mathematics and modulo art (pdf, 3.6mb)

YuMi Deadly Maths Past Project Resource

Mathematics and Modulo Art

YUMI DEADLY CENTRE School of Curriculum

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Acknowledgement

We acknowledge the traditional owners and custodians of the lands in which the mathematics ideas for this resource were developed, refined and presented in professional development sessions.

YuMi Deadly Centre

The YuMi Deadly Centre is a Research Centre within the Faculty of Education at Queensland University of Technology which aims to improve the mathematics learning, employment and life chances of Aboriginal and Torres Strait Islander and low socio-economic status students at early childhood, primary and secondary levels, in vocational education and training courses, and through a focus on community within schools and neighbourhoods.

“YuMi” is a Torres Strait Islander word meaning “you and me” but is used here with permission from the Torres Strait Islanders’ Regional Education Council to mean working together as a community for the betterment of education for all. “Deadly” is an Aboriginal word used widely across Australia to mean smart in terms of being the best one can be in learning and life.

YuMi Deadly Centre’s motif was developed by Blacklines to depict learning, empowerment, and growth within country/community. The three key elements are the individual (represented by the inner seed), the community (represented by the leaf), and the journey/pathway of learning (represented by the curved line which winds around and up through the leaf). As such, the motif illustrates the YuMi Deadly Centre’s vision: Growing community through education.

More information about the YuMi Deadly Centre can be found at http://ydc.qut.edu.au and staff can be contacted at [email protected].

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© QUT YuMi Deadly Centre Electronic edition 2013

School of Curriculum QUT Faculty of Education

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Queensland University of Technology

MATHEMATICS AND MODULO ART

Tom Cooper – QUT

Tony Watson – Edith Cowan University

YuMi Deadly Maths Past Project Resource © 2013 QUT YuMi Deadly Centre

Page iii Mathematics and Modulo Art

CONTENTS

PAGE

MODULO ART – WORKSHEET 1 ................................................................. 1

MODULO ART – WORKSHEET 2 ................................................................. 7

MODULO ART – WORKSHEET 3 ............................................................... 10







Mathematics and Modulo Art Page 1

INTRODUCTORY THEORY TO MODULO ARITHMETIC Any pair of integers is said to be congruent modulo 5 if they differ by a multiple of 5. For example, 3 and 18 are congruent, modulo 5, because 3 and 18 differ by 15, which is a multiple of 5. We write this as:

3 18 (mod 5)

1. Give four other pairs of integers that are congruent, modulo 5: (Check your answers with your teacher.) 2. Are the following pairs of integers congruent, modulo 5? i. 11, 1 YES / NO

ii. 0, 19 YES / NO

iii. -12, 13 YES / NO

iv. 6, 6 YES / NO Circle YES or NO in each case, and check your answers at the end of this sheet. 3. In general, two integers a and b are said to be congruent modulo an integer n, if and

only if the difference between a and b is a multiple of n. This is written as:

a b (mod n)

Examples: i. 2 9 (mod 7) since the difference between 2 and 9 is 7 and this is a multiple of 7.

ii. 47 11 (mod 6) since the difference between 47 and 11 is 36 and this is a multiple of 6.

iii. 21 21 (mod 79) since the difference between 21 and 21 is 0 and this is a multiple of 79 (as 0 = 0 x 79).

MODULO ART – WORKSHEET 1

’ ’

’

’



Circle TRUE or FALSE for each of the following:

i. 1 6 (mod 4) TRUE FALSE

ii. 21 11 (mod 5) TRUE FALSE

iii. 168 -32 (mod 25) TRUE FALSE

iv. - 68 -59 (mod 2) TRUE FALSE (Check your answers at the end of this sheet.)

4. Find three integers congruent to:

i. 7 (mod 4) ii. -8 (mod 6) iii. 0 (mod 29)

(Check your answers with your teacher.)

5. Complete the following:

i. 8 (mod 6) ii. -7 (mod 11)

iii. 10 -6 (mod ) iv. 11 27 (mod ) (Check your answers with your teacher.)

6. Find two integers modulo to which the following pairs of integers are congruent:

i. 1, 11 ii. -7, 26 iii. 16, 128

(Check your answers with your teacher.)



7. The most important congruence for our purposes is the congruence (mod n) between a number and its remainder after division by n. For example:

21 3 (mod 6) Because 3 is the remainder after dividing 21 by 6

Likewise, 27 1 (mod 13)

And 28 0 (mod 7) Using this congruence we can construct modular addition tables and modular multiplication tables. Tables modulo 4 are given below:

+ 0 1 2 3 x 0 1 2 3 x 1 2 3

0 0 1 2 3 0 0 0 0 0 1 1 2 3

1 1 2 3 0 1 0 1 2 3 2 2 0 2

2 2 3 0 1 2 0 2 0 2 3 3 2 1

3 3 0 1 2 3 0 3 2 1

Addition (mod 4) Multiplication Multiplication (mod 4) with zero (mod 4) without zero In these illustrations, the integer 1 appears in the fourth row of the grid of the addition (mod 4) table because:

3 + 2 = 5 1 (mod 4) The integer 0 appears in the second row of the grid of the multiplication (mod 4) without zero table because:

2 x 2 = 4 0 (mod 4) Similarly for all other integers in the grids. Make sure you see how these tables are obtained. Note: The grid of the tables is the part within the lines. That is, for the multiplication (mod 4) Table without zero, the grid is shown on the right. This is called a 3 x 3 grid.

1 2 3

2 0 2

3 2 1



8. Construct the tables (mod 5) below:

+ 0 1 2 3 4 x 0 1 2 3 4 x 1 2 3 4

0 0 1

1 1 2

2 2 3

3 3 4

4 4

(See the end of this sheet for solutions.) In general, modular tables (mod n) are constructed from the addition or multiplication of the integers 0, 1, 2, …, n-1 followed by finding the remainder after division by n. Multiplication tables (mod n) may be constructed without the zero.

9. Complete the three tables following:

i.

Multiplication (mod 21) without zero



ii. Addition (mod 12) iii.

Multiplication (mod 19) without zero (Check your answer with your teacher.)



10. i. Shade or colour all the squares containing a 1 in each of the above tables.

ii. Is the resulting pattern pleasing?

iii. Using a different colour, shade all the squares containing a 0 in the above tables.

iv. Construct a larger table and repeat steps i. and ii. with your larger table. 11. i. Can you detect any difference in pattern between:

(a) the shading for all the addition tables and the multiplication (mod 5 and mod 19)

tables without zero; and (b) the shading for the tables other than those above?

ii. Would this difference be the same if we only shaded the squares containing a 1 ?

iii. Is there a difference for any other numbers?

iv. Can you generalise a property for all numbers that distinguishes: (a) the addition tables and the multiplication (mod 5 and 19) tables without zero, from (b) the rest?

v. Are there any other differences between (a) and (b) above? 12. ANSWERS TO WORKSHEET ONE

2. i. YES ii. NO

iii. YES iv. NO

3. i. FALSE ii. TRUE

iii. TRUE iv. FALSE



GENERATING PATTERNS 1. Here is a 4 x 4 grid. Try to place 0, 1, 2 and 3 in the grid so that each row and each column has just one 0, one 1, one 2, and one 3.

There are many ways in which this can be done. Grids like this are called Latin Squares. (You may even find that your Latin Square is also a magic square – or nearly.)

2. Is this the difference you found between tables of type (a) and (b) in question 9? (on worksheet 1). 3. These Latin Squares can be used to make mathematical poster designs. This is how:

(i) Choose a simple design, perhaps like (ii) By using different colourings or shading different parts we can let this design represent the numbers 0, 1, 2 and 3. For example: 0 could become 1 could become 2 could become 3 could become

(iii) Now simply replace the numbers in our Latin square with their corresponding designs. For example:

This becomes

Latin Square Pattern 1

The grid on which this pattern appears is called a standard grid. Make sure that you see how the pattern was obtained. Note: The grid that we used here was the addition (mod 4) table. Did you recognise

it? Using modular tables is one method of obtaining Latin Squares. In fact, all addition (mod n ) tables are Latin Squares. Also all multiplication (mod p) tables without zero, where p is a prime number, are Latin Squares.


0 1 2 3

1 2 3 0

2 3 0 1

3 0 1 2



4. Use your own colours on the same Latin Square grid on the same design to make your own pattern from a 4 x 4 grid:

Latin Square Pattern The patterns obtained in the above way may be repeated, rotated or reflected to form an attractive poster design. For example, see what happens to Pattern 1 when it is (i) repeated: (ii) rotated 90°

(iii) reflected

Pattern 1

repeat

repeat repeat

Pattern 1

rotated

ro

tate

d

Pattern 1

reflect

reflect reflect



5. Notice in the above example that reflection and rotation give the same poster. Repeat, rotate and reflect your pattern:

Pattern repeat Repeat repeat

Pattern rotate Rotate rotate Pattern reflect Reflect reflect



PATTERN VARIATION

1. Suppose we use the same design for 0, 1, 2 and 3 as we did for pattern 1, but change the Latin Square. This is what happens:

Design Latin Square Pattern

repeated rotated reflected

Note: For these examples reflection gives a different poster to rotation.

2. The basis for number designs can be different to the one we have used so far:

Some other design forms are:

Different designs for different numbers can be produced by: reflecting: rotating:


0 2

1 3

0 1 2 3

3 2 1 0

2 3 0 1

1 0 3 2



3. Experiment with designs of your own. Some grids are given below. You might like to try one of the following design forms:

(a) (b) (c) (d) (i) repeat

0 1

2 3 Latin Square Repeated Pattern



(ii) rotate

0 1

2 3

Latin Square Rotated Pattern (iii) reflect

0 1

2 3 Latin Square Reflected Pattern



4. Of course grids and Latin Squares need not only be 4 x 4. They can be 8 x 8 or 5 x 5, or any size you wish (as long as each number in the square appears once and only once in each row and column).

Using the design for the numbers 0 to 6 and the Latin square given below, complete the pattern on the 6 x 6 standard grid. (This Latin square is the addition (mod 7) table.) (Try using different colours).

0 1 2

3 4 5

6

0 1 2 3 4 5 6

1 2 3 4 5 6 0

2 3 4 5 6 0 1

3 4 5 6 0 1 2

4 5 6 0 1 2 3

5 6 0 1 2 3 4

6 0 1 2 3 4 5



5. Using the design for the numbers 0 to 4 and the Latin Square given below, complete the reflected pattern.

0 1 2 3 4

Pattern (standard 5 x 5 grid reflected)

0 1 2 3 4

3 4 0 1 2

1 2 3 4 0

4 0 1 2 3

2 3 4 0 1



6. The designs need not be ones in which sections are coloured in. They can be anything your imagination can think up. And the patterns can be rotated on a corner.

Using the design and Latin square below, complete the repeated pattern.

0 1 2 3

Pattern (standard 4 x 4 grid repeated)

0 1 2 3

3 0 1 2

2 3 0 1

1 2 3 0



7. Using the design and Latin Square given below, complete the repeated pattern which has been turned on one corner.

Pattern (standard 4 x 4 grid repeated and rotated 45° )

0 1 2 3

1 3 0 2

2 0 3 1

3 2 1 0



8. Try a pattern of your own.

Of course a grid may be repeated (or rotated or reflected) more than 3 times. In the above example a 4 x 4 grid is repeated to get the 8 x 8 pattern which in turn is rotated to get the 16 x 16 pattern.

9. Make a large poster of your own design using these methods. (Do a small example first to see if

you like the pattern.) A design made by us follows.



Here is a Lift Out mathematics poster for the classroom wall, drawn from the following Modulo Art design.

1

2

3

4

1 2 3 4

4 1 2 3

3 4 1 2

2 3 4 1



Patterns similar to the examples in Part I can be based on modular addition or multiplication tables instead of Latin Squares. 1. (i) Construct a 4 x 4 grid for multiplication (mod 4) with zero: X 0 1 2 3 0 1

2

3

(ii) Replace the numbers 0 1 2 3 in the table above with the following designs and complete the table below using rotation once you have generated the basic 4 x 4 grid. x 1 2 3 4 5




2. (i) Construct the 5 x 5 grid for multiplication 1

(mod 6) without zero (i.e. no zero column or 2

Row in the grid – see diagram). 3

4

5 (ii) Replace the numbers in

the table with the following designs and, using the grid below, complete the mathematical poster. 0 1 2 3 4 5

3. Construct the multiplication table (mod 6) with zero, replace the numbers with designs of your choice and

reflect to make a larger pattern.

4. Construct the addition table (mod 4) without zero, replace the numbers with designs of your choice and repeat to make a larger pattern.



CONVERGING SEGMENT GRIDS FOR POSTER DESIGN Interesting effects can be achieved with variations from a non-standard grid. (The standard grids have all cells equal in area.) The converging segment grid, as its name implies, consists of cells whose areas converge. 1. A common variation is achieved by doubling the width of each row and column from the first square. e.g. Converging segment grid: ( 4 x 4 case )

8a 4a 2a a

2. A variation based on the divine proportion principle is shown below. The original square of side ‘a’ has a

diagonal of 2a, which forms the width of the second row and column. The diagonal of the square now formed is used to create the third column and row, etc.

Step 1 Step 2 Step 3

diagonal 2a new diagonal for

width of third colum

You may find a compass very useful here. 0

3. Using the given designs, replace the numbers in the given Latin Square with the designs on the following 1 converging segment grid. Be careful!

2

3

Check your pattern at the end of this worksheet before continuing.


8a 4a 2a a

a

2a

a new diagonal for width of

fourth column

0 1 2 3

1 2 3 0

2 3 0 1

3 0 1 2



4. Now reflect your pattern for Exercise 3: Check your pattern at the end of this worksheet. 5. Try rotating the following 5 x 5 converging segment grid using the given design and Latin Square.

4

6. Try to make up your own large grid and colour it brightly.

0

2

1

3

0 1 2 3 4

1 2 3 4 0

2 3 4 0 1

3 4 0 1 2

4 0 1 2 3



Solution to Exercise 3:

Solution to Exercise 4:



KALEIDOSCOPIC GRIDS FOR POSTER DEGISN The kaleidoscopic grid produces a very effective pattern when rotated and reflected. Two examples appear below: (i) Kaleidoscopic grid ( 4 x 4 case ) (ii) Circular kaleidoscopic grid ( 4 x 4 case )

This grid uses radii which divide the square side into equal proportions.

a a a

Fitting the patterns from a square into the cells of a kaleidoscopic grid can present some problems. To shade in the quadrilaterals a method similar to the method for the square cells is used; however, the triangles are more difficult. It is made somewhat easier by considering the triangle as half a square. For example: To illustrate this we can use the Latin Square from Worksheet 5, question 3, on a kaleidoscopic grid.


is represented

on as

, i.e.

and is

represented on as , i.e.



If this design is reflected, the following results:

What would have been the result if the design had been rotated instead of reflected? Try it and see. 1. Construct a multiplication table (mod 5) without zero. Select your pattern designs for the digits and

complete the first quadrant of the kaleidoscopic grid below. Then rotate and reflect the designs to complete the whole poster. Before you complete the whole poster check the solution at the end of this worksheet to make sure that you allocated the correct digit to each cell.



2. Complete the following circular kaleidoscopic pattern using your own design and Latin Square modular table.



3. Try making a large poster using your own Latin Square or modular table for your classroom wall. Don’t be restricted to only coloured designs or 4 x 4 grids. There are lots of possibilities.

Here are two extra grid possibilities:

Solution for allocation of digits to question 1 of the worksheet using multiplication and mod 5 without zero on a kaleidoscopic grid:



CIRCULAR RESIDUE DESIGNS To construct circular modulo patterns we return to our modular tables. As an example, consider the ninth row of the multiplication table (mod 19) without zero. For this row:

9 x 1 = 9, 9 x 2 = 18, 9 x 3 = 27 8 (mod 19), and so on.

The final number reached is called the residue of the respective multiple of 9. 1. Complete the table below:

x 9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

residue (mod 19)

9

18

8

Next draw a circle and divide its circumference evenly into 18 sections, numbered 1 to 18. (As mod 19 is being considered.) Join 1 to 9

2 to 18 3 to 8 and so on, as in the table above.

2. Complete the chords in the above circle, according to the table.

3. The chords now divide the circle into sections. Complete the pattern by colouring in alternate

areas – in this case the triangular areas. See the end of this worksheet for the solution. This pattern is called a (19, 9) residue design. In general (n, m) residue designs are constructed by dividing the circumference of a circle into n-1 equal arcs, labelling the points of division 1, 2, . . ., n-1 Drawing a chord from each point to its m-multiple (mod n), and shading in alternate areas produces the residue design.




4. Now it’s your turn. Complete a (21, 10) residue design as follows:

(a) Complete the following table

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

x 10 10 20 30 40 200

residue (mod 21)

10

20

9 11

(b) Draw chords from 1 to 10, 2 to 20, 3 to 9, . . ., 20 to 11. (c) Shade alternate areas (in this example shade the triangles.)



5. Complete a (19, 2) design on the following circle. Be careful! The alternate areas are not triangles in this case. Try using more than one colour to shade the regions.

(See the end of this sheet for solution.)

6. Use the circle below to construct another residue design, modulo 21:



Solution to Question 1:

x 9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

residue (mod 19)

9

18

8

17

7

16

6

15

5

14

4

13

3

12

2

11

1

10

Solution to Question 5:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

x 2 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

residue (mod 19)

2

4

6

8

10

12

14

16

18

20

3

5

7

9

11

13

15

17



MORE RESIDUE DESIGNS The final worksheet contains a number of exercises on circular residue designs. 1. Complete the following residue designs. Try using more than one colour.

(a) (7, 3) - divide the circle into 6 parts using a compass of arc equal to the length of the radius.

(b) (13, 5) - divide the circle into 12 parts using a compass of arc equal to half the length of the radius.

(c) (19, 18) - divide the circle into 18 parts using a compass of arc equal to one-third the length of the radius.

As the number of equal arcs increases so does the complexity of practically constructing the design. The necessity of colouring alternate areas to produce a pleasing pattern is no longer necessary. For example, consider the following ( 65, 2 ) residue designs: To overcome practical problems, it is best to decide to divide the circle into 32 or 64 parts; or 48 or 96 parts. Dividing into 32 or 64 parts can be achieved by dividing the circle into 2 parts (with a diameter), then halving each arc (4 parts), halving again (8 parts), having again (16 parts) halving again (32 parts) and halving again (64 parts). Dividing into 48 or 96 parts can be achieved by dividing the circle into 6 parts (using the radius), halving each arc (12 parts), halving again (24 parts), halving again (48 parts) and halving again (96 parts).

2. Try some of these circular residue designs: (a) (17, 9) - use colour

(b) (65, 3) - colour unnecessary

(c) (97, 3) - colour unnecessary




3. Try to produce a large circular poster.

If you are using a large number or arcs you can produce a large poster on wood by putting nails at each point of division and using string for the arcs. (A form of string sculpture will result.)

4. Some very beautiful patterns have been produced by rotating, reflecting or repeating circular patterns (not necessary a 90° rotation each time) or by changing the circle to a quadrant and

then rotating the quadrant. Try one of these.

Experiment – try combining square and circular designs. Try using wood, nails and coloured string. You now have the mathematical bases – the only limit is your own imagination!

REFERENCES Forseth, S. and Troutman, A., “Using Mathematical Structures to Generate Artistic designs”, The

Mathematics Teacher, May 1974, pp. 393-397. Locke, P., “Residue Designs”, The Mathematics Teacher, March 1972, pp. 260-263. What is the difference between method and device? A method is a device which you can use twice.

G. Polya

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