Tessellations

What is a tessellation?

• Repeating shapes or patterns that cover a surface without gaps or overlaps.

• The shapes in a tessellation can be geometric like squares and triangles, or shaped like animals and people. Surfaces can be square like a page or round like a CD. It can be flat or cover a 3D object like a lampshade, shoebox, or soccer ball.

The Three Regular Tessellations'Regular' means that all sides are equal, and internal angles are equal. The 3 regular geometric shapes that

tessellate are the equilateral triangle, square and hexagon.

Semi-Regular Tessellations• Archimedean tessellations have two properties:

– They are formed by two or more types of regular polygon, each with the same side length

– Each vertex has the same pattern of polygons around it.

Tessellations in Nature

Man-Made Tessellations

M.C. Escher

• born in Leeuwarden in Holland on June 17th, 1898

• youngest of 4 brothers• usually referred to by his initials which stand

for Maurits Cornelis• began drawing pen and paper art • visited the Alhambra where he became inspired to tessellate

David Bailey's World of Escher-like Tessellations

Symmetry

• A figure, picture, or pattern is said to be symmetric if there is at least one symmetry that leaves the figure unchanged.

• For example, the letters in

ATOYOTA• These form a symmetric pattern: if you draw a

vertical line through the center of the "Y" and then reflect the entire phrase across the line, the left side becomes the right side and vice versa. The picture doesn't change.

• There are FOUR types of symmetry.

Rotation

• To rotate an object means to turn it around. Every rotation has a center and an angle.

• http://illuminations.nctm.org/LessonDetail.aspx?ID=U138

Translation

• To translate an object means to move it without rotating or reflecting it. Every translation has a direction and a distance.

• http://illuminations.nctm.org/LessonDetail.aspx?ID=L474

Reflection

• To reflect an object means to produce its mirror image. Every reflection has a mirror line. A reflection of an "R" is a backwards "R".

• http://www.mathsisfun.com/geometry/symmetry-reflection.html

Glide Reflection• A glide reflection combines a reflection with a

translation along the direction of the mirror line. Glide reflections are the only type of symmetry that involve more than one step.

• http://illuminations.nctm.org/LessonDetail.aspx?ID=L475

The Alphabet Classify all the capital letters in English (in their simplest forms) according to their symmetries. For example, "A"

has a reflection in a vertical line, and "R" has no symmetry

A B C D E F G H I J

K L M N O P Q

R S T U V W X Y Z