tess el lations ss

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Tessellation

If many copies of a shape can be used to cover asurface, without leaving any gaps between them,

then we say that the shape will tessellate.

The pattern that is formed is called a tessellation.

A tessellation is a pattern of repeating figures that fittogether with NO overlapping orempty spaces.

Tessellations are formed using transformation.

Transformations:

Translation**

Rotation**

Reflection**

Dilation

HAMZAH NUN @ IPG KTHO


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Tessellations all around us

Look for tessellations in walls, patios and pavements.



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Common shapes can be

arranged in unusual ways

Sometimes an unusual

shape will tessellate



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Sometimes 2 or more different shapes will tessellate.



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Tessellation with Pentominoes



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Tessellation with 7-Pin Polygons



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Tessellations by M.C. Escher



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Make Your Own

Use this as a template to create your tessellation.



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Make Your Own



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Make Your Own



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Examples of Tessellation Art



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Project Directions1. Start with a template piece Must be a REGULAR polygon (squares work well)

2. Choose one of the following transformations Translation

Glide Reflection (translation with reflection) Rotation

Mid-point Rotation

3. Practice with template on computer paper.

4. Trace your final product on to white card stock (9x12).5. Color and decorate

6. Mount onto construction paper



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Translation Tessellation (EASY)For simple translation tessellations, your starting polygonshould have opposite sides that are parallel and congruent.

Squares, hexagons, and parallelograms work best.



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1.

Draw a design on

one side of the

square and slideit to opposite

side.

Tape the cutout

pieces to opposite

sides. Slide(translation) the

pattern when

tracing.

Start with a

square

2.3. 4.

Draw another

design on the

adjacent side ofthe square and

slide it to

opposite side.

Translation Tessellation (HARD)You can create more complex designs starting with square

tessellations and making changes on both pairs of sides.



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Depending how you

decide to color your

tessellation, a verysimple design can

have a very creative

result.



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Glide Reflection TessellationFor glide reflection tessellations, your polygons should

have opposite sides that are parallel and congruent.

Squares, hexagons, and parallelograms work best.

*You can make this one more difficult by

cutting out two pieces from different sides

and doing a glide reflection for both.



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Rotation TessellationFor rotation tessellations, the adjacent sides of thepolygon must be congruent. Squares, equilateraltriangles, regular hexagons, and rhombi work best.

*You can make this one more difficult by cutting out two

pieces from different sides and doing a rotation tessellationfor both.HAMZAH NUN @ IPG KTHO


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Midpoint Rotation Tessellations Triangles, squares, and quadrilaterals work best for

this type.

*You can make this one more difficult by cutting out two

pieces from different sides and doing a mid-point rotationfor both.HAMZAH NUN @ IPG KTHO


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Example:

Rotational

Tessellation



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In nature and life, they appear often:

Honeycombs...

Mud flats...

In games, like checkers..



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Patterns

Very basic tessellations are simply a repeating pattern.

The rule is that you have to make sure that the shapes fit togetherleaving no open space on the page.



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Professional TessellationDesigns

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


h h i f h
http://www.cromp.com/tess/people/T31.shtml


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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 repeatingpatterns and no overlapping pieces.

Note: this pentagon does not fitinto the vertex therefore it is

not a regular tessellation.

http://library.advanced.org/16661/background/polygons.htmlhttp://library.advanced.org/16661/background/regular.polygons.htmlhttp://library.advanced.org/16661/background/regular.polygons.htmlhttp://library.advanced.org/16661/background/polygons.html


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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.

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Kepler.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Kepler.html


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This is NOT a regular polygon tessellation,

because...

The plane is not filled at thevertex, because there is spaceleft over.

vertex

space

A regular polygon tessellation, can be changed using alterations to the sides

of the polygon. These alterations are called transformations.HAMZAH NUN @ IPG KTHO


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