Tessellations

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: What is a tessellation?

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

X-ray Crystallography This picture depicts the transformation of eight points in an array to make a very small crystal lattice, which tessellates. Study of Science

This image suggests the relationship between tessellations, symmetry, and x-ray crystallograpy. more science….

Hydrogen Peroxide, among other chemicals, has this tessellating, crystal structure. and even more science….

The mathematics of the Regular Tessellation... A regular polygon tessellation is constructed from regular polygons.polygon Regular polygons have equal sides and equal angles.Regular polygons 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.

A famous mathematician, Kepler studied tessellations and noted the regular tessellations or (tilings) of the plane.Kepler There are only three regular tessellations; one of triangles one of squares and one of hexagons.

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

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.

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

Let’s try together….. a translation or slide….

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

Let’s try together….. a reflection, flip or mirror image

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

Let’s try together….. a rotation or turn