Becca Stockford Lehman

Tessellate: to form or arrange small squares or blocks in a checkered or mosaic pattern Tessellate: to form or arrange small squares or blocks in a checkered or mosaic pattern Comes from the Latin word tessella which refers to the small square pieces used in mosaics Comes from the Latin word tessella which refers to the small square pieces used in mosaics Another word for tessellation is tiling Another word for tessellation is tiling Tiling: filling a flat space in with individual tiles ensuring there are no gaps or overlaps Tiling: filling a flat space in with individual tiles ensuring there are no gaps or overlaps

Regular Regular Semi-regular Semi-regular

Tessellations made up of congruent regular polygons Tessellations made up of congruent regular polygons Arrangement of nonoverlapping polygonal tiles surrounding a common vertex is called a vertex figure Arrangement of nonoverlapping polygonal tiles surrounding a common vertex is called a vertex figure Three regular polygons can tessellate in the Euclidean plane Three regular polygons can tessellate in the Euclidean plane-triangles-squares-hexagons (180 (N-2))/N is the angle at each vertex of a regular polygon, N being the number of sides (180 (N-2))/N is the angle at each vertex of a regular polygon, N being the number of sides

Semi-regular tessellations are formed from regular polygons Semi-regular tessellations are formed from regular polygons The arrangement at every vertex point is identical The arrangement at every vertex point is identical Unlike regular tessellations you can use more than one regular polygon Unlike regular tessellations you can use more than one regular polygon There are eight semi-regular tessellations There are eight semi-regular tessellations

To determine the name of a tessellation- To determine the name of a tessellation-  Look at a vertex within the tessellation  Look at one of the polygons touching that vertex  Count the sides of that polygon and that is the first number  Repeat until you have included the sides of every polygon touching the vertex

Tiling's with irregular polygons – as long as the polygons are congruent the plane can be tiled with Tiling's with irregular polygons – as long as the polygons are congruent the plane can be tiled with -any triangle tile -any quadrangular tile, convex or not -certain pentagonal tiles -certain hexagonal tile Can also tile a plane with nonconvex polygons

Heesch’s Tile- every tile is identical but appear in positions that are non-identical Heesch’s Tile- every tile is identical but appear in positions that are non-identical Aperiodic tilings – do not repeat themselves Aperiodic tilings – do not repeat themselves Honeycombs and crystals- 3-dimensinal tessellations Honeycombs and crystals- 3-dimensinal tessellations

Maurits Cornelius Escher: artist who created artistic tilings Maurits Cornelius Escher: artist who created artistic tilings Based on modifications of known tiling patterns Based on modifications of known tiling patterns

Use of reflections and rotations Use of reflections and rotations Transitional symmetry : moving the shape over some unit(s) Transitional symmetry : moving the shape over some unit(s) - 2 types, up-down and left-right Glide symmetry: combination of translation and a reflection Glide symmetry: combination of translation and a reflection

Frieze Groups- Frieze is a narrow band along the top of a wall which in the past were decorated with repeating geometrical patterns Frieze Groups- Frieze is a narrow band along the top of a wall which in the past were decorated with repeating geometrical patterns Seven groups Seven groups  Hop: only translations  Sidle: translations with reflections in vertical lines  Jump: translations and one horizontal reflection

(4) Step: translations and glides (5) Spinning hop: translations with rotational symmetries of 180 degrees (6) Spinning sidle: translations, glides, vertical reflections and rotational symmetries of 180 degrees (7) Spinning jump: translations, vertical reflections, one horizontal reflection and rotational symmetries of 180 degrees

 cannot tessellate over infinite plane (other)

Books- Mathematics Dr. Richard Elwes pg Mathematics Dr. Richard Elwes pg Mathematical Reasoning for Elementary Teachers. Calvin T. Long and Duane W. DeTemple, pg Mathematical Reasoning for Elementary Teachers. Calvin T. Long and Duane W. DeTemple, pg Websites l l l l