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Tessellations

Discover the beauty and intricacy of tessellations, repeated patterns that fill surfaces without gaps. Learn about famous artists, regular polygon tessellations, and common transformations.

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Tessellations

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  1. Tessellations

  2. What Is a Tessellation? • A shape or tile that repeats to fill a surface without any gaps or overlaps. • The name comes from the word tessella, the small square tile used in ancient Roman mosaics.

  3. What is a tessellation? Formal definition: A careful juxtaposition of elements into a coherent pattern sometimes called a mosaic or tiling. Simply put: A tessellation is like a puzzle that repeats a particular pattern. For example:

  4. In nature and life, they appear often: Honeycombs... Mud flats... In games, like checkers..

  5. Patterns • Very basic tessellations are simply a repeating pattern. • The rule is that you have to make sure that the shapes fit together leaving no open space on the page.

  6. Professional Tessellation Designs Sun and Moon M.C. Escher Fish Robert Ingalls • 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?

  7. 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.

  8. 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.

  9. This is NOT a regular polygon tessellation, because... vertex The plane is not filled at the vertex, because there is space left over. space A regular polygon tessellation, can be changed using “alterations” to the sides of the polygon. These alterations are called transformations.

  10. 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.

  11. Translations - a slide this side the alteration moves here Get practice template and try creating a translation….

  12. Reflections - mirror images the alteration flips here Get practice template and try creating a reflection….

  13. Rotations - turns the alteration here rotates around this vertex Get practice template and try creating a rotation...

  14. Extras ... Is this a regular tessellation? What kind of tessellation is this? Tessellation internet activity Cool site for interactive Javascript tessellations. Constructing tessellations - Claris/Appleworks, HyperCard, HyperStudio, Geometry Sketchpad, Windows Paint and more.

  15. Links • 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/ The end ...

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