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

1. On the plant leaf’s boundary, “jupe à godets” andisometric embeddings Sergei Nechaev LPTMS (Orsay); FIAN (Moscow)

2. lettuce (Lactuca sativa) What are the geometrical reasons for theboundary of plant leaves to fluctuate in the directiontransverse to the leaf’s surface? How to describe the corresponding stable profile?

3. Some more examples…

4. Hierarchical construction of an unbounded hyperbolic surface jupe à godets – “surface à godets” (SG)

5. 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:

6. Bounded vs open hyperbolic surfaces (M. Esher)

7. Precise formulation of the problem • 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. • Our aimconsists in the embedding a 4-branching Cayley tree (isometrically covering the SG) into a 3D Euclidean metric space with a signature {+1,+1,+1}.

8. Example Tesselate isometrically the open disc |ζ | < 1with all images of the some triangle. This way one gets a graph isometrically embedded into the unit disc endowed with the Poincaré metric This structure is invariant under conformaltransforms of the Poincaré disc onto itself

9. 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) isuniform in a surface of a two-dimensional hyperboloid. It can be embedded into a 3D space with aMinkovski metric, i.e. with a metric tensor of signature {+1, +1,−1}.

10. The main statement Tesselate isometrically the disc |ζ | < 1with all images of zero-angled rectangle AζBζA’ζCζ. Connecting the centres of the neighbouringrectangles, one gets a 4-branching Cayley treeisometrically 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)whichmaps a flat squareAzBzA’zCzin the complex plane z to a circular zero-angledrectangleAζBζA’ζCζin the Poincaré disc|ζ|<1.

11. The main statement (continuation) The relief of the correspondingsurface is encoded in the so-called coefficient of deformation, coinciding with the Jacobian of the conformal transformz(ζ): Finding explicit form of z(ζ) is our main goal.

12. Remark: In which sense our surface is optimal? All the images of the zero-angled rectangle AζBζA’ζCζin thecomplex plane ζ have the same area. The areas of elementary cellsAzBzA’zCz tessellating the plane zare Thus, all zero-angled rectangles have the same area Sin ζ. Consequence. Under requested constraints the surface J(z)=|z’(ζ)|2is minimal.

13. Explicit construction by conformal maps

14. Explicit expression in terms of elliptic functions

15. 3D plots of SG Sample plots of for 0 <m<mmax and 0<ϕ<p/2 mmax = 1.2 mmax = 1.5 mmax = 1.8

16. The boundary of the SG is a fractal 0<ϕ<p/2 mmax = 1.5 mmax = 1.8 mmax = 2.1 mmax = 2.4

17. SG is a “continuous Cayley tree” Plot the “normalized” surface This surface can be considered as a “continuous analog” of the Cayley tree.