The Mystery of the Golden Number

1. The Mystery of the Golden Number

2. Leonardo of Pisa (1170 – 1250 AD) was an Italian mathematician. He is sometimes called Fibonacci. Fibonacci is famous for helping to spread the use of Hindu Arabic numbers in Europe. These numbers replaced the Roman number system.

3. Italy

4. There is a special sequence of numbers called the Fibonacci sequence because Fibonacci wrote about this sequence. 0 , 1 , 1 , 2 , 3 , 8 , 13 , 21 … 5,

6. The Fibonacci numbers can be used to calculate the value of a special number called the golden number. The golden number cannot be written down exactly because it is a decimal number that goes on for ever without any recurring pattern. Because the golden number cannot be written down exactly we use the 21st letter of the Greek alphabet,  or phi to represent the golden number.  to 20 decimal places

7. The Greek Alphabet

8. This is  to 20 decimal places 1.618 033 988 749 894 848 20

9. This is  to 1000 decimal places 1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137484754088075386891752126633862223536931793180060766726354433389086595939582905638322661319928290267880675208766892501711696207032221043216269548626296313614438149758701220340805887954454749246185695364864449241044320771344947049565846788509874339442212544877066478091588460749988712400765217057517978834166256249407589069704000281210427621771117778053153171410117046665991466979873176135600670874807101317952368942752194843530567830022878569978297783478458782289110976250030269615617002504643382437764861028383126833037242926752631165339247316711121158818638513316203840052221657912866752946549068113171599343235973494985090409476213222981017261070596116456299098162905552085247903524060201727997471753427775927786256194320827505131218156285512224809394712341451702237358057727861600868838295230459264787801788992199027077690389532196819861514378031499741106926088674296226757560523172777520353613936

11. Egypt It is thought that the Egyptians may have known about  over 2500 years

12. Some people think that the Egyptians used  in the design of the Great pyramid of Giza which was completed about 2500 years ago.

13. Golden Rectangles

14. Any rectangle that has a length that is  times the width is called a golden rectangle l w This is a golden rectangle because l =   w

16. Greece People in Greece knew about golden rectangles about 2500 years ago.

17. The Parthenon is a temple that was built about 2500 years ago on the Acropolis in Athens, Greece.

18. Many golden rectangles can be found in the design of the Parthenon.

20. France The west face of the cathedral of Notre Dame in Paris which was completed in the 13th century contains many golden rectangles.

23. India The Taj Mahal In Agra in India was completed in around 1648 AD.Many golden rectangles can be found in the Taj Mahal.

26. Constructing a Golden Rectangle

27. Draw a square of any size.

28. Mark the midpoint of one side of the square. Extend the same side of the square.

29. Put the pencil on this corner Put the point of the compasses on the midpoint Draw an arc from the corner of the square to the extend line.

30. Draw the golden rectangle

31. Mexico Guatemala Honduras The Mayan people have lived in the countries that are now Mexico, Guatemala and Honduras for thousands of years.

32. It is thought that the Mayan people used golden rectangles in their architecture. This is part of the Maya ruins of Copan in Honduras.

33. Golden Triangles

34. Angles Acute angle Less than 90º Right angle 90º Obtuse angle Between 90º and 180º

35. Triangles Equilateral Triangle Isosceles Triangle Right Angled Triangle Scalene Triangle

36. There are two special isosceles triangles that are called golden triangles. In both the length of the longer side is  times the length of the shorter side. Golden triangle with the two equal sides shorter than the third side. Golden triangle with the two equal sides longer than the third side.

37. This golden triangle has one obtuse angle and two acute angles. This golden triangle has three acute angles

38. Constructing Golden Triangles

39. Constructing a golden triangle with three acute angles

40. Draw a line 5 cm long Multiply the length of the line by 1.618 () Write down your answer correct to 1 decimal place. Your answer should be 8.09 which is 8.1 to 1 decimal place

41. Set the compasses so that the point and the pencil are 8.1cm apart. Put the point of the compasses on the right hand side of the line and draw an arc above the middle of the line.

42. Keep the compasses at the same setting Put the point on the left hand side of the line and draw an arc to cross the first arc.

43. Carefully join the ends of the line to the point where the two arcs cross. You now have a golden isosceles triangle. Estimate the size of each angle. Measure the three angles of the triangle

44. 36º 72º 72º

45. Golden triangle with one obtuse and two acute angles.

46. Draw a line 10 cm long Divide the length of the line by 1.618 () Write down your answer correct to 1 decimal place. Your answer should be 6.1804 … which is 6.2 to 1 decimal place.

47. Set the compasses to 6.2 cm. Put the point of the compasses on the right hand side of the line and draw an arc above the middle of the line.

48. Keep the compasses at the same setting Put the point on the left hand side of the line and draw an arc to cross the first arc.

49. Carefully join the ends of the line to the point where the two arcs cross. You now have a golden isosceles triangle. Estimate the size of the three angles. Measure the three angles of the triangle

50. 108º 36º 36º