Geometry, Patterns and Art Tessellations Geometry, Patterns and Art

What is a Tessellation? Write down everything that comes to your mind when you think of the word tessellation and look at these pictures.

Tessellations Video 30 minutes Complete Anticipation Guide prior to viewing and after viewing

Form or arrange small squares in a checked or mosaic pattern

Tessellations can be extended in the plane infinitely in every direction

Tessellation From the Latin tessella, which was the small, square stone or tile used in ancient Roman mosaics. Plane Tessellation – a pattern made up of one or more shapes, completely covering a surface without any gaps or overlaps.

Historical References Moorish Artists Floors, walls, ceilings Pottery, tapestries, carpets Metal work, wood carvings, stained glass Crochet, lace work, quilts Architecture

Interesting Variations M.C. Escher

M.C. Escher Dutch Inspired by Moorish mosaics in the Alhambra in Granada Altered geometric shapes into birds, fish, reptiles, people Made woodcuts 1898 - 1972

Op Art Designs Created by contemporary artists from the 1960’s such as Victor Vasrely and Bridget Riley. Geometrically precise patterns that vibrate and flicker

Scavenger Hunt Go on-line and find an example of a tessellation used for some practical purpose – for a floor, building, textile, etc. Bookmark it for reference to use later on. Answer Question 1 on the handout.

Regular Polygons Three regular polygons tessellate in the Euclidean plane (2D or Flat) Triangles Squares Hexagons ( 6 sides)

Basic Tessellation Samples Which figures line up? Which do not? What happens if you combine 6 triangles?

Interior Angle Measure Shape Angle Measure in degrees Triangle 60 Square 90 Pentagon 108 Hexagon 120 More than six sides More than 120 degrees Why don’t other figures tessellate or are able to tile the plane?

Interior Measure The sum, s, of the measures of the interior angle of a regular polygon with n sides is given by s = (n - 2) 180º Triangle: s = (3 - 2) 180º = 180º Square: s= (4 - 2) 180º = 360º Hexagon: s = (6 - 2) 180º = 720º How do you find the measure of each interior angle for a regular triangle, square and hexagon?

Answer to Question The interior angle must be an exact divisor of 360 degrees. For all the others, the interior angles are not exact divisors of 360 degrees (a circle), and therefore the figures cannot tile the plane

Scavenger Hunt Worksheet Answer questions 2 - 5 on the Scavenger Hunt handout.

Naming a Tessellation A tessellation of square is named “4.4.4.4”. Choose a vertex, then look at one of polygons that touches the vertex. How many sides does it have? Move around the vertex in either direction, finding the number of sides of the polygons until you get back to the polygon you started with. How many polygons did you count?

Naming a Tessellation

Name a Tessellation of regular congruent hexagons and triangles

Semi-Regular Tessellations Definition: Formed by regular polygons Arrangement of polygons at every vertex point is identical There are 8 Semi-regular tessellations

Semi-regular Tessellations Examples

Basic Tessellation Information Handout All the information from the previous slides and additional examples of semi-regular polygons are on this handout. http://mathforum.org/sum95/suzanne/whattess.html

Scavenger Hunt Worksheet Complete Questions 6 & 7

Basic Tessellations Practice Create some basic tessellations on this website. http://shodor.org/interactivate/activities/Tessellate/?version=1.6.0_06&browser=MSIE&vendor=Sun_Microsystems_Inc.

Symmetries Symmetries create patterns that help us organize our world conceptually. Symmetric patterns occur in nature, and are invented by artists, craftspeople, musicians, choreographers, and mathematicians. In mathematics, the idea of symmetry gives us a precise way to think about this subject.

Plane Symmetries Those that take place on a flat plane, but the ideas generalize to spatial symmetries too. Plane symmetry involves moving all points around the plane so that their positions relative to each other remain the same, although their absolute positions may change. Symmetries preserve distances, angles, sizes, and shapes.

Rotation, Translation, Reflection, & Glide Reflection A pattern is symmetric if there is at least one symmetry that leaves the pattern unchanged. Next slides are an overview of rotation, translation, reflection, and glide reflection including diagrams illustrating the four symmetries.

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".

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

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.

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

Scavenger Hunt Worksheet Answer questions 8 & 9 on the Scavenger Hunt worksheet.

Problem 1 Make a symmetric pattern by starting with an asymmetric shape (a letter is fine) and repeating a single translation over and over (also translate it backwards). That is, decide on a direction and distance for your translation (for example, 5 cm to the right). Translate your letter 5 cm to the right, then translate the new letter 5 cm to the right, etc. Also translate the original letter 5 cm. to the left, etc. Did you get any other types of symmetries (reflections, glide reflections, or rotations) in the process?

Problem 2 Make a symmetric pattern by starting with an asymmetric shape and repeating a single glide reflection over and over (also glide it backwards). That is, pick a reflection line and a translation in a direction parallel to the reflection line. Keep applying the same glide reflection to the new shapes that you generate until you run out of paper. Did you get any other types of symmetries (reflections, translations, or rotations) in the process?

Final Activity Create your own tessellation. Electronic or on paper Use the print resource material Use internet resources Write a paragraph that describes the purpose of your tessellation, the polygons use, name, the symmetries and the process you used to create it.

Closure Using the puzzle pieces discuss the geometry of tessellations Triangle Hexagon Escher