On the plant leaf’s boundary, “jupe à godets” and

isometric embeddings

Sergei Nechaev

LPTMS (Orsay); FIAN (Moscow)

What are the geometrical reasons for the boundary of plant leaves to fluctuate in the direction transverse to the leaf’s surface?

lettuce (Lactuca sativa)

How to describe the corresponding stable profile?

Some more examples…

Hierarchical construction of an unbounded hyperbolic surface

jupe à godets – “surface à godets” (SG)

We are interested in the embedding of unbounded (open) surfaces only!

Hyperbolic surfaces with a boundary can be isometrically embedded in a Euclidean space. The pseudosphere (surface of revolution of the tractrix) is an example of such an embedding:

cos sin

sin sin

cos log tan / 2

x u u

y u v

z v v

Bounded vs open hyperbolic surfaces (M. Esher)

Precise formulation of the problem

1.The SG, being the hyperbolic structure, admits Cayley trees as possible discretizations. Cover the SG by a lattice – the 4-branching Cayley tree. The Cayley trees cover the SG isometrically, i.e. without gaps and selfintersections, preserving angles and distances.

2.Our aim consists in the embedding a 4-branching Cayley tree (isometrically covering the SG) into a 3D Euclidean metric space with a signature {+1,+1,+1}.

Example

Tesselate isometrically the open disc |ζ | < 1 with all images of the some triangle. This way one gets a graph isometrically embedded into the unit disc endowed with the Poincaré metric

*2

* 2(1 )

d dds

0*0 1

This structure is invariant under conformal transforms of the Poincaré disc onto itself

The stereographic projection of an open disc gives the 2D hyperboloid with the metric tensor

where is the hyperbolic distance.

The tessellation of the Poincaré disc by circular triangles (rectangles) is uniform in a surface of a two-dimensional hyperboloid. It can be embedded into a 3D space with a Minkovski metric, i.e. with a metric tensor of signature {+1, +1,−1}.

2

1 0,

0 sinhikg 2 2 2 2sinh ds d d

1ln

1

r

r

The main statement

Tesselate isometrically the disc |ζ | < 1 with all images of zero-angled rectangle AζBζA’ζCζ. Connecting the centres of the neighbouring rectangles, one gets a 4-branching Cayley tree isometrically covering the unit Poincaré disc.

The requested isometric embedding of a 4-branching Cayley tree into a 3D Euclidean space is realized via conformal transform z=z(ζ ) which maps a flat square AzBzA’zCz in the complex plane z to a circular zero-angled rectangle AζBζA’ζCζ in the Poincaré disc |ζ|<1.

The main statement (continuation)

The relief of the corresponding surface is encoded in the so-called coefficient of deformation,coinciding with the Jacobian of the conformal transform z(ζ ):

Finding explicit form of z(ζ ) is our main goal.

2

Re Re

Re Im( ) '( )

Im Im

Re Im

z z

J zz z

2( ) /J dz d

Remark: In which sense our surface is optimal?

All the images of the zero-angled rectangle AζBζA’ζCζ in the complex plane ζ have the same area. The areas of elementary cells AzBzA’zCz tessellating the plane z are

Thus, all zero-angled rectangles have the same area S in ζ.

Consequence. Under requested constraints the surface J(z)=|z’(ζ )|2 is minimal.

z z z z

* *

A B A' C A B A' C( )S dzdz J z d d

Explicit construction by conformal maps

Explicit expression in terms of elliptic functions

( 1)41

0

1' ( 1)41

00

( 1)42

0

, 2 ( 1) sin(2 1)

,0, 2 ( 1) (2 1)

0, 2 ( 1)

ii n i n n

n

iii n i n n

n

ii n i n n

n

e e e n

d ee e n e

d

e e e

2 22 1 1 1 1' 2 2

1 242

( ) 4( ) 0, 0,

i ii i

i idzJ e e

d i

3D plots of SG

Sample plots of for 0 <<max and 0<ϕ</2 ( , )J

max = 1.2

max = 1.5

max = 1.8

1 Imln ; arctan

1 Re

r

r

The boundary of the SG is a fractal

0 .2 0 .4 0 .6 0 .8 1 .0 1 .2 1 .40 .2 0 .4 0 .6 0 .8 1 .0 1 .2 1 .40 .2 0 .4 0 .6 0 .8 1 .0 1 .2 1 .40 .2 0 .4 0 .6 0 .8 1 .0 1 .2 1 .4

max = 1.5 0<ϕ</2max = 1.8max = 2.1max = 2.4

SG is a “continuous Cayley tree”

Plot the “normalized” surface

This surface can be considered as a “continuous analog” of the Cayley tree.

* 2

1( ) ( )

(1 )g J

exp( )

1 Imln ; arctan

1 Re

r i

r

r