Overview
1.Lamé curves and Universal Natural Shapes


2.Applications: some examples


3.The geometry of means and proporti...
I. Universal Natural Shapes
Descartes
“What I would like to present to the public is a science with
wholly new foundations which will enable us to ans...
Gabriel Lamé’s superellipses
                                2     2
                           ⎛ x⎞ ⎛ y⎞
                ...
Piet Hein’s supereggs
Nature’s superellipses
Gielis curves and -transformations
                                   1
                               −
           ⎛ 1 m ...
⎡ x ⎤ ⎡ ρ1 cosϕ.ρ2 cosθ ⎤
3D − supershape(ϕ,θ)  ⎢ y ⎥ = ⎢ ρ1 cosϕ.ρ2 sinθ ⎥
                       ⎢⎥ ⎢                  ...
Complex or simple?
Snowflakes, fluids, diatoms....
Flowers, shells, phyllotaxis
Starfish and sea life
(Bio)molecules
                                                                                                           ...
Spheres, of course
Natural shapes & conic sections




“That we can construct an abstract, purely
geometrical theory of morphogenesis,
indepe...
Leopold Verstraelen
“The basic shapes of the highly diverse creatures, objects and phenomena,
as they are observed by huma...
II. Applications
Computer graphics



                           Visualisation of sounds


Masks for video




                            ...
All shapes in less than 4kb
CAD/CAE/CAM


                                                                 CAD/CAE



                              De...
Imaging, Search & Sound




Modeling and counting   Specific search algorithms
     blood cells
Koiso & Palmer

• CMC surfaces: surfaces with constant mean curvature

• CAMC surfaces: with constant anisotropic mean cur...
Equilibrium shapes for far-from-
   equilibrium conditions?
III. Means and more
Pascal’s Triangle
•Many beautiful connections, like Fibonacci series, powers of
 two, connection to means......




•Pasca...
Stifel




Yanghui   Mersenne
Lamé in the triangle
           (a + b) 2 = a 2 + 2ab + b 2
           a 2 + b 2 = (a + b) 2 − 2ab
           a 3 + b 3 = ...
The geometry of means
Means for geometers

•Gaussian curvature K = square of geometric mean
•Mean curvature H = arithmetic mean

•Euler’s inequa...
Higher-order means
               • AM1/3.HM2/3 =a.b

               • AM2/3.HM1/3 =a.b

               • GM1/3.GM2/3 =a.b...
Means and the Triangle
                 a 3 + b 3 = (a + b) 3 − (3a 2b + 3ab 2 )
                 (a + b) 3 = (a 3 + b 3 )...
n-cubes and n-volume
                a +b =c
                 n      n     n




 Conservation of “n-volume” when going ar...
n-cubes and means
•Binomial expansion: cubes and beams
•if you have the volume of a beam, you can make an
 n-cube with sid...
René Descartes
“.....others try to express these proportions n ordinary algebraic term
by means of several different dimen...
Calculating with cubes
•You can make same dimension for all:

                                           )(               ...
The old notation for numbers
•Used by Barrow, Stevin,......
•When using the unit element
           x3 = x ⊗ x ⊗ x
       ...
IV. Fusion in Plants
In cacti and succulents
In flowers (e.g. Asclepiads)
Question of Karl J. Niklas:
        Can supershapes describe fusion in flowers?

Constraining of growth through supershapes...
Answer: weighted addition
        r = a CF + (1-a) DF
The flower model
three basic strategies and combinations thereof
The deeper meaning:
        arithmetic and geometric means, once again

                Geometric mean   Arithmetic mean
 ...
“Thus number may be said to rule the
world of quantity and the four rules of
  arithmetic may be regarded as the
     comp...
Addition and multiplication,
               means
• Against the flow   Aeθ + Be-θ

• Fixed number raised to a variable powe...
Addition and multiplication,
                means
• Alternatively, a variable raised to a fixed power

             Functi...
Universal Natural Shapes
Universal Natural Shapes
Universal Natural Shapes
Universal Natural Shapes
Universal Natural Shapes
Universal Natural Shapes
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Universal Natural Shapes

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Lecture given at the University of Louvain in March 08 for the Simon Stevin Institute for Geometry

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Universal Natural Shapes

  1. 1. Overview 1.Lamé curves and Universal Natural Shapes 2.Applications: some examples 3.The geometry of means and proportion; n-cubes 4.Fusion in Plants: a simple model using means and proportion ........ and some interesting connections throughout 2
  2. 2. I. Universal Natural Shapes
  3. 3. Descartes “What I would like to present to the public is a science with wholly new foundations which will enable us to answer every question that can be put about any kind of quantity whatsoever, whether continuous or discontinuous, each according to its nature..... In this way I hope to demonstrate that in the case of continuous quantity, certain problems can be solved with straight line and circles alone, that others can be solved only with curves other than circles, but which can be generated by a single motion and which can therefor be drawn using a new compass which I do not believe to be any less accurate than, and just a geometrical as, the ordinary compass which is used to draw circles”.
  4. 4. Gabriel Lamé’s superellipses 2 2 ⎛ x⎞ ⎛ y⎞ ⎜ ⎟ +⎜ ⎟ =1 ⎝ A⎠ ⎝ B⎠ AB + =1 xy xy + =1 AB x y + =1 A B
  5. 5. Piet Hein’s supereggs
  6. 6. Nature’s superellipses
  7. 7. Gielis curves and -transformations 1 − ⎛ 1 m n2 1 m n3 ⎞ n1 ρ = ⎜ cos ϕ + − sin ϕ ⎟ .f (ϕ) ⎝A 4 B4⎠ Flowers, leaves, cells, phyllotaxis, eggs, shells, DNA, sand dollars.....
  8. 8. ⎡ x ⎤ ⎡ ρ1 cosϕ.ρ2 cosθ ⎤ 3D − supershape(ϕ,θ)  ⎢ y ⎥ = ⎢ ρ1 cosϕ.ρ2 sinθ ⎥ ⎢⎥ ⎢ ⎥ ⎢ z ⎥ ⎢ ρ2 sinθ ⎥ ⎣⎦ ⎣ ⎦
  9. 9. Complex or simple?
  10. 10. Snowflakes, fluids, diatoms....
  11. 11. Flowers, shells, phyllotaxis
  12. 12. Starfish and sea life
  13. 13. (Bio)molecules Human Rhinovirus 16 Cyclophilin: Isometric Decagonal Lattice 000200 y Glu15 [1 0 0 0, 2] PV1 red chains [-2 1 -2 -1, 2] PV2 green chains [3 0 2 1, 2] [0 1 0 0, 2] [-3 -1 -2 -3, 2] C PV3 blue chains ! PV4 black chains [3 0 1 2, 2] [1 -1 2 -1, 2] r P ° 1 -1 1 1 1 1 [-1 -1 -1 -1, 2] A 1 x [3 1 1 3, 2] !3 r ° Q [-1 2 -1 1, 2] ! Indexed [2 1 0 3, 2] Ico-Dodecahedral Form D [-3 -2 -1 -3, 2] [0 0 1 0, 2] [1 2 0 3, 2] r0 = a = c [-1 -1 1 -2, 2] [0 0 0 1, 2] z [-1 2 -1 1, 2] [2 1 0 3, 2] 0 0 0 0 -2 0 Glu15 2r ° x 4r ° 1 1 1 -1 1 -1 -1 1 -1 1 -1 -1 Glu15 0 0 0 0 -2 0 [-1 2 -1 1,-2] [2 1 0 3,-2] HRV16 Hadfield et al. Structure 5 (1997) 427-441 (1aym) . – p.29/35 Cyclo. Iso-decagonal Ke et al., Current Biology Structure, 2 (1994) 33-44 . – p.23/35 Facet-like Snow Crystal with Growth Lattice Regular hexagons with center and vertices at points of the growth lattice BH 114.8 bh114.8-83 (fig4b) . – p.4/35
  14. 14. Spheres, of course
  15. 15. Natural shapes & conic sections “That we can construct an abstract, purely geometrical theory of morphogenesis, independent of the substrate of forms and the nature of forces that create them, might seem difficult to believe, especially for the seasoned experimentalist who is always struggling with an elusive reality” (René Thom)
  16. 16. Leopold Verstraelen “The basic shapes of the highly diverse creatures, objects and phenomena, as they are observed by humans, either visually or with the aid of sophisticated apparatus, can essentially, either singular or in combinations, be considered as derived from a limited number of special types of geometric figures. From Greek science up to the present this is probably the most important subject of natural philosophy. ...When we return to circles, these are the most symmetrical among all planar curves, describing growth from a central point with perfect isotropy. By applying the appropriate Gielis’ transformations (which are technically determined by just a few parameters), this results in an immediate and accurate description of the symmetries and shapes of e.g. flowers or hexagons in viscous fluids or honeycombs.”
  17. 17. II. Applications
  18. 18. Computer graphics Visualisation of sounds Masks for video Fractals
  19. 19. All shapes in less than 4kb
  20. 20. CAD/CAE/CAM CAD/CAE Deformation of solids in liquids Heat Shields for Spacecraft EM-waveguides: antenna’s with arbitrary cross section
  21. 21. Imaging, Search & Sound Modeling and counting Specific search algorithms blood cells
  22. 22. Koiso & Palmer • CMC surfaces: surfaces with constant mean curvature • CAMC surfaces: with constant anisotropic mean curvature • Delaunay surfaces: surfaces of revolution of constant mean curvature (catenoid & plane for H = 0, cylinder, unduloid and nodoid for H ≠ 0) • Anisotropic Delaunay surfaces: CAMC catenoid for example • http://www.isu.edu/~palmbenn/
  23. 23. Equilibrium shapes for far-from- equilibrium conditions?
  24. 24. III. Means and more
  25. 25. Pascal’s Triangle •Many beautiful connections, like Fibonacci series, powers of two, connection to means...... •Pascal’s name, but is much older
  26. 26. Stifel Yanghui Mersenne
  27. 27. Lamé in the triangle (a + b) 2 = a 2 + 2ab + b 2 a 2 + b 2 = (a + b) 2 − 2ab a 3 + b 3 = (a + b) 3 − (3a 2b + 3ab 2 ) a n + b n = (a + b) n − (..........................) = c n From n > 2, a, b, c, n cannot be expressed in integers Therefore: the modulo part (which is detracted) is non-integer.
  28. 28. The geometry of means
  29. 29. Means for geometers •Gaussian curvature K = square of geometric mean •Mean curvature H = arithmetic mean •Euler’s inequality: K  H2 •This is number theory’s cornerstone GM  AM
  30. 30. Higher-order means • AM1/3.HM2/3 =a.b • AM2/3.HM1/3 =a.b • GM1/3.GM2/3 =a.b • AMi/n.HMn-i/n =a.b • AMn-i/n.HMi/n =a.b • GMi/n.GMn-i/n =a.b
  31. 31. Means and the Triangle a 3 + b 3 = (a + b) 3 − (3a 2b + 3ab 2 ) (a + b) 3 = (a 3 + b 3 ) + (3a 2b + 3ab 2 ) •the Lamé-part of an expansion consists of “pure” numbers. •the modulo-part consists of the various means between two numbers a and b •Casorati-curvature does not take into account the modulo-part (the mean curvature does)
  32. 32. n-cubes and n-volume a +b =c n n n Conservation of “n-volume” when going around a shape, area when n = 2
  33. 33. n-cubes and means •Binomial expansion: cubes and beams •if you have the volume of a beam, you can make an n-cube with sides M1/n •For example: M1/ 3 = 3 a 2b Volume = ( 3 ab 2 ) 3 •Then you have only cubes, not beams
  34. 34. René Descartes “.....others try to express these proportions n ordinary algebraic term by means of several different dimension and shapes. The first they call the root, the second the square, the third the cube, the fourth the biquadrratic. These expressions have, I confess, long misled me... All such names should be abandoned as they are liable to cause confusion in our thinking. For though a magnitude may be termed a cube or biquadratic, it should never be represented to the imagination otherwise than as a line or a surface. What above all, requires to be noted is that the root, the square, the cubes etc. are merely magnitudes in continued proportion”
  35. 35. Calculating with cubes •You can make same dimension for all: )( ) ( 3 3 x +x +x=x + (1.x ) + 3 2 3 2 2 3 3 (1 x) •These are the geometric means between x and the unit element “Just as the symbol c1/3 is used to represent the side of a cube, a3 has the same dimension as a2b” René Descartes
  36. 36. The old notation for numbers •Used by Barrow, Stevin,...... •When using the unit element x3 = x ⊗ x ⊗ x x 2 = x ⊗ x = x ⊗ x ⊗1 [ ] n (n −1) n n Compare : x with : x .1 •the number one, or a unit distance is what we always need for comparison • All one needs to do is calculate the means between the number and the unit element
  37. 37. IV. Fusion in Plants
  38. 38. In cacti and succulents
  39. 39. In flowers (e.g. Asclepiads)
  40. 40. Question of Karl J. Niklas: Can supershapes describe fusion in flowers? Constraining of growth through supershapes as constraining functions r = SF * f(φ) = CF * DF
  41. 41. Answer: weighted addition r = a CF + (1-a) DF
  42. 42. The flower model three basic strategies and combinations thereof
  43. 43. The deeper meaning: arithmetic and geometric means, once again Geometric mean Arithmetic mean Weighted arithmetic mean WAM Relations Area based on AM Area based on GM GM AM Numbers a and w1.a + w2.b √a.b (a+b)/2 = GM ≤ AM ((a+b)/2)2 a.b b άCF + (1- ά)DF Flowers, DF DF.CF ≤ ((DF ((DF+CF)/2)2 DF.CF √ (DF.CF) (DF+CF)/2 and CF +CF)/2)2 w1+w2= ά+(1- ά)=1 κ1cos2 φ+ κ2 sin2 φ K ≤ H2 √(κ1 κ2) (κ1+ κ2)/2 Surfaces, κ1 κ2 = K H2 = ((κ1+ κ2)/2)2 (Euler’s theorem) k1 and k2 = √K =H (Euler’s inequality) w1+w2 = cos2φ+sin2φ =1 The flower model connects to the deepest notions in mathematics; many results from the geometry of surfaces can be used for the flower model
  44. 44. “Thus number may be said to rule the world of quantity and the four rules of arithmetic may be regarded as the complete equipment of the mathematician” James Clerk Maxwell
  45. 45. Addition and multiplication, means • Against the flow Aeθ + Be-θ • Fixed number raised to a variable power Functions Polar plane XY-plane eθ and e-θ Addition & Arithmetic Logarithmic spiral Catenary mean Multiplication & Geometric Circle Straight line mean
  46. 46. Addition and multiplication, means • Alternatively, a variable raised to a fixed power Functions Expression Graph xn and ym Addition & Arithmetic xn + y m Lamé curves / superellipses mean xn.ym = C Multiplication & Power functions, superparabola y = C xn/m Geometric mean

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