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On the plant leaf’s boundary, “jupe à godets” and isometric embeddings Sergei Nechaev LPTMS (Orsay); FIAN (Moscow) lettuce (Lactuca sativa) What are the geometrical reasons for the boundary of plant leaves to fluctuate in the direction transverse to the leaf’s surface? 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: x cos u sin u y sin u sin v z cos v log tan v / 2 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 * d d 2 ds * 2 (1 ) This structure is invariant under conformal transforms of the Poincaré disc onto itself 0 1 * 0 The stereographic projection of an open disc gives the 2D hyperboloid with the metric tensor gik 1 0 0 sinh 2 , ds2 d 2 sinh2 d 2 1 r where ln is the hyperbolic distance. 1 r The tessellation of the Poincaré disc by circular triangles (rectangles) is uniform in a surface of a twodimensional hyperboloid. It can be embedded into a 3D space with a Minkovski metric, i.e. with a metric tensor of signature {+1, +1,−1}. 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 4branching 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 zeroangled rectangle AζBζA’ζCζ in the Poincaré disc |ζ|<1. The main statement (continuation) The relief of the corresponding surface is encoded in 2 the so-called coefficient of deformation, J ( ) dz / d coinciding with the Jacobian of the conformal transform z(ζ ): Re z Re 2 J ( ) z '( ) Im z Re Re z Im Im z Im Finding explicit form of z(ζ ) is our main goal. 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 S Az Bz A'z Cz dzdz * A B A' C J ( z )d d * 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. Explicit construction by conformal maps Explicit expression in terms of elliptic functions dz( ) 4 ' J ( ) 2 1 0, e 4 d i 2 1 , ei 2e 1' 0, ei 1i 1 i 2 i 2 2 0, e 1i 1 i 2 i i 4 n i n ( n 1) ( 1) sin(2n 1) e n 0 d1 , ei 2 0, ei 2e d i 4 2e i 4 0 n i n ( n 1) ( 1) e n 0 n i n ( n 1) ( 1) (2 n 1) e n 0 2 3D plots of SG Sample plots of J ( , ) for 0 <<max and 0<ϕ</2 1 r Im ln ; arctan 1 r Re max = 1.2 max = 1.5 max = 1.8 The boundary of the SG is a fractal 0.2 0.4 0.6 0.8 max = 2.4 1.5 1.8 2.1 1.0 1.2 1.4 0<ϕ</2 SG is a “continuous Cayley tree” Plot the “normalized” surface g ( ) J ( ) 1 (1 ) * 2 r exp(i ) 1 r Im ln ; arctan 1 r Re This surface can be considered as a “continuous analog” of the Cayley tree.