1. also known as The Golden Mean The Golden Section

3. The square on the hypotenuse is equal to the sum of the squares on the other two sides 5 4 3 3 2 + 4 2 = 5 2 9 + 16 = 25

6. 11 2

7. Each number is the sum of the 2 previous numbers

10. 2

11. How many golden triangles can you see? 36 72 36 72 108 D A B C E

12. AB = 5 AC = 4 BC = 3 PQ BP = φ= φ

13. AB = BC = CA AL = BL AM = CM

17. When 3 equal circles touch each other in sequence, and a larger semicircle, the ratio of the radius of the semicircle to the diameter of the small circles is φ.

18. The area of an annulus defined by the radii, a and b (b < a) is equal to the area of an ellipse with major and minor axis radii equal to a and b, respectively, when a/b = φ a bb a

19. Two examples of flower heads containing multiple seeds in spiral arrays

20. 21 clockwise spirals 13 anticlockwise spirals

23. Storms and galaxies are often spirals but are they Golden Ratio spirals?

26. The Golden End