TESSELLATIONS

History of Tessellations• Many ancient cultures

have used tessellations.

• Johannes Kepler conducted one of the first mathematical studies of tessellations.

• E.S. Fedorov proved an aspect of tiling in 1891.

Escher Quote

“I try in my prints to testify that we live in a beautiful and orderly world, not in a chaos without norms, even though that is how it sometimes appears. My subjects are also often playful: I cannot refrain from demonstrating the nonsensicalness of some of what we take to be irrefutable certainties. It is, for example, a pleasure to deliberately mix together objects of two and three dimensions, surface and spatial relationships, and to make fun of gravity.”

-M.C. Escher

M.C. Esher• Born June 17, 1898 in the Dutch province of

Friesland.• He began studying tiling around 1935.• He created his most famous tessellation,

“Reptiles” in 1943.• Escher drew his last tessellation in July of

1969.• He died at the age of 73 in March of 1972.

What Is a Tessellation?• A tessellation is a repeating pattern of

distinct shapes without any holes or gaps in the pattern.

• Tessellations cover an entire plane.• Tiling is often another term used for

tessellation patterns.

How many types are there?There are three main types of tessellations: * Regular

* Semi-Regular * Demi-Regular

Regular Tessellations• A regular tessellation

is a pattern only using one regular polygon shape.

• A regular polygon is any many sided shape that has sides of equal length and angles or equal measure.

Semi-Regular Tessellations• A semi-regular

tessellation is a pattern consisting of more than one type of regular polygon.

• The vertex arrangement is the same throughout the entire pattern.

Demi-Regular Tessellations A demi-regular tessellation is

a pattern of regular polygons in which there are two or three different polygon arrangements.

How are Tessellations Made?• Tessellations are made through a group of

techniques called transformations.

• Transformations are techniques used to take a shape and match it exactly to another.

• There are four types of transformations.

Transformations• Translations: when a shape is moved in any

direction.• Reflections: a mirror image that is made of

the original shape.• Rotations: the original shape is rotated

around a central point.• Glide reflections: the figure that results after

reflection and translation.

Vertex Figure and Dual Tessellations• Vertex figures are created by connecting the

midpoints of all the sides that meet at a vertex point.

• Dual tessellations are when the vertex figures are connected and they create another tessellation on top of the original tessellation.

• It also can be done by connecting the centers of the polygons that meet at the vertex point.

Naming of Conventions• A tessellation is named by looking at a vertex

point and finding how many polygons touch the vertex point. Conventions are named based on the type of polygons that touch the vertex point. The convention number represents the number of sides of each polygon.

• Examples:tessellation of squares- 4-4-4-4tessellation of hexagons- 6-6-6

Tessellations Around the House

Tessellations can be found in quilts, floor tiling, and wallpaper.

Tessellations in Nature

snake skin

armadillo armorspider web

Tessellations in Sports

Tessellations in Architecture

Islamic Minaret

Islamic Arch

Medieval Window

Islamic Floor Tiling

Afghanistan Mosaic

M.C. Escher Art

• 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 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/

Links

• Math. Com - Tessellation linkshttp://test.math.com/students/wonders/tessellations.html