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The Golden Ratio

The Golden Ratio. Begin by drawing a square with sides of 10cm. 10 cm. 10 cm. The Golden Ratio. Make a 5 cm mark on the top and bottom of the square and connect with a line. 10 cm. x. 10 cm. x. 5 cm. The Golden Ratio. Notice that the square has been divided into two equal rectangles.

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The Golden Ratio

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  1. The Golden Ratio Begin by drawing a square with sides of 10cm 10 cm 10 cm

  2. The Golden Ratio Make a 5 cm mark on the top and bottom of the square and connect with a line 10 cm x 10 cm x 5 cm

  3. The Golden Ratio Notice that the square has been divided into two equal rectangles 10 cm x 10 cm x 5 cm

  4. The Golden Ratio Now draw a diagonal in the 2nd rectangle as shown 10 cm x 10 cm x 5 cm

  5. The Golden Ratio Set your compass to the length of this diagonal 10 cm x 10 cm x

  6. The Golden Ratio Use the compass to draw an arc as shown 10 cm x 10 cm And extend the bottom line of the square to meet the arc x

  7. The Golden Ratio Carefully measure the line AB 10 cm x Write down yourmeasurement in centimetres to 1 decimalplace 10 cm x A B ? cm

  8. The Golden Ratio Now complete the rectangle Write down the ratio of long side : short side 10 cm x ??.? : 10 10 cm Express this as a fraction ??.? 10 x A B ? cm And convert to a decimal ??.? ÷ 10 =

  9. The Golden Ratio Next we will perform an EXACT calculation for the ratio of …… long side : short side E Consider the diagonal DE 10 cm Use Pythagoras in triangle DCE to find DE (try it now) 10 cm a² = b² + c² DE² = DC² + CE² A B DE² = 5² + 10² D C 5 Notice that DB = DE So AB = AD + DB =

  10. The Golden Ratio Next we will perform an EXACT calculation for the ratio of …… long side : short side E 10 cm So the ratio is… long side : short side AB : AC 10 cm : 10 And as a fraction canbe expressed as A B D C 5 and as a decimal is 1.61803

  11. 10 cm E 10 cm A B C D 5 The Golden Ratio Now look closely at the smaller rectangle which has formed inside the large one as shown… The long side, CE is obviously 10 cm Can you determine an expression for the short side CB? CB = DB – DC

  12. 10 cm E 10 cm A B C D 5 The Golden Ratio We can now find the ratio of long side : short side for this small rectangle… long side : short side CE : CB Convert the ratio into a fraction and then to a decimal as before (try it now)

  13. 10 cm E 10 cm A B C D 5 The Golden Ratio We can now find the ratio of long side : short side for this small rectangle… long side : short side CE : CB As a fraction is … And as a decimal is = 1.61803

  14. 10 cm E 10 cm A B C D 5 The Golden Ratio So BOTH rectangles….. The BIG one

  15. 10 cm E 10 cm A B C D 5 The Golden Ratio So BOTH rectangles….. and the SMALL one have long side : short side In the SAME ratio of 1.61803 And they share a common side This ratio is very special and is given the symbol Φ (Phi) It is commonly referred to as The Golden Ratio

  16. A B C The Golden Ratio Here is another, faster way to find the Golden Ratio. Consider the same rectangle, drawn the same way. Let the distance AC = x And the distance CB = 1 Remember that the sides of the big rectangle must be in the SAME ratio as the sides of the small rectangle (they are SIMILAR rectangles) x 1 So AB:AC = AC:CB Write this relationship in fraction fromthen substitute the lengths in. (try it now)

  17. A B C x 1 The Golden Ratio You should now have the following working…. By coss-multiplying, find a quadratic equation to solve.(try it now) You should end up with the equation

  18. A B C x 1 The Golden Ratio This quadratic cannot be readily factorised so try using the quadratic formula to solve it. (try it now)

  19. A B C x 1 The Golden Ratio Find the two solutions in decimal form to 5 decimal places. Any comments about the two answers? So x = 1.61803 or x = -0.61803 We can discount the negative answer as it makes little sense in terms of a length. The negative answer does however contain exactly the same digits after the point as the positive answer which is a bit strange!

  20. The Golden Ratio More to explore http://www.mathsisfun.com/numbers/golden-ratio.htmlhttp://www.bbc.co.uk/learningzone/clips/the-beauty-of-the-golden-ratio/9017.html Also, the negative answer is interesting and worth finding out more about. Fibonacci numbers are directly related to the Golden Ratio – find out why. K.Rybarczyk 2011

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