Tessellations, Repetition and Design

Tessellations Ch 4, page 1

Tessellations, Repetition and Design

Repetition and Design

Repetition is a key concept in mathematics. For example, when we measure we repeat the unit joining them together. When we cover areas we repeat the unit. The unit is joined without gaps and without change of size.

Repetition is also a key concept in design. Repetition is used for effect in art for aesthetic reasons and to strengthen some aspect or mood which is being portrayed. For example, the repetition of rectangular grey buildings and bare trees can give a drab, dull look to a painting of a city. Repetition is used in architecture because of the ease of repeating features such as rectangular windows but also to make a building cohesive.

Traditional cultures use repetition for several reasons. Symmetrical and repeated designs are easy to repeat when weaving.

Designs from Tongan tapa

These designs are first made with flexible palm fronds as a kupesi on boards and then marked onto the tapa. In Tonga the tapa is laid on top and the design rubbed on. This is then strengthened by painting. People work as a group to make large tapa cloths. The squares are varied to give different effects. Variations can be obtained by using colour in different places.

Figure 1. Tongan tapa designs.

Design From China

The following design is made by halving and halving an isosceles right-angled triangle found in the tangram set.

Figure 2. Repeating half triangles.

Woven Patterns

Weaving is common in many cultures. The following designs come from PNG.

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Weaving patterns from England

Figure 3. Woven mats from PNG and English weave patterns.

Patterns from Circles

Other repeated designs can be seen in the making of Paisley variations.

Figure 4. Paisley designs using circles.

Islamic and Arabic Designs

While most Islamic and Arabic designs are developed from circular patterns, they frequently have one or more shapes tessellating.

Figure 5. Islamic and Arabic art.

Tessellations

Tessellations are tilings of the same shape. The tiles join together without gaps or overlaps and with a pattern that allows the tessellation to continue in both directions infinitely. If two shapes are used, it is called a semi-tessellation. Tessellations work because angles at a point add up to 360°. Certain regular polygons will tessellate and other special polygons will also tessellate. Triangles and quadrilaterals can tessellate as well as other shapes.

Often the special shapes that tessellate are actually based on transferring a section of a tessellating shape from one side to another so that the area is conserved. These are often called nibble tessellations and they were used as starting points in Escher’s art.

When two regular shapes are used together to cover space, they are called semi-regular tessellations. Tessellations are important as they form the basis of area measurements, they are needed to make walls and floors of buildings, they are found in nature, and they can be used effectively in artistic creations. Many cultures have developed interesting tessellating and other patterns.

Tiles with certain designs can be tessellated to produce a myriad of interesting patterns and continuous curve designs. For example, see Figure 6 below.

Tessellations of 2D shapes are a way of considering area. Tessellations of 3D shapes are a way of considering volume.

Figure 6. A tile that will provide interesting tessellations.

Learning Tasks for the Reader

Tessellation Activities

/ ·  From using pattern block tiles, what shapes do you know tessellate? Are they all regular?
·  Use small right-angled isosceles triangles to make one of the Tongan designs and then make up your own.
·  Discuss the angles in shapes that tessellate. Particularly develop a range of patterns for rectangles and then discuss some of the others.
·  Investigate whether all triangles will tessellate and why. Cut out at least 10 scalene triangles that are all congruent. Try tessellating these triangles by carefully following a pattern of placing them by slides and rotations and by matching the side lengths. What do the angles at any point add up to? Take one of the triangles, tear of the corners and place them together to form a straight line. See chapter 2 for the movements needed to try and tessellate shapes.
·  Investigate whether all quadrilaterals will tessellate.
·  What are semi-tessellations? Illustrate.
·  Use the pentomino shapes. Do they tessellate?
·  Do regular pentagons tessellate? Do any pentagons tessellate (try the house shape and look for the special ones on the web)
·  Take a square card and cut out a section. Tape this onto the opposite side. Trace around your new template and make your nibbled square tessellation. Try a few nibbles and make something artistic. Try working with equilateral triangles and rectangles.
·  Look at the page of designs from different cultures and find out how they are developed.
·  Which of the pentomino tiles tessellate? How do you know?
·  Look at Escher’s work .
/ ·  Why do triangles and quadrilaterals tessellate?
·  Why don’t regular pentagons tessellate?
·  Define a tessellation.
·  Define a semi-regular tessellation. Illustrate some.
/ ·  Give some of the lessons at the end of the chapter.
·  Why is it important to teach about tessellations?
·  Where are they mentioned in the syllabus?

Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics Kay Owens