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n Yesterday’s mention of breeding livestock, and the earliernmention of the nautilus here has put me in mind of ratios and sequences. Let’sndo a little thought experiment: instead of stick insects, let’s substitutenrabbits. Assume I want to start breeding rabbits for fun and profit, so I popndown to Bob’s Bunny Shop and buy my first pair of rabbits.
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nNow thesenimaginary rabbits don’t reach maturity until they are two months old, so fornthe first month I have one pair of rabbits, and for the second month I stillnonly have one pair. Then, in month three, they produce a pair of baby rabbits,nso now I have two pairs of rabbits. In month four, the first pair produces yetnanother pair, so now I have three pairs of rabbits. Month five gets me anothernpair from Pair 1 but now the pair born in month three start to breed, so I getna pair from them too, giving me a total of five pairs.
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nAnd so it goes,nthroughout the next year, with pairs producing more pairs after a two monthnwait. Here’s a little table to illustrate the progress of my rabbit-breedingnprogramme:
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nMonthn |
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nGenerations of Rabbitsn |
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| n
n
n |
n
n1st
n |
n
n2nd
n |
n
n3rd
n |
n
n4th
n |
n
n5th
n |
n
n6th
n |
n
nTotaln |
| n
nOne
n |
n
n1
n |
n
n
n |
n
n
n |
n
n
n |
n
n
n |
n
n
n |
n
n1
n |
| n
nTwo
n |
n
n1
n |
n
n
n |
n
n
n |
n
n
n |
n
n
n |
n
n
n |
n
n1
n |
| n
nThree
n |
n
n1
n |
n
n1
n |
n
n
n |
n
n
n |
n
n
n |
n
n
n |
n
n2
n |
| n
nFour
n |
n
n1
n |
n
n2
n |
n
n
n |
n
n
n |
n
n
n |
n
n
n |
n
n3
n |
| n
nFive
n |
n
n1
n |
n
n3
n |
n
n1
n |
n
n
n |
n
n
n |
n
n
n |
n
n5
n |
| n
nSix
n |
n
n1
n |
n
n4
n |
n
n3
n |
n
n
n |
n
n
n |
n
n
n |
n
n8
n |
| n
nSeven
n |
n
n1
n |
n
n5
n |
n
n6
n |
n
n1
n |
n
n
n |
n
n
n |
n
n13
n |
| n
nEight
n |
n
n1
n |
n
n6
n |
n
n10
n |
n
n4
n |
n
n
n |
n
n
n |
n
n21
n |
| n
nNine
n |
n
n1
n |
n
n7
n |
n
n15
n |
n
n10
n |
n
n1
n |
n
n
n |
n
n34
n |
| n
nTen
n |
n
n1
n |
n
n8
n |
n
n21
n |
n
n20
n |
n
n5
n |
n
n
n |
n
n55
n |
| n
nEleven
n |
n
n1
n |
n
n9
n |
n
n28
n |
n
n35
n |
n
n15
n |
n
n1
n |
n
n89
n |
| n
nTwelve
n |
n
n1
n |
n
n10
n |
n
n36
n |
n
n56
n |
n
n35
n |
n
n6
n |
n
n144
n |
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nSuper. After twelve months, Inhave 144 pairs of rabbits. Not bad for the first year. But look at the totalsncolumn of the right. Do you notice anything about the sequence? Each pair ofnconsecutive numbers add up to the number below them: 1+1=2, 1+2=3, 2+3=5,n3+5=8, 5+8=13 and so on.
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nThis is called a Fibonacci sequence, afternLeonardo Pisano, also called Leonardo Fibonacci (son of Gugleilmo Bonacci – Fi(lius)-Bonacci)nor simply Fibonacci, an Italian mathematician who introduced the Hindu/Arabicnnumeral system into Europe, supplanting the system of Roman numerals. In 1202,nhe published his Liber Acubi (Book of Calculation), in which he used thensequence that now bears his name as an example (he did not discover thisnsequence) using the hypothetical breeding of rabbits (which I’ve just pinchednabove).
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nThere are lots of things you canndo with the Fibonacci sequence, (some of which are mind-boggling complicated),nbut some of the most aesthetically pleasing are the relationships found innnatural objects. Instead of rabbits, let’s look at squares.
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nTake a singlensquare.
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nThen add another square of the same size. That gives us a rectangle onenunit wide and two units high. Like the bunnies in the table, 1+1 has given usntwo.
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nSo add another square, with its sides equal to two units. And now we havenanother rectangle of two units by three units, so we can add another squarenwith sides of three units.
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nThe resulting rectangle measures 3 x 5, so we cannadd yet another square of 8 units.
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nAnd so it goes, adding squares of increasingnside length.
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nThen, if you get yourself a compass (try this with a ruler and anbit of paper), draw an arc in the first square with a radius of one unit.
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nAddnan arc in the second square, also with a radius of one unit.
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nIn the thirdnsquare, draw an arc with a radius of two units, and so on.
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nThe result is anFibonacci spiral.
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nThis is a logarithmic spiral, where the proportional increasenin each quarter turn is equal to the square-side radius of the arc. This is thenspiral found in the shells of molluscs, as the growth allows for a larger shellnwithout altering the shape of the shell. (The Golden Spiral, anothernlogarithmic spiral, increases by the ratio of phi (Φ) – more of which later –nand so would necessarily alter the shape of a shell).
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nOnce you start lookingnfor these spirals, you see them everywhere.
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| Cretaceous Silver Ammonite from Madagascar |
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nNautilus shells we have seen, andnthe same spiral occurred in their extinct relatives, the ammonites (ammonitesnget their name from the similarity of their shape to the ram’s horns of thenEgyptian god. Ammon).
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| Ammon |
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nIt appears in other shells – this conch shell has one.
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nAsndoes this murex shell.
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nYou can find them on your dinner plate – here is anRomanesco cauliflower.
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nYou can find them in the sky – here are a couple ofnspiral galaxies.
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nThey are in pinecones and pineapples.
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nBut, and this isnimportant, there is nothing mystical or magical about these spirals – they arenentirely natural phenomena. They have not been put there, they havenevolved over millions of years. They are the result of an underlying geometrynin the universe, as nature produces the most economical solution to problems innterms of effort and materials. The garden is already beautiful enough as it isn– there is no need to go and invent fairies at the bottom of it.
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