tessellations complete

8/13/2019 Tessellations Complete

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Tessellationspangea.tec.selu.edu/~versek/Tessellations%20Complete.ppt


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If many copies of a shape can be used to cover asurface, without leaving any gaps between them, thenwe say that the shape will tessellate.The pattern that is formed is called a tessellation.

Simply put: A tessellation is likea puzzle that repeats aparticular pattern.

For example:

What is a tessellation?


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Tessellations in Nature

snake skin

armadillo armorspider web


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Tessellations are all around us:

Look for tessellations in walls, patios and

pavements.


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Tessellations are all around us:

Common shapes can be

arranged in unusual ways

Sometimes an unusual

shape will tessellate


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


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Tessellations Around the House

Tessellations can be found

in quilts, floor tiling, and

wallpaper.


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X-ray Crystallography

This picture depicts the

transformation of eightpoints in an array to make

a very small crystal lattice,

which tessellates.

Study of Science


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This image suggests the

relationship betweentessellations, symmetry,

and x-ray crystallograpy.

more science.


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Hydrogen Peroxide,among otherchemicals, has thistessellating, crystal

structure.

and even morescience.


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Math in Tessellations Many different geometric shapes can be used

to create tessellations.

Which of the shapes below would fit withitself to make a good tessellation design?


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The mathematics of the

Regular Tessellation...

A regular polygontessellation is constructed fromregular polygons.

Regular polygonshave equal sides and equal angles.The regular polygons must fill the plane at eachvertex, with repeating patterns and no overlappingpieces.

Note: this pentagon does notfit into 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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Which other regularpolygons do youthink can tile theplane?

Regular tiling


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Triangle?

Triangles

The interior angle of every equilateral triangle is 60. If wesum the angles around a vertex, we get 60 + 60 + 60 +

60 + 60 + 60 = 360 again!.

Yep! How many

triangles tomake 1 complete

rotation?


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Will pentagons work?

Pentagons

The interior angle of a pentagon is 108, and 108 + 108+ 108 = 324.


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Hexagons?

Hexagons

The interior angle is 120, and 120 + 120 +

120 = 360. How many hexagons to make 1 complete

rotation?


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Heptagons?Octagons?

Heptagons

Not without getting overlaps.In fact, all polygons with more than six sides willoverlap.

l ili


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So, the only regular polygons that tessellate

the planeare triangles, squares and hexagons.

That was an easy game. Lets make it a bitmore rewarding.

Regular tiling


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A famous mathematician, Keplerstudiedtessellations and noted the regulartessellations 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 filledat the vertex, becausethere is space left over.

vertex

space

A regular polygon tessellation, can be changed usingalterations to the sides of the polygon. Thesealterations are called transformations.


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Three Common Transformations1. Translation, which is a slide of one side of thepolygon.

2. Reflection, which is a flip or mirror image of oneside of the polygon.

3. Rotation, which is a turn of a side around onevertex of the polygon.


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Translations- a slide

moves here

Get practice template and try creating a translation.

the alteration

this side


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Lets try together.. a translation or slide.


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Reflections- mirror images

flipshere

the alteration

Get practice template and try creating a reflection.


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Lets try together.. a reflection, flip or mirror image


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Rotations- turns

the alteration

rotates aroundthis vertex

Get practice template and try creating a rotation...

here


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Lets try together.. a rotation or turn


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Tessellations by M.C. Escher


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Make Your Own Escher

Tessellations 1Step 1: Start with a simpleshape 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.


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Tessellations 1Use this as a template to create your tessellation.


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Tessellations 2You could start with a hexagon and cut semi-circlesfrom three of the sides and fix them to the opposite

sides like this.


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Make Your Own EscherTessellations 2


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Make Your Own EscherTessellations 3

Start with a square and cut segments from two sides

and fix them to the opposite sides like this.


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You Can Be aTessellation 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 acorner.

Step 3:cut out the pieces and attach them withtape to the opposite sides from where they came.

You have now created your tessellation template.


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Seeing a Figure Looking at your tessellation template, study the

sides and the shape to decide what sort of figureyou 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 toocrowded.


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Putting It All Together Take your tessellation

template and trace iton to a piece of paper.

When tracing you can

use different types ofsymmetry to change

your picture. Be sure that your

whole page is filledwith a repeatingpattern.

Colour in your picturecreatively.

Elephants

Translation

symmetry

Elves and

ElephantsCombinations ofsymmetry


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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.
http://localhost/var/www/apps/conversion/tmp/scratch_10/quiz1.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_10/quiz2.ppthttp://forum.swarthmore.edu/sum95/suzanne/tess.intro.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_10/tesselation.htmlhttp://home6.inet.tele.dk/bergmann/10galleri/idx10.htmhttp://home6.inet.tele.dk/bergmann/10galleri/idx10.htmhttp://localhost/var/www/apps/conversion/tmp/scratch_10/tesselation.htmlhttp://forum.swarthmore.edu/sum95/suzanne/tess.intro.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_10/quiz2.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_10/quiz1.ppt


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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 wasfamous for his tessellation art.http://WorldOfEscher.com/gallery/

Science Universitys 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
http://library.advanced.org/16661/http://forum.swarthmore.edu/sum95/suzanne/tess.intro.htmlhttp://forum.swarthmore.edu/sum95/suzanne/tess.intro.htmlhttp://forum.swarthmore.edu/sum95/suzanne/links.htmlhttp://worldofescher.com/galleryhttp://www.scienceu.com/geometry/articles/tiling/tilings.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_10/end.ppthttp://forum.swarthmore.edu/library/topics/transform_ghttp://test.math.com/students/wonders/tessellations.htmlhttp://test.math.com/students/wonders/tessellations.htmlhttp://test.math.com/students/wonders/tessellations.htmlhttp://test.math.com/students/wonders/tessellations.htmlhttp://test.math.com/students/wonders/tessellations.htmlhttp://forum.swarthmore.edu/library/topics/transform_ghttp://localhost/var/www/apps/conversion/tmp/scratch_10/end.ppthttp://www.scienceu.com/geometry/articles/tiling/tilings.htmlhttp://worldofescher.com/galleryhttp://forum.swarthmore.edu/sum95/suzanne/links.htmlhttp://forum.swarthmore.edu/sum95/suzanne/tess.intro.htmlhttp://forum.swarthmore.edu/sum95/suzanne/tess.intro.htmlhttp://library.advanced.org/16661/


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