The document discusses Lamé curves and universal natural shapes. It describes how shapes found in nature like flowers, shells, cells, and molecules can be described using curves generated by a single variable like Gabriel Lamé's superellipses or Gielis curves. These curves use a limited number of parameters to model a wide variety of complex shapes. The document also discusses applications of these curves in fields like computer graphics, visualization, modeling cells and molecules, and modeling the fusion of plant structures using means and proportions.

2. 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

3. I. Universal Natural Shapes

4. 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”.

5. Gabriel Lamé’s superellipses
2 2
⎛ x⎞ ⎛ y⎞
⎜ ⎟ +⎜ ⎟ =1
⎝ A⎠ ⎝ B⎠
AB
+ =1
xy
xy
+ =1
AB
x y
+ =1
A B

6. Piet Hein’s supereggs

7. Nature’s superellipses

9. 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.....

10. ⎡ x ⎤ ⎡ ρ1 cosϕ.ρ2 cosθ ⎤
3D − supershape(ϕ,θ)  ⎢ y ⎥ = ⎢ ρ1 cosϕ.ρ2 sinθ ⎥
⎢⎥ ⎢ ⎥
⎢ z ⎥ ⎢ ρ2 sinθ ⎥
⎣⎦ ⎣ ⎦

12. Complex or simple?

14. Snowflakes, fluids, diatoms....

15. Flowers, shells, phyllotaxis

16. Starfish and sea life

17. (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

18. Spheres, of course

19. 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)

20. 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.”

21. II. Applications

22. Computer graphics
Visualisation of sounds
Masks for video
Fractals

23. All shapes in less than 4kb

24. CAD/CAE/CAM
CAD/CAE
Deformation of solids in liquids
Heat Shields for Spacecraft
EM-waveguides: antenna’s with arbitrary cross section

25. Imaging, Search & Sound
Modeling and counting Specific search algorithms
blood cells

26. 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/

27. Equilibrium shapes for far-from-
equilibrium conditions?

28. III. Means and more

29. Pascal’s Triangle
•Many beautiful connections, like Fibonacci series, powers of
two, connection to means......
•Pascal’s name, but is much older

30. Stifel
Yanghui Mersenne

31. 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.

32. The geometry of means

33. 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

34. 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

35. 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)

36. n-cubes and n-volume
a +b =c
n n n
Conservation of “n-volume” when going around a shape,
area when n = 2

37. 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

38. 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”

39. 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

40. 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

41. IV. Fusion in Plants

42. In cacti and succulents

43. In flowers (e.g. Asclepiads)

44. 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

45. Answer: weighted addition
r = a CF + (1-a) DF

46. The flower model
three basic strategies and combinations thereof

48. 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

49. “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

50. 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

51. 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