiling Maker
http://www.ScienceU.com/geometry/handson/trigrp/trigrp.cgi?group5=on

Resources

Math Forum Suzanne Alejandre: Tessellations
http://mathforum.org/sum95/suzanne/tess.intro.html

Middle Level Mathematics: M.C. Escher: The Poet of the Impossible

Introduction to Tilings
http://www.scienceu.com/geometry/articles/tiling/

Native American Geometry Designs
http://www.earthmeasure.com/Designs/index.html

Islamic Architecture
http://islamicart.com/main/architecture/index.html

STEPS

1. Explore the works of M.C. Escher on the Web sites provided, or in a book or poster, to identify similarities and differences in the patterns he created. Begin by showing students samples of Escher's work and facilitate a discussion about the nature of patterns. Students can also work in small groups to discuss their ideas about Escher's patterns. In groups, students can categorize the different kinds of tessellations they find using their knowledge of symmetry, similarity, and congruency. Student definitions may not use exact mathematical vocabulary. For example, a student might say a shape "moved from one location to another" to describe "translation across the plane." Ask each group to write a definition of the word tessellation. Have the groups share their definitions with the whole class.

2. Hand out copies of the Student Organizer, "Getting into Shapes," below, with which students will clarify the vocabulary associated with tessellations by exploring animated examples provided at The Seventeen Wallpaper Tilings (http://www.ScienceU.com/geometry/articles/tiling/wallpaper.html) and Run the Program at the interactive Boxer Math site (http://www.boxermath.com/plp/modules/online/workshop/toolbox/mosaictool3) to create designs of their own online. This Web-based activity will help students clarify the vocabulary associated with tessellations.

3. Have students explore the relationship between two-dimensional patterns and three-dimensional objects by using Interactive Triangle Tiling Maker (http://www.ScienceU.com/geometry/handson/trigrp/trigrp.cgi?group5=on). Provide time for students to review the background information provided at Triangle Tilings and Polyhedra (http://www.ScienceU.com/geometry/articles/tritile/) or provide a guided exploration of the site by displaying it using a single computer connected to a LCD projector or large screen monitor.

 Student Organizer: Getting into Shapes Check out the animations that demonstrate the seventeen different types of tiling symmetry. Follow the link to Boxer Math Toolbox and create a tessellation of your own. Can you identify the pattern unit in your tessellation? The Seventeen Wallpaper Tilings http://www.ScienceU.com/geometry/articles/tiling/wallpaper.html Boxer Math Toolbox http://www.boxermath.com/plp/modules/online/workshop/toolbox/mosaictool3 Triangle Tilings and Polyhedra http://www.ScienceU.com/geometry/articles/tritile/ Interactive Triangle Tiling Maker http://www.ScienceU.com/geometry/handson/trigrp/trigrp.cgi?group5=on It's Time to Tessellate! Use the tools provided by your teacher to create your own original tessellation. Can you come up with other ways to tessellate a pattern?

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