A MAGIC FORMULA OF NATURE
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1 A MAGIC FORMULA OF NATURE
2 Mathematics & Real life The descrition of the forms is one of the major roblems of biology. Is mathematics able to give a suort? Mathematics is the language of Science and Tachnology
3 Mathematics & Real life Mathematics is the language of Science and Tachnology No human inquiry is true science if it does not ass from mathematical demonstrations Trattato sulla Pittura, Leonardo Da Vinci (452-59).
4 Mathematics & Real life Mathematics is the language of Science and Tachnology The Universe can not be understood if not reviously one learns to understand the language and to know the characters in the which is written He is written in the mathematical language, and the characters are triangles, circles and other geometric figures, without these it is a vain circumvention for an obscure labyrinth. Il Saggiatore, Galileo Galilei ( ).
5 Mathematics & Real life Johan Gielis (American Journal of Botany 2003) roosed a formula that can describe a wide range of natural shaes m m ρ= R( ϕ) cos ϕ + sin ϕ a 4 b 4 2 3
6 Gielis suerformula Bottom u: to discover the the main idea behind To down: to understand the role of each arameter m m ρ= R( ϕ) cos ϕ + sin ϕ a 4 b 4 2 3
7 Gielis suerformula Bottom u: to discover the the main idea behind Product of two functions m m ρ= R( ϕ) cos ϕ + sin ϕ a 4 b 4 2 3
8 Gielis suerformula Bottom u: to discover the the main idea behind Let us concentrate our attemition on the second function assuming constant the first one m m ρ= cos ϕ + sin ϕ a 4 b 4 2 3
9 Gielis suerformula Bottom u: to discover the the main idea behind Assume the three ower arameters coincide m m ρ= cos ϕ + sin ϕ a 4 b 4 2 3
10 Gielis suerformula Bottom u: to discover the the main idea behind Assume the three ower arameters coincide m m ρ= cos ϕ + sin ϕ a 4 b 4
11 Gielis suerformula Bottom u: to discover the the main idea behind Equivalent formulation m m ρ= cos ϕ + sin ϕ a 4 b 4 m m cos ϕ + sin ϕ a 4 b 4 ρ = m m = ρcos ϕ + ρsin ϕ a 4 b 4
12 Gielis suerformula Bottom u: to discover the the main idea behind m m = ρcos ϕ + ρsin ϕ a 4 b 4
13 Gielis suerformula Bottom u: to discover the the main idea behind Re-scale the variable m m = ρcos ϕ + ρsin ϕ a 4 b 4
14 Gielis suerformula Bottom u: to discover the the main idea behind Re-scale the variable m=4 = cos sin a ρ ϕ + b ρ ϕ
15 Gielis suerformula Bottom u: to discover the the main idea behind x y =ρsinϕ =ρcosϕ From olar to cartesian coordinates = ρ cosϕ + ρsin ϕ a b
16 Gielis suerformula Bottom u: to discover the the main idea behind x y =ρsinϕ =ρcosϕ From olar to cartesian coordinates x a b = + y
17 Gielis suerformula Bottom u: to discover the the main idea behind Re-scale the two viariablex a=b= x a b = + y
18 Gielis suerformula Bottom u: to discover the the main idea behind Well known equation = x + y 2 2 = x + y key idea
19 Mathematics & Real life Botton u: to discover the the main idea behind To down: to understand the role of arameters m m ρ= R( ϕ) cos ϕ + sin ϕ a 4 b 4 2 3
20 Gielis suerformula Bottom u: to discover the the main idea behind Let us start from the key idea = x + y 2 2 = x + y
21 The squared circle Lamé circumference r = x + y Gabriel Lamé ( ) revolutionized this view
22 The squared circle Lamé circumference r = x + y For a long time the circle and the square have been considered as "oosed" figures. Gabriel Lamé ( ) revolutionized this view
23 The squared circle Lamé circumference r = x + y For a long time the circle and the square have been considered as "oosed" figures. Gabriel Lamé ( ) revolutionized this view
24 The squared circle Lamé circumference r = x + y For a long time the circle and the square have been considered as "oosed" figures. Gabriel Lamé ( ) revolutionized this view L Euclidean L infinuty MAX Manhattan =0 0<< = <<2 =2 >2 ->infinity
25 Suer ellises In the real life r x = + a y b Piet Hein (959) Sergel's Torg, Stockholm
26 Suer ellises In the real life r x = + a y b Piet Hein (959) Sergel's Torg, Stockholm = 5/2 a/ b= 6/5
27 Suer ellises In the real life r x = + a y b Piet Hein ( ) glasses, lates, desk lams
28 Suer ellises In the real life r x = + a y b bamboo cane
29 First ste to suerformula From Cartesian to Polar coordinates x =ρsinϕ y =ρcosϕ x y = + = ρsin ϕ + ρcosϕ a b a b
30 First ste to suerformula From Cartesian to Polar coordinates Rodonee Grandi s roses Luigi Guido Grandi (67-742) ρ= Rsin( ωϕ)
31 First ste to suerformula From Cartesian to Polar coordinates x =ρsinϕ y =ρcosϕ x y = + = ρsin ϕ + ρcosϕ a b a b ρ = sin ϕ + cosϕ a b
32 First ste to suerformula From Cartesian to Polar coordinates x =ρsinϕ y =ρcosϕ x y = + = ρsin ϕ + ρcosϕ a b a b ρ= sin ϕ + cosϕ a b /
33 First ste to suerformula From Cartesian to Polar coordinates ρ ρ= cosϕ + sin ϕ a b reresent the length of the vector ray corresonding to angle the local minima and maxima lay a fundamental for the figure shae ϕ
34 First ste to suerformula From Cartesian to Polar coordinates ρ ρ= cosϕ + sin ϕ a b reresent the length of the vector ray corresonding to angle the local minima and maxima lay a fundamental for the figure shae They corresond to the minimum and maximum oints of the recirocal function ϕ ρϕ ( ) ρ ϕ ϕ I / ρϕ ( ) / ρ ϕ ϕ ( ) ( ) 0 0 I
35 First ste to suerformula From Cartesian to Polar coordinates ρ ρ= cosϕ + sin ϕ a b reresent the length of the vector ray corresonding to angle the local minima and maxima lay a fundamental for the figure shae They corresond to the minimum and maximum oints of the recirocal function ( ) f = / ρ = cosϕ + sin ϕ ϕ
36 First ste to suerformula From Cartesian to Polar coordinates ρ= cosϕ + sin ϕ a b f ρ f Functions admits 4 minimum oints and 4 maximum oints for every value of arameter
37 Second ste to suerformula Fase arameter m m ρ= cos ϕ + sin ϕ a 4 b 4
38 Second ste to suerformula Fase arameter m m ρ= cos ϕ + sin ϕ a 4 b 4 m integer m= m=2 m=3 m=4 m=5 m=6 m=0 m=20
39 Second ste to suerformula Fase arameter m m ρ= cos ϕ + sin ϕ a 4 b 4 m rational m=/2 m=3/2 m=5/3 2 sins 2 sins 3 sins
40 Second ste to suerformula Fase arameter m m ρ= cos ϕ + sin ϕ a 4 b 4 m irrational m=e 3 sins 7 sins 4 sins
41 Third ste to suerformula Power Parameters i 2 3 m m ρ= cos ϕ + sin ϕ a 4 b 4 The number of ossible shaes increase greatly assuming different values for the exonents Each arameter roduces the effect of a non-linear transformation.
42 Third ste to suerformula Power Parameters i m= 3 = = = m m ρ= cos ϕ + sin ϕ a 4 b 4 m= 7 = 0.5 = 0.5 = m= 6 = = 0 = m= 7 = 0.5 = 0.5 =
43 The suerformula m m ρ= R( ϕ) cos ϕ + sin ϕ a 4 b Two remarkable articular cases: k m R( ϕ ) =ϕ R( ϕ ) = cos 2 ϕ
44 The suerformula m m ρ= R( ϕ) cos ϕ + sin ϕ a 4 b Two remarkable articular cases: R( ϕ ) =ϕ m R( ϕ ) = cos 2 ϕ
45 The suerformula m m ρ= ϕ cos ϕ + sin ϕ a 4 b 4 First case: sirals 2 3 m= 6 = = = 00 0 ϕ 2π 2 3 m= 0 = = = 8 0 ϕ 8π m= 4 = = = 00 0 ϕ 8π 2 3 m= 6 = = 2 3 = 00 0 ϕ 5π m= 0 = 50 = = 8 0 ϕ 6π m= 0 = 0,5 = = 0 ϕ 6π
46 The suerformula m m ρ= R( ϕ) cos ϕ + sin ϕ a 4 b Two remarkable articular cases: R( ϕ ) =ϕ m R( ϕ ) = cos 2 ϕ
47 The suerformula m m m ρ= cos ϕ cos ϕ + sin ϕ 2 a 4 b 4 Second case: flowers 2 3 m = 5 = = = 2 3 m = 0 = = = 2 3 m = 6 = = = m = 5 = = = m = 8 = 5 = 0.3 = 2 3 m = 2 = 0. = 5 = 2 3
48 The suerformula m m ρ= R( ϕ) cos ϕ + sin ϕ a 4 b a= b= 0 m= 5 = 3 = 2 2 = 0 ϕ 2 2π a= b= 0 m= 5 = 3 = = ϕ 2 2π a= b= 0 m= 5 2 = 3 = 5 = 0 ϕ 2 2π a = b= m= 0 2 = 3 = 5 = 8 0 ϕ 4 2π R 2.55 ( ϕ ) =ϕ a= b= m= 6 2 = 0 3 = = 00 0 ϕ 4 2π R 2.4 ( ϕ ) =ϕ
49 The suerformula m m ρ= R( ϕ) cos ϕ + sin ϕ a 4 b a= b= 0 m= 5 = 3 = 2 2 = 0 ϕ 2 2π a= b= 0 m= 5 = 3 = = ϕ 2 2π a= b= 0 m= 5 2 = 3 = 5 = 0 ϕ 2 2π a = b= m= 0 2 = 3 = 5 = 8 0 ϕ 4 2π R 2.55 ( ϕ ) =ϕ a= b= m= 6 2 = 0 3 = = 00 0 ϕ 4 2π R 2.4 ( ϕ ) =ϕ The code a b m 2 3 k
50 Code comuter grahic Classis cartoons
51 Code comuter grahic Comuter images
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58 Thank you very much for your attention
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