Mathematical Art

A 40 point circle mapping

A drawing to download and try from theMathematics Department @

www.knightswoodsecondary.org

n 7n + 10 (mod 40)

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Right click the circle image from our download web page and save it to your hard drive. Open the image (it’s a jpeg) in a graphics program (download IrfanView using the link on the maths web page if you don’t have a suitable graphics program). Print out an A4 (portrait) size copy of the image on your printer. Now just follow these instructions!

n 7n + 10 (mod 40)

Write in the mapping here

Look at the table at the foot of your printout

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Look at the mapping closely

n 7n + 10 (mod 40)

For n = 1, you do 7 x 1 + 10 = 17

So 1 maps to 17

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The Analysis

n 7n + 10 (mod 40)

17

Fill in the answers in the table

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Now consider n = 2

n 7n + 10 (mod 40)

For n = 2, you do 7 x 2 + 10 = 24

So 2 maps to 24

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The Analysis

n 7n + 10 (mod 40)

17

Put this answer below the 2 in the table

24

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Now consider n = 3

n 7n + 10 (mod 40)

For n = 3, you do 7 x 3 + 10 = 31

So 3 maps to 31

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The Analysis

n 7n + 10 (mod 40)

17

Put this answer below the 3 in the table

24 31

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Now consider n = 4

n 7n + 10 (mod 40)

For n = 4, you do 7 x 4 + 10 = 38

So 4 maps to 38

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The Analysis

n 7n + 10 (mod 40)

17

Put this answer below the 4 in the table

24 31 38

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Now consider n = 5

n 7n + 10 (mod 40)

For n = 5, you do 7 x 5 + 10 = 45

45 is too big (there are only 40 numbers on our circle) so you must subtract the mod

number

45 - 40 = 5Press spacebar when ready

The Analysis

n 7n + 10 (mod 40)

17

Put this answer below the 5 in the table

24 31 38 5

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Now complete the rest of the mapping for yourself

Carefully look for patterns which will help you

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The Analysis

Press spacebar to complete the table (try it yourself first)

n 7n + 10 (mod 40)

17 24 31 38 5 30

10

12 19 26 33 40 7 14 21 28 35 2 9 16

37 4 11 18 25 32 39 6 13 20 27 34 1 8 15 22 29 36

23

3

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Now look at the circle abovethe table

You are about to start the drawing

Make sure you use a long rulerand pencil

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The Analysis

The first mapping is 1 17

n 7n + 10 (mod 40)

17 24 31 38 5 30

10

12 19 26 33 40 7 14 21 28 35 2 9 16

37 4 11 18 25 32 39 6 13 20 27 34 1 8 15 22 29 36

23

3

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Join 1 to 17on the circle

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The Analysis

The next mapping is 2 24

n 7n + 10 (mod 40)

17 24 31 38 5 30

10

12 19 26 33 40 7 14 21 28 35 2 9 16

37 4 11 18 25 32 39 6 13 20 27 34 1 8 15 22 29 36

23

3

Score out as you draw lines

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Join 2 to 24on the circle

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The Analysis

The next mapping is 3 31

n 7n + 10 (mod 40)

17 24 31 38 5 30

10

12 19 26 33 40 7 14 21 28 35 2 9 16

37 4 11 18 25 32 39 6 13 20 27 34 1 8 15 22 29 36

23

3

Score out as you draw lines

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Join 3 to 31on the circle

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The Analysis

The next mapping is 4 38

n 7n + 10 (mod 40)

17 24 31 38 5 30

10

12 19 26 33 40 7 14 21 28 35 2 9 16

37 4 11 18 25 32 39 6 13 20 27 34 1 8 15 22 29 36

23

3

Score out as you draw lines

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Join 4 to 38on the circle

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The Analysis

The next mapping is 5 5

n 7n + 10 (mod 40)

17 24 31 38 5 30

10

12 19 26 33 40 7 14 21 28 35 2 9 16

37 4 11 18 25 32 39 6 13 20 27 34 1 8 15 22 29 36

23

3

Score out as you draw lines

Press spacebar when ready

The Analysis

Since 5 joins to itself there is no line to draw.. Just score it out

n 7n + 10 (mod 40)

17 24 31 38 5 30

10

12 19 26 33 40 7 14 21 28 35 2 9 16

37 4 11 18 25 32 39 6 13 20 27 34 1 8 15 22 29 36

23

3

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Now complete the rest of the drawing for yourself

Be very careful not to make mistakes and be as neat as you

possibly can

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Press spacebar when readyPress spacebar when ready

Press spacebarto completethe drawing

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Can you see anylines of reflectionsymmetry?

Press spacebarto see symmetry

Lets colour a shape

And reflect in lines of symmetryPress spacebar when ready

Choose another shape

And reflect in lines of symmetryPress spacebar when ready

Now, change of colour

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Another shape

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Keep using the samemethod

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You can choose yourown shapes

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Keep on colouring

You can extendlines outside the circle Press spacebar when ready

Here’s the same trickagain

Now add somebackground Press spacebar when ready

And bring up the colour

Rotate and label

Mathematical Artn7n+10(mod40)Internet downloadMr Rybarczyk 2002

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I hope you enjoyed this piece of Mathematical Art. Why not try another drawing for yourself. Just choose a different mapping, complete the table, do the drawing and colour in.

Good Luck!