• The word tessera in Latin means a small stone cube.• They were used to make up tessellata - the mosaic pictures forming floors and tilings in Roman buildings• A tessellation or a tiling is a way to cover a floor with shapes so that there is no overlapping or gaps A Roman floor mosaic
Tessellations Around the House• Tessellations can be found in quilts, floor tiling, and wallpaper.
Tessellations in Nature snake skin spider web armadillo armor
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.
Quote of Escher “I try in my print 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. the nonsensicalness of some of what we take to be irrefutable certainties.” -----M.C. Escher
• Born June 17, 1898 in the Dutch province of Friesland.• He began studying tilings 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.
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.
• 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.
• 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 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.
• 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.