Tessellations Lindsay Stone Fara Mandelbaum Sara Fazio Cori McGrail Laura Welch Jason Miller

Definition Tessellation – a careful juxtaposition of elements into a coherent pattern sometimes called 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 definite interior and a definite exterior                                                                               

History Mathematics 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 the relationship between tessellations,symmetry, and X-ray crystallography

M.C. Escher (1898 – 1972) -Created over 100 tessellated patterns -Work involves topology, optical illusions, hyperbolic tessellations, and other advanced mathematical topics -Escher’s tilings were designed to resemble recognizable objects -Escher’s work with tilings of the plane embodies many ideas that scientists and mathematicians discovered only after Escher did

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 fit the vertex…therefore it is not a regular tessellation

There are only 3 regular tessellations One of triangles One of squares One of hexagons

This is NOT a regular polygon tessellation because….. The plane is not filled at the vertex because there is a space left over vertex space 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 the alteration moves here

Reflections – “flip” the alteration flips here

Rotation – “turn” the alteration here rotates around this vertex

Steps 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 or more 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 it with 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 Tessellations Definition – are tessellations of more than one type of regular polygons such that the polygon arrangement at each vertex is the same

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

Semi-Regular Tessellation’s 3.12.12 4.6.12 4.8.8

Semi-Regular Tessellation’s 3.4.6.4 3.6.3.6 3.3.3.3.6

Semi-Regular Tessellation’s 3.3.3.4.4 3.3.4.3.4

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

Duals and Vertex Configurations Duals - 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 vertex creating a new shape 6^3 3^6 4^4 3^6 6^3 4^4

In life tessellations appear all around us…. Mud Flats Honeycombs Hydrogen Peroxide Checkers

Gallery

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