Tessellation Project

Today we will discuss the requirements and expectations for your Tessellation projects and you will receive a brief introduction to the different types of tessellations. This is an individual project.

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According to the NCTM Standards geometry "provides an opportunity for students to experience the creative interplay between mathematics and art." This project is specifically designed to allow students to demonstrate their ability to create art using transformational geometry. Many high school students are familiar with the work of M. C. Escher, and are very excited to get the chance to create tessellations, especially since they get to use computers. This lab-based unit will take approximately 5 class days and a couple of hours outside class for the students to complete. This project is worth one test grade.

Escher Artist...

Maurits Cornelis Escher (1898 – 1972) was a Dutch artist famous for his repetitive, interlocking pattern. His works look like paintings but were done by woodcarving and lithographs.

Escher ’ s designs are made from variations on tiling patterns called

tessellations.

A floor covered by square tiles is an example of a tessellation of squares.

Explore...

Spend some time exploring examples. Click on each link below to explore.

Google Images: Escher Escher Artist Tessellations

Explore...

Spend some time exploring examples. Click on each link below to explore.

Triangle Tessellation Quadrilateral Tessellation Tessellating Alphabet Visit Applet to create your own regular tessellation: http://www.shodor.org/interactivate/activities/Te ssellate/

Transformations

Tessellations can be modified by using

transformations

. As you know, transformations are movements of geometric figures. One transformation, commonly used to create tessellations is a slide, or

translation

, of a figure.

Translations

Translations

You can create more complex designs starting with square tessellations and making changes on both pairs of sides.

Translations

Depending how you decide to color your tessellation, a very simple design can have a very creative result.

Glide Reflections

For glide reflection tessellations, polygons should have opposite sides that are parallel and congruent – squares, hexagons, parallelograms.

Glide Reflections

By reflecting and gliding over more than one side, you can create a more complex tessellation.

Glide Reflections

Adding coloring and other features will enhance the artwork.

Rotations

Adjacent sides must be congruent – squares, equilateral triangles, regular hexagons, rhombi

Rotations

Or your could perform a midpoint rotation. Note: More than one side may be altered for more challenging designs. Coloring one side of the pattern will help prevent accidental flipping during tracing.

Rotational Tessellation

"Nets" vs. "Templates"

The “net” is the field or background of basic shapes from which the template is made. If the net tessellates, so will the template. The template is repeated (translated, rotated, glide-reflected) to make the tessellation. For the project, the net will have sides less than 3 inches.

"Nets" vs. "Templates"

The template is used to trace and repeat the design. Thus any inaccuracy is either the net or the template will be magnified as the design is repeated, and the interlocking of the edges will not be precise, or the vertices will not meet. Thus your net and template needs to be VERY accurate, or your design will not tessellate well.

For simple tessellations of polygons, the templates are the polygons or the combination or polygons that make the tessellation. To make a more complex design, experiment with your template, adding details and features as you wish to create something recognizable, with depth, that is more than just a shape. Often, trying to find a design within your template will prompt you to alter the template. This was Escher’s sequence.

Suggestions

For translation, rotation and altered hexagon tessellations, nets are complex. Escher Artist’s nets will be the most complex.

To accurately copy a shape by tracing an index card cutout, remember that when you trace around the cutout, the drawing will be slightly larger. It takes practice to use this technique, so make plenty of trial runs. Draw the tessellation lightly so you can make adjustments before completing. It is acceptable to darken the lines to cover aberrations. Be careful, though, since the design will look best if these lines are all the same.

Suggestions

Try out several designs, by cutting and taping paper together until you find something you like.

When you have decided on a design, create your template on a stiff material – heavy cardstock or a file folder seem to work well for creating a sturdy template that can be traced over and over.

Be creative. Your design should not look like any of the designs in the packet or in this presentation.

Remember: Finding a design online and copying it is plagiarism.

Work Days

o You will have 5 days in class to work on your project. Monday – Friday o You will receive a participation grade for these days.

o You must bring all necessary materials with you. There are markers (however colored pencils might create a nicer product), scissors, white computer paper, and tape in the classroom. Points will be deducted from your participation grade if you do not have your materials with you.

o You may want to cut out several regular polygons that tessellate and bring them with you to work on creating your template during class.

Project Notes

o Your project is due on

Monday March 11th

at the beginning of class. Grading criteria is on your handout.

o On the due date, you must turn in your completed tessellation, your template (attached to the back of your tessellation), your summary write-up, and your grading checklist.

o Do NOT wait until the last minute!!!!!

Grading

ONLINE EXPLORATION: 10 points Check off each site on your checklist as you visit the webpages.

DESIGN NET: 10 Points Draw a picture of your net on the back of your grading checklist.

TEMPLATE: 10 Points Partial sketch of completed design, with some indication of color. This does not have to match the final design exactly, to allow for creativity and problem solving in the execution phase, but should be pretty close to the net.

WRITE UP: 10 points Step by step explanation of the creation of your template, as well as a paragraph about your experience with this project. http://mste.illinois.edu/courses/ci336kt/garrison/tesexpl.html

FINAL DESIGN: 60 points possible

Difficulty Escher Artist

15 points Rotation, Glide Reflection

Pretty Good

10 points Translation, Non-regular polygons

Short Cut

5 points Regular Polygons

Technicality

Complex design, changing all original sides, no side exceeds 3 inches Simple template, changing some of the sides.

Very simple, or shapes do not tessellate

Decoration

Many details, colorful, creative, recognizable design Some details, somewhat colorful or creative, design not instantly recognizable Few or no details, little or no color, design unnoticeable

Appearance

Centered, crisp shapes, neatly colored, smudge free Not centered, pencil marks showing, messy coloring, partial shapes Not mounted, inconsistent template, perfunctory coloring