2. Mathematical Art A 40 point circle mapping A drawing to download and try from the Mathematics Department @ www.knightswoodsecondary.org n 77n + 10 (mod 40) Click spacebar to continue

3. 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!

4. n 77n + 10 (mod 40) Write in the mapping here Look at the table at the foot of your printout Press spacebar when ready

5. Look at the mapping closely n 77n + 10 (mod 40) For n = 1, you do 7 x 1 + 10 = 17 So 1 maps to 17 Press spacebar when ready

6. The Analysis n  7n + 10 (mod 40) 17 Fill in the answers in the table Press spacebar when ready

7. Now consider n = 2 n 77n + 10 (mod 40) For n = 2, you do 7 x 2 + 10 = 24 So 2 maps to 24 Press spacebar when ready

8. The Analysis n  7n + 10 (mod 40) 17 Put this answer below the 2 in the table 24 Press spacebar when ready

9. Now consider n = 3 n 77n + 10 (mod 40) For n = 3, you do 7 x 3 + 10 = 31 So 3 maps to 31 Press spacebar when ready

10. The Analysis n  7n + 10 (mod 40) 17 Put this answer below the 3 in the table 2431 Press spacebar when ready

11. Now consider n = 4 n 77n + 10 (mod 40) For n = 4, you do 7 x 4 + 10 = 38 So 4 maps to 38 Press spacebar when ready

12. The Analysis n  7n + 10 (mod 40) 17 Put this answer below the 4 in the table 243138 Press spacebar when ready

13. Now consider n = 5 n 7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 = 5 Press spacebar when ready

14. The Analysis n  7n + 10 (mod 40) 17 Put this answer below the 5 in the table 2431385 Press spacebar when ready

15. Now complete the rest of the mapping for yourself Carefully look for patterns which will help you Press spacebar when ready

16. The Analysis Press spacebar to complete the table (try it yourself first) n  7n + 10 (mod 40) 17 2431385 30 10 12192633407142128352916 37411182532396132027341815222936 23 3 Press spacebar when ready

17. Now look at the circle above the table You are about to start the drawing Make sure you use a long ruler and pencil Press spacebar when ready

18. The Analysis The first mapping is 1  17 n  7n + 10 (mod 40) 17 243138530 10 12192633407142128352916 37411182532396132027341815222936 23 3 Press spacebar when ready

19. Join 1 to 17 on the circle Press spacebar when ready

20. The Analysis The next mapping is 2  24 n  7n + 10 (mod 40) 17 243138530 10 12192633407142128352916 37411182532396132027341815222936 23 3 Score out as you draw lines Press spacebar when ready

21. Join 2 to 24 on the circle Press spacebar when ready

22. The Analysis The next mapping is 3  31 n  7n + 10 (mod 40) 17 243138530 10 12192633407142128352916 37411182532396132027341815222936 23 3 Score out as you draw lines Press spacebar when ready

23. Join 3 to 31 on the circle Press spacebar when ready

24. The Analysis The next mapping is 4  38 n  7n + 10 (mod 40) 17 243138530 10 12192633407142128352916 37411182532396132027341815222936 23 3 Score out as you draw lines Press spacebar when ready

25. Join 4 to 38 on the circle Press spacebar when ready

26. The Analysis The next mapping is 5  5 n  7n + 10 (mod 40) 17 243138530 10 12192633407142128352916 37411182532396132027341815222936 23 3 Score out as you draw lines Press spacebar when ready

27. The Analysis Since 5 joins to itself there is no line to draw.. Just score it out n  7n + 10 (mod 40) 17 243138530 10 12192633407142128352916 37411182532396132027341815222936 23 3 Press spacebar when ready

28. Now complete the rest of the drawing for yourself Be very careful not to make mistakes and be as neat as you possibly can Press spacebar when you have finished Press spacebar when ready

29. Press spacebar to complete the drawing Press spacebar when ready

30. Can you see any lines of reflection symmetry? Press spacebar to see symmetry

31. Lets colour a shape And reflect in lines of symmetry Press spacebar when ready

32. Choose another shape And reflect in lines of symmetry Press spacebar when ready

33. Now, change of colour Press spacebar when ready

34. Another shape Press spacebar when ready

35. Keep using the same method Press spacebar when ready

36. You can choose your own shapes Press spacebar when ready

37. Keep on colouring You can extend lines outside the circle Press spacebar when ready

38. Here’s the same trick again Now add some background Press spacebar when ready

39. And bring up the colour

40. Rotate and label

42. Mathematical Art n  7n+10(mod40) Internet download Mr Rybarczyk 2002 Press spacebar when ready

43. 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!