Tessellations/ Tiling

Becca Stockford Lehman

TessellationsTessellate: 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

Another word for tessellation is tiling

Tiling: filling a flat space in with individual tiles ensuring there are no gaps or overlaps

Types of TesselationsRegular

Semi-regular

Regular TessellationsTessellations made up of congruent regular polygons

Arrangement of nonoverlapping polygonal tiles surrounding a common vertex is called a vertex figure

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

Semi-regular TessellationsSemi-regular tessellations are formed

from regular polygons

The arrangement at every vertex point is identical

Unlike regular tessellations you can use more than one regular polygon

There are eight semi-regular tessellations

Naming TessellationsTo determine the name of a

tessellation-

(1)Look at a vertex within the tessellation

(2)Look at one of the polygons touching that vertex

(3)Count the sides of that polygon and that is the first number

(4)Repeat until you have included the sides of every polygon touching the vertex

Examples with Regular Tessellations

4.4.4.4 3.3.3.3.3.3

6.6.6

Examples with Semi-regular Tessellations

3.3.3.3.63.3.3.4.4

3.4.6.4

4.6.12

3.6.3.6

3.3.4.3.4

3.12.12

4.8.8

Other TessellationsTiling'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 tileCan also tile a plane with nonconvex

polygons

Heesch’s Tile- every tile is identical but appear in positions that are non-identical

Aperiodic tilings – do not repeat themselves

Honeycombs and crystals- 3-dimensinal tessellations

Escher Tilings Maurits Cornelius Escher: artist who

created artistic tilings

Based on modifications of known tiling patterns

Creation of TessellationsUse of reflections and rotations

Transitional symmetry : moving the shape over some unit(s)

- 2 types, up-down and left-right

Glide symmetry: combination of translation and a reflection

Tessellations in architectureFrieze Groups- Frieze is a narrow band along

the top of a wall which in the past were decorated with repeating geometrical patterns

Seven groups

(1) Hop: only translations

(2) Sidle: translations with reflections in vertical lines

(3) 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

Can you name this tessellation? What type of tessellation is it (regular, semi-regular, other)?

3.4.3.12 cannot tessellate over infinite plane (other)

Sources:

Books-

Mathematics 1001. Dr. Richard Elwes pg 108-114

Mathematical Reasoning for Elementary Teachers. Calvin T. Long and Duane W. DeTemple, pg 861-868

Websites-

http://mathforum.org/sum95/suzanne/whattess.html

http://www.mathsisfun.com/geometry/tessellation.html