MATHEMATICS BEHIND FLOWER PETALS
Have you ever observed a flower from the mathematical point of view? Have you ever given a thought on why flowers are aesthetically appealing? In nearly all the flowers, the number of petals is one of the numbers that occur in the strange sequence 1, 2,3,5,8, 13, 21, 34, 55, 89, 144. They form the beginning of the so-called Fibonacci series, in which each number is the sum of the two that precedes.
In order to verify this, we collected 30 different samples of flowers from Somaiya Vidyavihar Campus and counted the number of petals present in each flower. These data were then added to excel from which we observed that 60% of the 30 samples of flowers considered had petal counts in the above Fibonacci sequence.
No of flowers | Fibonacci | Non Fibonacci | 30 | 18 | 12 | Proportion | .6 | .4 | Z value | 1.09 | | P value | .1366>.05 | Null acceptance |
As we can see the above table, the p value is greater than .05, (level of significance) the alternate hypothesis that the number of petals in flowers follows Fibonacci series is not conclusively proved. However even the proportion of 0.65 would have proved the Fibonacci hypothesis. We feel that sample size of 30 is insufficient.
GOLDEN RATIO
What is it exactly?
Think of any two numbers. Make a third by adding the first and second, a fourth by adding the second and third, and so on. When you have written down about 20 numbers, calculate the ratio of the last to the second from last. The answer should be close to 1.6180339887. What makes the golden ratio special is the number of mathematical properties it possesses. The golden ratio is the only number whose square can be produced simply by adding 1 and who’s reciprocal by subtracting 1. The golden ratio is also difficult to pin down: it's the most difficult to express as any kind of fraction and its digits - 10 million of which were computed in 1996 - never repeat.
In flowers there is a circular region at the centre of the tip of the shoot of a growing plant, of tissue with no special features, called the apex. Around the apex, one by one, tiny lumps form, called primordia. Each primordium migrates away from the apex — or more accurately the apex grows away from the lump, leaving it behind — and eventually the lump develops into a leaf, petal, or the like. Moreover, the general arrangement of those features is laid downright at the start, as the primordia form. The spirals that are most apparent to the human eye are not fundamental. The most important spiral is formed by considering the primordia in their order of appearance. What human eye catches is the Fibonacci spirals which are formed from primordial which appear near each other in space. The primordia are equally spaced- angularly the angle being equal to 137.5(the golden angle) so as to fill the available space most efficiently. The golden angle is obtained by multiplying 360o with most irrational number known as golden ratio(φ) and its value is (sqrt(5)+1)/2.
Golden angle=360* (1-1/φ) φ = 1.6180….
Significance of golden ratio with context to flowers: * The available space is filled efficiently. * Each petal is placed to allow for the best possible exposure to sunlight and other factors. * Gives aesthetic appeal to the flower.
What else is Golden ratio associated with? * physics of black holes * growth of Quasi crystals * Design of pyramid * paintings * Architectural desi 