Tessellations Starting slide…. Needed - Computer with overhead computer projector, with power point software and internet….

What is a tessellation? If many copies of a shape can be used to cover a surface, without leaving any gaps between them, then we say that the shape will tessellate. The pattern that is formed is called a tessellation. Simply put: A tessellation is like a puzzle that repeats a particular pattern. Read…. To students…. Click anywhere when finished... For example:

In nature and life, they appear often: Honeycombs... Mud flats... Read headings… Click when slide finishes... In games, like checkers..

Tessellations in Nature snake skin spider web armadillo armor

Tessellations are all around us: Look for tessellations in walls, patios and pavements.

Tessellations are all around us: Common shapes can be arranged in unusual ways Sometimes an unusual shape will tessellate

Tessellations are all around us: Sometimes 2 or more different shapes will tessellate.

Tessellations Around the House Tessellations can be found in quilts, floor tiling, and wallpaper.

Study of Science X-ray Crystallography This picture depicts the transformation of eight points in an array to make a very small crystal lattice, which tessellates. Read to students… click when completed...

more science…. This image suggests the relationship between tessellations, symmetry, and x-ray crystallograpy. Read to students… click when completed...

and even more science…. Hydrogen Peroxide, among other chemicals, has this tessellating, crystal structure. Read to students… click when completed...

Math in Tessellations Many different geometric shapes can be used to create tessellations. Which of the shapes below would fit with itself to make a good tessellation design?

The mathematics of the Regular Tessellation... A regular polygon tessellation is constructed from regular polygons. Regular polygons have equal sides and equal angles. The regular polygons must fill the plane at each vertex, with repeating patterns and no overlapping pieces. Read to students… Note... links for student presentation.. Two links take you to internet... Polygon examples and Regular polygon examples Important concept… filling the vertex... click when completed... Note: this pentagon does not fit into the vertex… therefore it is not a regular tessellation.

Regular tiling Which other regular polygons do you think can tile the plane?

Triangles Triangle? Yep! How many triangles to make 1 complete rotation? The interior angle of every equilateral triangle is 60º. If we sum the angles around a vertex, we get 60º + 60º + 60º + 60º + 60º + 60º = 360º again!.

Pentagons Will pentagons work? The interior angle of a pentagon is 108º, and 108º + 108º + 108º = 324º.

Hexagons Hexagons? The interior angle is 120º, and 120º + 120º + 120º = 360º. How many hexagons to make 1 complete rotation?

Heptagons Heptagons? Octagons? Not without getting overlaps. In fact, all polygons with more than six sides will overlap.

Regular tiling So, the only regular polygons that tessellate the plane are triangles, squares and hexagons. That was an easy game. Let’s make it a bit more rewarding.

A famous mathematician, Kepler studied tessellations and noted the regular tessellations or (tilings) of the plane. There are only three regular tessellations; one of triangles one of squares READ... Note… internet link ... to Kepler…. Click when finished…. and one of hexagons.

This is NOT a regular polygon tessellation, because... vertex The plane is not filled at the vertex, because there is space left over. space Read slide… Space at vertex should be noted. click when completed... A regular polygon tessellation, can be changed using “alterations” to the sides of the polygon. These alterations are called transformations.

Three Common Transformations 1. Translation, which is a slide of one side of the polygon. 2. Reflection, which is a flip or mirror image of one side of the polygon. 3. Rotation, which is a turn of a side around one vertex of the polygon. Read slide… There are other transformations … really combinations of these three… click when completed...

Translations - a slide this side the alteration moves here click when completed... Get practice template and try creating a translation….

Let’s try together….. a translation or slide…. Hand out worksheet and have students complete first translation... Use pen tool to draw translation. First on small square… then on series of squares click when completed... ….

Reflections - mirror images the alteration flips here Reflections….. Go through with audience…. Demo on board… check audience for correctness. Get practice template and try creating a reflection….

Let’s try together….. a reflection, flip or mirror image Have students complete reflection on worksheet along with PowerPoint… Use pen tool to draw reflection. First on small square… then on series of squares….

Rotations - turns the alteration here rotates around this vertex Go through with audience…. Demo on board… check audience for correctness. Please note: students have most difficulty here…with rotations… we limit rotations to 90 degree increments. Get practice template and try creating a rotation...

Let’s try together….. a rotation or turn Have students complete using third space on worksheet… Use pen tool to model…. click when completed...

Tessellations by M.C. Escher

Make Your Own Escher Tessellations 1 Step 1: Start with a simple shape that will tessellate e.g. a rectangle. Step 2: Remove a shape or shapes from one side of the rectangle and fix them to the opposite side.

Make Your Own Escher Tessellations 1 Use this as a template to create your tessellation.

Make Your Own Escher Tessellations 2 You could start with a hexagon and cut semi-circles from three of the sides and fix them to the opposite sides like this.

Make Your Own Escher Tessellations 2

Make Your Own Escher Tessellations 3 Start with a square and cut segments from two sides and fix them to the opposite sides like this.

Make Your Own Escher Tessellations 3

Professional Tessellation Designs Sun and Moon M.C. Escher Fish Robert Ingalls Many artists have created master works of art using the simple rules of tessellations. M.C. Escher and Robert Ingalls are among many tessellation artists. Can you you spot the repeating shape?

You Can Be a Tessellation Artist You need to first create a tessellating template. Step 1: using grid paper cut out a perfect square. Step 2: with a pencil make a line on two adjacent sides of the square. Be sure not to draw through a corner. Step 3: cut out the pieces and attach them with tape to the opposite sides from where they came.  You have now created your tessellation template.

Seeing a Figure Looking at your tessellation template, study the sides and the shape to decide what sort of figure you might see. The person who designed this form saw an elephant and an elf. Are there any other different forms that you might see? When looking for the shape be sure not to see too much detail as it can make the art work too crowded.

Putting It All Together Take your tessellation template and trace it on to a piece of paper. When tracing you can use different types of symmetry to change your picture. Be sure that your whole page is filled with a repeating pattern. Colour in your picture creatively. Elephants Translation symmetry Elves and Elephants Combinations of symmetry

Extras ... Is this a regular tessellation? What kind of tessellation is this? Link to with the first two titles to have students complete “assessment”…. Is this a tessellation… What kind of tessellation is this… Java script internet site… great for whole class instruction… May need more class time to have students complete their own tessellations using a variety of software tittles.. Tessellation internet activity Cool site for interactive Javascript tessellations. Constructing tessellations - Claris/Appleworks, HyperCard, HyperStudio, Geometry Sketchpad, Windows Paint and more.

Links Totally Tessellated - ThinkQuest winner - great site, instruction, information. http://library.advanced.org/16661/ Tessellations Tutorials - Math Forum site http://forum.swarthmore.edu/sum95/suzanne/tess.intro.html - site for construction of tessellations. http://forum.swarthmore.edu/sum95/suzanne/links.html - great list of tessellation links Math. Com - List of good tessellation links http://test.math.com/students/wonders/tessellations.html World of Escher site - commerical site with gallery of Dutch artist,Escher who was famous for his tessellation art. http://WorldOfEscher.com/gallery/ Science University’s Tilings Around Us Site. http://www.ScienceU.com/geometry/articles/tiling/tilings.html References…. Links… lots of good activities and materials available for additions activities. Link to end show… for slide show of tessellation…. Press escape to end slide show… and return to this page.. Other links from Forum. http://forum.swarthmore.edu/library/topics/transform_g/ The end ...

Paper Folding Movie Click to play the movie.