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Thursday, March 2, 2023

from humble Cabbage - to - complex Fibonacci numbers !!

How good were you in Mathematics in schools !  :   a²+b² = (a+b)² - 2ab  !!  - how much do you like Cabbage in your food ?  - the humble cabbage – encompasses several cultivars of Brassica oleracea, a leafy green, red (purple), or white (pale green) biennial plant grown as an annual vegetable crop for its dense-leaved heads.  The word “cabbage” comes from the French word caboche, which means “head.” That makes sense given the shape. And the metaphor exists in English, where we talk about a head of cabbage.

Some of its cousins are more interesting !! and tastier !!!  -  Cauliflower is one of several vegetables in the species Brassica oleracea in the genus Brassica. It is an annual plant that reproduces by seed. Typically, only the head is eaten – the edible white flesh is sometimes called "curd" (with a similar appearance to cheese curd). The cauliflower head is composed of a white inflorescence meristem. Cauliflower heads resemble those in broccoli, which differs in having flower buds as the edible portion.

Its much elitist cousin is - Broccoli an edible green plant in the cabbage family  whose large flowering head, stalk and small associated leaves are eaten as a vegetable. Broccoli is classified in the Italica cultivar group of the species Brassica oleracea. Broccoli has large flower heads, usually dark green, arranged in a tree-like structure branching out from a thick stalk which is usually light green. The mass of flower heads is surrounded by leaves. Broccoli resembles cauliflower, which is a different but closely related cultivar group of the same Brassica species.  It is eaten either raw or cooked and is used in soups too.

Acharya Pingala  was an ancient Indian poet and mathematician, the author of the Chandaśāstra (also called the Pingala-sutras), the earliest known treatise on Sanskrit prosody.   Fibonacci (1170 – 1240)  also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano   was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, Fibonacci, was made up in 1838 by the Franco-Italian historian Guillaume Libri  and is short for filius Bonacci ('son of Bonacci').

Fibonacci popularized the Indo–Arabic numeral system in the Western world primarily through his composition in 1202 of Liber Abaci (Book of Calculation). He also introduced Europe to the sequence of Fibonacci numbers, which he used as an example in Liber Abaci.  In mathematics, the Fibonacci numbers, commonly denoted Fn, form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1 - the first few values in the sequence are:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.

The Fibonacci numbers were first described in Indian mathematics, as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, later known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci.

Romanesco broccoli   is an edible flower bud of the species Brassica oleracea, which also includes regular broccoli and cauliflower.  It is chartreuse in color,  and has a form naturally approximating a fractal. Romanesco broccoli has a nutty flavor and a firmer texture than regular broccoli when cooked. 

The Romanesco superficially resembles a cauliflower, but it has a visually striking fractal form.  It  is chartreuse in color, with the form of a natural fractal. Nutritionally, romanesco is rich in vitamin C, vitamin K, dietary fiber, and carotenoids.  The inflorescence (the bud) is self-similar in character, with the branched meristems making up a logarithmic spiral, giving a form approximating a natural fractal; each bud is composed of a series of smaller buds, all arranged in yet another logarithmic spiral. This self-similar pattern continues at smaller levels. The pattern is only an approximate fractal since the pattern eventually terminates when the feature size becomes sufficiently small. The number of spirals on the head of Romanesco broccoli is a Fibonacci number.

Interesting !  - some photos of  Romanesco broccoli  taken from twitter. 

With regards – S. Sampathkumar



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