Lindsay StoneFara Mandelbaum Sara Fazio Cori McGrail Laura Welch Jason Miller

                  

                                                            

DefinitionTessellation – a careful juxtaposition of elements into a coherent pattern sometimescalled a mosaic or tiling

Example:

                  

                                                            

Tessellations are different from patterns because patterns usually do not have distinct closed shapes

A closed shape is a shape that has a definiteinterior and a definite exterior

HistoryMathematics-Johannes Kepler 1619-Russian crystallographer E.S. Fedorov 1891

Science-X-ray Crystallography

This picture is a transformation of eight points in an array to make a very small crystal lattice which tessellates

Science Continued….

This image suggests therelationship between tessellations,symmetry,and X-ray crystallography

M.C. Escher (1898 – 1972) -Created over 100 tessellated patterns

-Escher’s tilings were designed to resemble recognizableobjects

-Escher’s work with tilings of the plane embodiesmany ideas that scientists and mathematicians discovered only after Escher did

-Work involves topology, optical illusions,

hyperbolic tessellations, and other advanced

mathematical topics

Sun and Moon• Uses birds to transform

day into night

• In this image the white birds bring forth the sun while the dark birds carry the moon and the stars. Day and night fight each other for attention but fit seemlessly together.

Symmetry Drawing No. 71• Symmetry Drawing

No. 71 is one of his most complex with 12 different birds forming a rectangle in this image

Regular Tessellations-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 fitthe vertex…therefore it is not aregular tessellation

There are only 3 regular tessellations

One of triangles

One of squares

One of hexagons

This is NOT a regular polygon tessellation because…..

vertex

space

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

A regular polygon tessellation,can be changed by 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, “move”

2. Reflection – flip or mirror image of one side of the polygon

3. Rotation – turn of a side around one vertex of a polygon

Translation – “slide”this side

moves here the alteration

Reflections – “flip”

the alteration

flipshere

Rotation – “turn”

the alteration

rotates aroundthis vertex

here

                   

                           

Steps to name an arrangement of regular polygonsaround a vertex

1. To name an arrangement of regular polygons around a vertex, first find the regular polygon with the least number of sides. 2. Then find the longest consecutive run of this polygon, that is, two

ormore repetitions of this polygon around the vertex. 3. Next, indicate the number of sides of this regular polygon. For example, to name a triangle with 3 sides, we name it 3 and follow itwith a period (.). If you find more than one consecutive "run" of this polygon, then name it twice, i.e., 3.3. 4. Proceeding in a clockwise or counterclockwise order, indicate the number of sides of each polygon as you see them in the arrangement. 5. Do remember to start with the longest consecutive run of the regular polygon with the shortest number of sides.

Semi-regular TessellationsDefinition – are tessellations of more than one type of regular polygons such that thepolygon arrangement at each vertex is the same

In order for the semi-regular tessellation to work, the interior angle sum must be equal to 360

number of sides

interior angle (degrees)

3 60

4 90

5 108

6 120

7 128

8 135

9 140

10 144

11 150

... ...

n 180(n-2)/n

Semi-Regular Tessellation’s

3.12.12 4.6.12

4.8.8

3.6.3.6 3.4.6.4

3.3.3.3.6

Semi-Regular Tessellation’s

3.3.3.4.4 3.3.4.3.4

Semi-Regular Tessellation’s

Demi-regular TessellationsDefinition – tessellations of regular polygons in which there are two or three different polygon arrangements

                                   

Duals and Vertex ConfigurationsDuals - connect the centers of the regular polygons around a vertex creating a new shape

Vertex Configurations – connect the midpoints of the sides of the regular polygons around a vertexcreating a new shape

3^6 4^4 6^3

3^64^4

6^3

Tessellations are found in our every day lives, just waiting to be discovered.

• Have you ever been in a building and noticed the pattern in the tile floors? Or, have you noticed the repeating, interlocking pattern of the landscaping stones in someone's back yard? These are both examples of tessellations in the world around us

Tessellations are found in our every day lives, just waiting to be discovered.

• Amazing that many of the buildings that we use on a daily basis can display such intricate tessellations in their brick work and tilings. Many families have kitchen floors and bathroom walls that are looked at daily which are full of tessellations.

Tessellations are found in our every day lives, just waiting to be discovered.

• Even children's play toys, like Legos and soccer balls, contain tessellations. Tessellations exist even in forms that we may not recognize as a work of art (and math).

In life tessellations appear all around us….

HoneycombsMud Flats

CheckersHydrogen Peroxide

Gallery

References• 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

• Math. Com - List of good tessellation linkshttp://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

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