Tessellations

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.

For example:

What is a tessellation?

In nature and life, they appear often:

Honeycombs...

Mud flats...

In games, like checkers..

Tessellations in Nature

snake skin

armadillo armorspider web

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.

X-ray Crystallography

This picture depicts the transformation of eight points in an array to make a very small crystal lattice, which tessellates.

Study of Science

This image suggests the relationship between tessellations, symmetry, and x-ray crystallograpy.

more science….

Hydrogen Peroxide, among other chemicals, has this tessellating, crystal structure.

and even more science….

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.

Note: this pentagon does not fit into the vertex… therefore it is not a regular tessellation.

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

Regular tiling

• Triangle?

Triangles

• 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!.

• Yep!• How many triangles to make 1 complete rotation?

• Will pentagons work?

Pentagons

• 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? Octagons?

Heptagons

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

• 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.

Regular tiling

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

and one of hexagons.

This is NOT a regular polygon tessellation,because...

The plane is not filled at the vertex, because there is space left over.

vertex

space

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.

Translations - a slide

moves here

Get practice template and try creating a translation….

the alteration

this side

Let’s try together….. a translation or slide….

Reflections - mirror images

flipshere

the alteration

Get practice template and try creating a reflection….

Let’s try together….. a reflection, flip or mirror image

Rotations - turns

the alteration

rotates aroundthis vertex

Get practice template and try creating a rotation...

here

Let’s try together….. a rotation or turn

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

• 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?

Sun and Moon

M.C. Escher

Fish

Robert Ingalls

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

Is this a regular tessellation?

What kind of tessellation is this?

Extras ...

Constructing tessellations - Claris/Appleworks, HyperCard, HyperStudio, Geometry Sketchpad, Windows Paint and more.

Tessellation internet activity

Cool site for interactive Javascript tessellations.

• Totally Tessellated - ThinkQuest winner - great site, instruction, information. http://library.advanced.org/16661/

• Tessellations Tutorials - Math Forum sitehttp://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

• 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

The end ...

• Other links from Forum.http://forum.swarthmore.edu/library/topics/transform_g/

Links

• Math. Com - List of good tessellation linkshttp://test.math.com/students/wonders/tessellations.html

Paper Folding Movie

Click to play the movie.