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

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 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”.

Gabriel Lamé’s superellipses

xA+yB= 1

Ax+By= 1

xA+

yB

= 1

xA

⎛⎝⎜

⎞⎠⎟2

+yB

⎛⎝⎜

⎞⎠⎟2

= 1

Piet Hein’s supereggs

Nature’s superellipses

Gielis curves and -transformations

Flowers, leaves, cells, phyllotaxis, eggs, shells, DNA, sand dollars.....

ρ =1Acosm

4ϕ

n2+−1Bsin m

4ϕ

n3⎛

⎝⎜⎞

⎠⎟

− 1n1. f (ϕ )

3D − supershape(ϕ,θ) xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

ρ1 cosϕ.ρ2 cosθρ1 cosϕ.ρ2 sinθρ2 sinθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Complex or simple?

Snowflakes, fluids, diatoms....

Flowers, shells, phyllotaxis

Starfish and sea life

(Bio)molecules

HRV16

Human Rhinovirus 16

0 0 0 2 0 0

0 0 0 0 -2 0

1 1 1 -1 1 -1

-1 1 -1 1 -1 -1

0 0 0 0 -2 0

1 -1 1 1 1 1

Indexed

Ico-Dodecahedral Form

Hadfield et al. Structure 5 (1997) 427-441 (1aym)

PV1 red chains

PV2 green chains

PV3 blue chains

PV4 black chains

. – p.29/35

bh114.8-83 (fig4b)

Facet-like Snow Crystal with Growth Lattice

Regular hexagons with center and vertices at points of the growth lattice

BH 114.8

. – p.4/35

Cyclo. Iso-decagonal

Cyclophilin: Isometric Decagonal Lattice

r°

!

!

1

!3r°

x

y

A

C

P

Q

D

[3 1 1 3, 2]

[-1 -1 -1 -1, 2]

[1 0 0 0, 2]

[0 1 0 0, 2]

[0 0 1 0, 2]

[0 0 0 1, 2]

[1 -1 2 -1, 2]

[1 2 0 3, 2]

[-3 -2 -1 -3, 2]

[3 0 1 2, 2]

[-2 1 -2 -1, 2]

[-1 2 -1 1, 2]

[-1 -1 1 -2, 2]

[2 1 0 3, 2]

[-3 -1 -2 -3, 2]

[3 0 2 1, 2]

Glu15

x

z

4r°

2r°

Glu15

Glu15

[2 1 0 3,-2]

[2 1 0 3, 2]

[-1 2 -1 1,-2]

[-1 2 -1 1, 2]

Ke et al., Current Biology Structure, 2 (1994) 33-44

r0 = a = c

. – p.23/35

Spheres, of course

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)

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.”

II. Applications

Computer graphics

Masks for video

Visualisation of sounds

Fractals

All shapes in less than 4kb

CAD/CAE/CAM

Heat Shields for Spacecraft Deformation of solids in liquids

EM-waveguides: antenna’s with arbitrary cross section

CAD/CAE

Imaging, Search & Sound

Modeling and counting blood cells

Specific search algorithms

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/

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......

•Pascal’s name, but is much older

Stifel

Yanghui Mersenne

Lamé in the triangle

From n > 2, a, b, c, n cannot be expressed in integers

Therefore: the modulo part (which is detracted) is non-integer.�

(a + b)2 = a2 + 2ab + b2

a2 + b2 = (a + b)2 − 2aba3 + b3 = (a + b)3 − (3a2b + 3ab2)an + bn = (a + b)n − (..........................) = cn

The geometry of means

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

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

Means and the Triangle

•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)

�

a3 + b3 = (a + b)3 − (3a2b + 3ab2)

�

(a + b)3 = (a3 + b3) + (3a2b + 3ab2)

n-cubes and n-volume

Conservation of “n-volume” when going around a shape, area when n = 2

�

an + bn = cn

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:

•Then you have only cubes, not beams�

M1/ 3 = a2b3

�

Volume = ( ab23 )3

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”

Calculating with cubes

•You can make same dimension for all:

•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

�

x 3 + x 2 + x = x 3 + (1.3 x 2)( )3 + (123 x)( )3

The old notation for numbers

•Used by Barrow, Stevin,......

•When using the unit element

•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

�

x 3 = x⊗ x⊗ xx 2 = x⊗ x = x⊗ x⊗1

�

Compare : xnwith : x(n−1).1n[ ]n

IV. Fusion in Plants

In cacti and succulents

In flowers (e.g. Asclepiads)

Constraining of growth through supershapes as constraining functions

r = SF * f(φ) = CF * DF

Question of Karl J. Niklas: Can supershapes describe fusion in flowers?

Answer: weighted additionr = 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 GM

Arithmetic mean AM Weighted arithmetic mean WAM Relations Area based on AM Area based on GM

Numbers a and b √a.b (a+b)/2 = w1.a + w2.b GM ≤ AM ((a+b)/2)2 a.b

Flowers, DF and CF √ (DF.CF) (DF+CF)/2

άCF + (1- ά)DF

w1+w2= ά+(1- ά)=1

DF.CF ≤ ((DF+CF)/2)2 ((DF+CF)/2)2 DF.CF

Surfaces, k1 and k2

√(κ1 κ2) = √K

(κ1+ κ2)/2 = H

κ1cos2 φ+ κ2 sin2 φ

(Euler’s theorem)w1+w2 = cos2φ+sin2φ = 1

K ≤ H2 (Euler’s inequality)

H2 = ((κ1+ κ2)/2)2 κ1 κ2 = K

The flower model connects to the deepest notions in mathematics; many results from the geometry of surfaces can be used for the flower model

“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

Addition and multiplication, means

• Against the flow Aeθ + Be-θ

• Fixed number raised to a variable power

Functions eθ and e-θ Polar plane XY-plane

Addition & Arithmetic mean

Logarithmic spiral Catenary

Multiplication & Geometric mean

Circle Straight line

Addition and multiplication, means

• Alternatively, a variable raised to a fixed power

Functions xn and ym Expression Graph

Addition & Arithmetic mean xn + ym Lamé curves / superellipses

Multiplication & Geometric mean

xn.ym = Cy = C xn/m Power functions, superparabola