paper patterns. 4. paper weaving

Paper Patterns. 4. Paper WeavingAuthor(s): William GibbsSource: Mathematics in School, Vol. 19, No. 5 (Nov., 1990), pp. 16-19Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214727 .

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4

4 4

p leaving

by William Gibbs School of Education,

Leeds University

In all countries and in all cultures people have woven strips of leaf or bark or wool to create mats and baskets with intricate patterns and designs. This activity can be success- fully simulated with strips of coloured paper and by varying the colour and order of the horizontal and vertical strips the fascinating variety of patterns created can be investigated.

Start by considering the simple weave in which each strand goes over and under alternately. What pattern will be created if the horizontal strips are of one colour, and the vertical strips of another.

The result, alternating squares of each colour is not surprising.

Fig. 1

However if both the horizontal and the vertical strips alternate in colour then the result is surprising;

~wse N

Fig. 2

It is satisfying first to predict the pattern and then to weave it. To make it easy to weave, cut strips of equal width from coloured paper, arrange the vertical strips in the pattern required and tape them along the top to the table or other firm surface. The crossing strips can then be woven in quickly and easily. (Pre-cut coloured gummed strips de- signed for making paper chains can be bought from good stationery shops).

Here is another interesting pattern created using the simple weave. If the vertical strips are arranged in a repeating order, black, black, white and the horizontal strips are woven in the order black, white, white then a new tessellation emerges;

Fig. 3

Many of the patterns woven traditionally display repeating patterns involving symmetry and tessellation. Here for example is the "Star" pattern, or "Shepherds check", well known to weavers throughout the world;

Fig. 4 The Star pattern.

16 Mathematics in School, November 1990



How is this tessellation created? It is an interesting challenge to try and deduce the weaving pattern used. In fact double strands of each colour alternate in both directions but although this result can be derived logically there is nothing like actually weaving it to confirm ones reason- ing.

This and other patterns prompt the questions; Can other tessellations be created? Can the weavings needed be described mathemat- ically? Can a simple algorithm be derived to determine the colour of any square in the pattern? If we concentrate on patterns created using two colours

then the order of the horizontal and vertical strips can be binary coded. For example the "Star" pattern which had the strips arranged black, black, white, white can be coded as 1100 on both axes with 1 standing for black and 0 for white.

Fig. 5

Here is another tessellation for which the code is 10010110 on both axes;

Fig. 6

Looking at the structure of the code that created this tessellation notice that it has two properties. Firstly the code can be divided into two halves one of which is the "complement" of the other; 1001 and 0110. Secondly, the code has two lines of symmetry. As the code is repeated when making the pattern all codes of 8 elements taken from this sequence are equivalent; 1001011010010110... ..01011010.. ..... 10100101. These two codes are equivalent and both are symmetrical.

Using these two ideas and creating sequences that have symmetry and complementary halves it is possible to create any number of tessellations;

Fig. 7 The Log Cabin pattern.

The pattern woven here is sometimes known as the Log Cabin design. Try the Binary equivalents of 43860, 699732 or 11187540 for even more elaborate designs of the same kind.

A simple algorithm for predicting the colour of any square in the pattern can be derived by considering the nature of the woven pattern. If the (1,1) square has the first vertical strip on top then so will (1,3), (1,5), (2,2), (3,5), (4,4) .....In general if (x + y) is even then the vertical strip will be on top, if (x + y) is odd then the horizontal strip will cover the vertical.

6

5

4

3

1 2. , 4 5i 6 "7 Fig. 8

The colour of the strip will depend on the modular nature of the code being used. For example a pattern with strips arranged in the order black, black, white (ie with code 110) can be considered as modular 3. The coordinates now have to be converted to this mode to discover the correct colour. For example consider the square represented by (8,6); 8 + 6 = 14. The result is even so the vertical strip is on top. Mode 3 is 2. As the code is 110 the 2nd strip is "1". The square is thus black.

This analysis of the weave provides an excellent basis for creating a computer program which will generate weaving patterns and allow for the investigation of tessellations.

Mathematics in School, November 1990 17



Fig. 9

To widen the investigation still further consider weaves with more than 2 colours. Here for example is another tessellation created using strips of three different colours arranged in sequence; shown above as Fig. 9.

Furthermore, the investigation can be extended to consider different types of weaving. So far the simple alternating "under-over" pattern has been the basis of all patterns. Most weaves in practice are more complicated than this. For example the "twill" weave involves weaving under and over two strips at a time; shown alongside as Fig. 10. Fig. 10 The Twill weave.

18 Mathematics in School, November 1990



Finally, consider patterns woven with three colours intersecting at 120 degrees. Here are three examples. The first is a common weave used to create fish traps. The second and third are basket weaves from India. The second weave presents a real challenge to recreate. Each colour goes under two and over one strip of colour and over two and under one of the other colour.

Fig. 11(a)

Fig. 11(b)

Fig. 11(c) The Diamond Weave.

Fig. 12 How the Diamond Weave is woven.

>LIST 10 MODE 5 20 PRINT "ENTER COLOURS AS" 25 PRINT "NUMBERS" 30 AA= 2 40 INPUT NN 50 INPUT PP 52 PRINT "SIZE OF UNIT?" 53 INPUT U 56 L= 1200/U:M= 800/U 60 N=NN:P=PP 80 FOR XX=0 TO L 90 GOSUB 600

100 P=PP 110 FOR YY= 0 TO M 120 GOSUB 650 130 TT= XX + YY 140 IF INT (TT/2)*2=TT GOTO 170 150 R=YC 160 GOTO 180 170 R=XC 180 GCOL 0,R+1 190 A=U 210 PLOT 69,A*XX,A*YY 220 DRAW (A*XX),(A*(YY +1)) 230 PLOT 85, (A*(XX+1)),(A*YY) 240 PLOT 85, (A*((XX+1)),(A*(YY+1)) 530 IF INKEY (-49)= -1 THEN GOTO 580 540 NEXT YY 550 NEXT XX 560 IF INKEY (-49)= -1 THEN GOTO 580 570 GOTO 560 580 CLS 590 GOTO 10 600 REM X COL 610 XC= N - (I NT(N/AA)*AA) 620 N= (N - XC)/AA 630 IF N=O THEN N=NN 640 RETURN 650 REM Y COL 660 YC= P- (INT(P/AA)*AA) 670 P= (P - YC)/AA 680 IF P= O THEN P=PP 700 RETURN

In this program the Binary weaving code is entered in its base ten equivalent. Thus to weave a 110 arrangement in both directions enter 6 and 6.

Mathematics in School, November 1990 19



paper patterns. 4. paper weaving

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