Creating Islamic Patterns from Folded Shapes

Year: 2009 Authors: E.B. Meenan; B.G. Thomas

Core claim

Paper folding offers a quick, accessible classroom alternative to compass-and-ruler constructions for making Islamic patterns while supporting cross-curricular learning.

Topics

Islamic patterns, paper folding, classroom geometry, cross-curricular art

Domains

geometry, symmetry, tessellations, two-dimensional shapes, Islamic art, pattern design, visual design, craft

Methods

folded-paper construction, workshop activities, pattern assembly, classroom discussion

Media

colored paper circles, A-size paper, backing paper or card, folding diagrams

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Bridges 2009: Mathematics, Music, Art, Architecture, Culture

Creating Islamic Patterns from Folded Shapes

E.B. Meenan* and B.G. Thomas School of Education* and School of Design University of Leeds Leeds, LS2 9JT e.b.meenan@leeds.ac.uk

Abstract

In this practical workshop participants will learn how to fold shapes including squares, equilateral triangles and hexagons from colored paper circles and A-size paper. The participants will then have the opportunity to make Islamic patterns for themselves using the folded shapes. This work can be extended to include participants designing and making their own patterns. There will also be opportunities to discuss implications for classroom practice, and ways in which the ideas can be used for students of varying interests and abilities.

Introduction

Geometry is at the heart of Islamic art. The designs and patterns use a finite number of geometric shapes that combine in many different ways. Traditionally, Islamic patterns are created through compass and ruler constructions but this method is often time consuming and can be complicated to oversee in the classroom. Students frequently do not exercise the precision required to achieve accurate results. An alternative method of creating Islamic patterns in the classroom is through manipulation of folded paper shapes cut from circles or A-size paper. Metric (A-sized) paper is mathematical paper system based on root 2 rectangles and is commonly used throughout the Europe, see [1, 2]. Squares, equilateral triangles, hexagons and rhombi can be quickly and easily folded from colored circles or A-size paper. Fixing these folded shapes onto large sheets of backing paper or card enables the quick assembly of eye-catching and colorful patterns. A photograph of pupils exploring how to create a tiling pattern from folded paper kites is shown in Figure 1.

Figure 1: Pupils exploring Islamic patterns using folded paper kites

Meenan and Thomas

The aim of this workshop is to ensure that participants become familiar with the methods for making a variety of folded shapes and using these to create Islamic patterns. This will act as a creative stimulus for designing other patterns, exploring the properties of the folded shapes and investigating how to fold other shapes. Participants will be encouraged to reflect on their own teaching which, in turn, will allow them to introduce new teaching strategies that help to:

• make cross-curricular connections between art and mathematics
• develop two-dimensional spatial awareness
• broaden students’ creative outlook and develop practical skills
• increase students’ motivation and enjoyment of mathematics

These activities assist in increasing pupils’ understanding of two-dimensional shapes, symmetry and pattern. The material in this workshop can also be combined into the teaching of a larger cross-curricular project across the art, religious education and mathematics departments, which could include topics such as symmetry, two-dimensional shapes, tessellations, paper folding and a study of Islam and Islamic geometric patterns. Further examples of patterns that can be created using folded paper shapes are available from [3].

A series of successful workshops have been delivered by Meenan and Thomas to school groups alongside an exhibition of tilings and polyhedra, presented at the University of Leeds [4]. Photographs from one such workshop are shown in Figure 2. A report on a workshop by the accompanying teacher and her class is available from [5].

Figure 2: Students in a practical paper-folding workshop

Geometry and Symbolism in Islamic Design

Patterns are symbolic in Islamic culture [6]. It is within the circle, the symbol of unity, that polygons, the building blocks of tiling patterns, are constructed. The circle has also been regarded as a symbol of eternity, without a beginning and without an end. The compass and ruler constructions of the three

Creating Islamic Patterns from Folded Shapes

fundamental shapes used in Islamic art – the equilateral triangle, the square and the hexagon – are shown in Figure 3.

Figure 3: Constructions of a) a triangle, b) a square and c) a hexagon

The equilateral triangle, the simplest regular polygon, consists of only three equal lines connected at three vertices. Two lines cannot enclose a plane – three are needed – and so three is the beginning. By tradition the triangle is a symbol of harmony and human consciousness. The square is often taken as the symbol to represent the earth and its four corners symbolically represent the four directions – north, south, east and west or the four states of matter – water/liquid, earth/solid, air/gas and fire/ether. The hexagon represents heaven [6]. Another symbol commonly found in Islamic decoration is the star. The star symbolizes equal distance in all directions from a central point. All stars, whether they have 6, 8, 10 or more points, can be created through division of a circle into equal parts. The center of the star is the center of the circle from which it was created, and its points touch the circumference of the circle. The rays of the star spread out in all directions making the star an appropriate symbol for the spread of Islam.

Repetition and variation are important aspects of Islamic design. A series of tiles may consist of only one or two shapes, although the patterns on the tiles themselves may be different. In other designs, a number of different shapes are combined to create a complex interlocking pattern, as shown in Figure 4.

Figure 4: A tiling pattern found at the Alcazar, Seville, Spain

In many Islamic patterns, different elements seem to dominate, depending on how the pattern is viewed. A simple example is shown in Figure 5. It can be considered to be a six-pointed star surrounded by six rhombi or as three rhombi surrounded by three six-pointed stars. For further examples and discussion of Islamic geometrical patterns and their constructions see [6, 7, 8].

Meenan and Thomas

Figure 5: Six-pointed star tiling

Creating Shapes and Tiling Patterns

Workshop participants will have the opportunity to fold two-dimensional shapes from colored paper circles or A-size paper. All the shapes are straightforward to accurately fold following the instructions and diagrams shown in Figures 6 - 9. Short video clips demonstrating the folding are available online at [9]. No previous experience is required and pleasing results can be achieved in a relatively short time.

a) Square

b) Equilateral triangle Figure 6: Folding instructions for a) a square an b) an equilateral triangle from ‘A’ size paper

Creating Islamic Patterns from Folded Shapes

a) Hexagon A size paper

fold in half, then in quarters and unfold

rotate paper

roll corners A and B along quarter fold lines to make lines AX and BX

rotate shape

fold edge to edge

fold edge to edge

fold corners to centre using quarter fold lines

flip over

regular hexagon

b) Octagon c) Kite

Figure 7: Folding instructions for a) a hexagon and b) an octagon and c) a kite from ‘A’ size paper

Meenan and Thomas

a) Square

Figure 8: Folding instructions for a) a square, b) an equilateral triangle and c) a rhombus from circles

Fold the circle in half. Open and fold the diameter back on itself to create two diameters at right angles.

Then fold in the four flaps as shown to a square.

b) Equilateral triangle

Mark the centre. (The centre can be found by folding the circle in half, then in half again.)

Fold in the edge to the centre.

Fold in the edge the centre again so that the new fold joins previous fold.

Fold in once again to the centre to create an equilateral triangle.

The flaps can be interleaved to give a triangle like this.

c) Rhombus

First fold two arcs to the centre.

The end of the third to the centre of the circle.

Do the same with the end of the other chord.

Tuck in the flaps. You now have a neat rhombus.

Creating Islamic Patterns from Folded Shapes

Regular hexagon

Fold in half and open, then fold across diameter and open

Fold across the vertical diameter to the centre

Do this at top and bottom

And unfold to give a circle with these creases

These creases can now be used to fold in the edge of the circle to create a hexagon which will have corners at A, B, C, D, E and F

Figure 9: Folding instructions for a hexagon from a circle

The finished hexagon

A selection of Islamic tiling patterns that can be made using folded shapes is illustrated in Figure 10. Decorative designs can be drawn onto the plain tiling patterns once constructed.

Figure 10: Islamic tiling patterns

Meenan and Thomas

Summary

This paper presents a novel approach to the teaching of two-dimensional shape and pattern. Setting these subjects in the cross-curricular context of Islamic art makes the topics more relevant to students. It has been observed by the authors that the students involved in the paper folding workshops have shown a deeper engagement with the subject. The material presented in this paper forms a component of a larger seminar, which was designed to promote the teaching of two and three-dimensional geometry in both a visual and practical way. This workshop was successfully targeted at 10 – 16 year old students of a wide range of abilities and was associated with the exhibition ‘Form, Shape and Space’ [1]. A freely available teacher’s booklet was developed to support this larger seminar and contains activity sheets on making polygons and polyhedra using compass and ruler constructions, as well as paper folding and pull-up net methods [10].

References

[1] K. Elam, Geometry of Design: Studies in Proportion and Composition, New York, Princeton Architectural Press. 2001. [2] W. Gibbs and E.B. Meenan, “Metric Paper Magic”, from TES Magazine [online]. Accessed 23/04/2009.] Available on the World Wide Web: <http: article.aspx?storycode=“346654” www.tes.co.uk=""> [3] W. Gibbs, “A collection of activities to help enrich mathematical learning”, from William’s Homepage [online]. Accessed 23/04/2009.] Available on the World Wide Web: <http: www.cyffredin.co.uk=""> [4] B.G. Thomas. Form, Shape and Space: An Exhibition of Tilings and Polyhedra, The University of Leeds International Textiles Archive, UK. 10 October 2007 – 16 May 2008. [5] C. Burland. “Aim in Leeds - Engaging Gifted & Talented Pupils in Mathematics Outside the Classroom”, from National Centre for Excellence in the Teaching of Mathematics [online]. [Accessed 25/02/2009.] Available on the World Wide Web: <http: default.aspx?page=“13&module=res&mode=100&resid=10069” www.ncetm.org.uk=""> [6] K. Critchlow, Islamic Patterns: An Analytical and Cosmological Approach, New York, Thames and Hudson. 1984. [7] J. Bourgoin, Arabic Geometrical Pattern Design, New York, Dover Publications. 1974. [8] D. Wade, Pattern in Islamic Art, Woodstock, New York, The Overlook Press. 1976. [9] E.B. Meenan. “MoreMathsGrads Videos”, from YouTube [online]. [Accessed 25/02/2009.] Available from the World Wide Web: <http: moremathsgrads="" user="" www.youtube.com=""> [10] E.B. Meenan and B.G. Thomas. “Form, Shape and Space: Teacher Booklet”. from National Centre for Excellence in the Teaching of Mathematics [online]. [Accessed 25/02/2009.] Available on the World Wide Web: <http: 271019="" files="" ulita+teachers+booklet.pdf="" www.ncetm.org.uk="">

Acknowledgments

The authors wish to thank the NCETM, particularly Tony Shepherd, for all their support and encouragement throughout this cross-curricular project. Thanks are also due to Margaret Chalmers at the University of Leeds International Textiles Archive and Anna Clapham at South Leeds City Learning Centre, for their support with the workshops and associated activities.</http:></http:></http:></http:></http:>