DEFINITION The golden ratio section of segment AB is said to be segment AC, with C included between A and B, mean proportional between the whole segment AB and the remaining part CB, meaning AB:AC = AC:CB. THE RATIO is the GOLDEN RATIO
Luca Pacioli De Divina Proportione Niccolò Tartaglia Trad elementi di Euclide IN VENETIA. Appresso Curtio Troiano 1565
Geometric contruction of the golden ratio section If AB is the given segment: -you draw the perpendicular of AB at point B -you choose a point on it D so that DB is half of AB -you draw the circumference and radius DB which is tangent to AB in B -you draw the ray AD and its points of intersection on the circumference are indicated by R and Q -you draw a circumference arc with A as its centre and radius AR which meets AB at point C Because of a known theorem segment AC is the golden ratio section of AB : AB:AC=AC:DB ACB D Q R
Calculation of the golden ratio Let’s look for the value of the ratio AB/AC=AC/CB Given AB= 1 and AC= x we have : X 2 +x-1=0 With the solution: X=0,618033……….. Golden ratio section 1/x = 1,618033…….. Golden ratio φ 1-XX 1
AE is the golden part of segment AD, the segment AF is the golden part of segment AE, the segment FG is the golden part of segment AF ecc…… SOME “CONSIDERABLE’’ FIGURES PENTAGON GOLDEN TRIANGLE GOLDEN RECTANGLE
THE FIBONACCI SEQUENCE 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ecc. Each digit is the sum of the two previous digits Fn-2 + Fn-1 = Fn WHAT IS THE LINK BETWEEN THE FIBONACCI SEQUENCE AND THE GOLDEN RATIO?