1 Graphical modeling using L-systems.- 1.1 Rewriting systems.- 1.2 DOL-systems.- 1.3 Turtle interpretation of strings.- 1.4 Synthesis of DOL-systems.- 1.4.1 Edge rewriting.- 1.4.2 Node rewriting.- 1.4.3 Relationship between edge and node rewriting.- 1.5 Modeling in three dimensions.- 1.6 Branching structures.- 1.6.1 Axial trees.- 1.6.2 Tree OL-systems.- 1.6.3 Bracketed OL-systems.- 1.7 Stochastic L-systems.- 1.8 Context-sensitive L-systems.- 1.9 Growth functions.- 1.10 Parametric L-systems.- 1.10.1 Parametric OL-systems.- 1.10.2 Parametric 2L-systems.- 1.10.3 Turtle interpretation of parametric words.- 2 Modeling of trees.- 3 Developmental models of herbaceous plants.- 3.1 Levels of model specification.- 3.1.1 Partial L-systems.- 3.1.2 Control mechanisms in plants.- 3.1.3 Complete models.- 3.2 Branching patterns.- 3.3 Models of inflorescences.- 3.3.1 Monopodial inflorescences.- 3.3.2 Sympodial inflorescences.- 3.3.3 Polypodial inflorescences.- 3.3.4 Modified racemes.- 4 Phyllotaxis.- 4.1 The planar model.- 4.2 The cylindrical model.- 5 Models of plant organs.- 5.1 Predefined surfaces.- 5.2 Developmental surface models.- 5.3 Models of compound leaves.- 6 Animation of plant development.- 6.1 Timed DOL-systems.- 6.2 Selection of growth functions.- 6.2.1 Development of nonbranching filaments.- 6.2.2 Development of branching structures.- 7 Modeling of cellular layers.- 7.1 Map L-systems.- 7.2 Graphical interpretation of maps.- 7.3 Microsorium linguaeforme.- 7.4 Dryopteris thelypteris.- 7.5 Modeling spherical cell layers.- 7.6 Modeling 3D cellular structures.- 8 Fractal properties of plants.- 8.1 Symmetry and self-similarity.- 8.2 Plant models and iterated function systems.- Epilogue.- Appendix A Software environment for plant modeling.- A.1 A virtual laboratory in botany.- A.2 List of laboratory programs.- Appendix B About the figures.- Turtle interpretation of symbols.

/pdf/the-algorithmic-beauty-of-plants-4f9yunhil9.pdf

The Algorithmic Beauty of Plants

The braid groups are infinite non-commutative groups naturally arising from geometric braids The aim of this article is twofold One is to show that the braid groups can serve as a good source to enrich cryptography The feature that makes the braid groups useful to cryptography includes the followings: (i) The word problem is solved via a fast algorithm which computes the canonical form which can be efficiently manipulated by computers (ii) The group operations can be performed efficiently (iii) The braid groups have many mathematically hard problems that can be utilized to design cryptographic primitives The other is to propose and implement a new key agreement scheme and public key cryptosystem based on these primitives in the braid groups The efficiency of our systems is demonstrated by their speed and information rate The security of our systems is based on topological, combinatorial and group-theoretical problems that are intractible according to our current mathematical knowledge The foundation of our systems is quite different from widely used cryptosystems based on number theory, but there are some similarities in design

/pdf/new-public-key-cryptosystem-using-braid-groups-4jt4e0sw77.pdf

New Public-Key Cryptosystem Using Braid Groups

The aim of this chapter is to generalize concepts and techniques of formal language theory to two dimensions. Informally, a two-dimensional string is called a picture and is defined as a rectangular array of symbols taken from a finite alphabet. A two-dimensional language (or picture language) is a set of pictures.

Two-dimensional languages

This paper selectively surveys contributions to major topics in pattern recognition since 1968. Representative books and surveys pattern recognition published during this period are listed. Theoretical models for automatic pattern recognition are contrasted with practical,, design methodology. Research contributions to statistical and structural pattern recognition are selectively discussed, including contributions to error estimation and the experimental design of pattern classifiers. The survey concludes with a representative set of applications of pattern recognition technology.

Patterns in pattern recognition: 1968-1974

A new method for generating pictures is presented and illustrated with examples. The idea is to generate a string of symbols using an L−system, and to interpret this string as a sequence of commands which control a "turtle". Suitable generalizations of the notions of the L−system and of a turtle are introduced. The resulting mathematical model can be used to create a variety of (finite approximations of) fractal curves, ranging from Koch curves, to classic space−filling curves, to relatively realistic−looking pictures of plants and trees. All these pictures are defined in a uniform and compact way.

http://algorithmicbotany.org/papers/graphical.gi86.pdf

Graphical applications of L-systems

Radial Grammars and Radial L-Systems

Graph grammars and array grammars for the description and analysis of two-dimensional structure have been extensively studied [2,4,11,19]. More than a decade ago, Rosenfeld [i0] argued the case for cycle grammars for generating patterns having various types of rotational symmetry [14,15]. Languages of cycle grammars consist of cycles ("necklaces") of symbols i.e. cyclically ordered sequences of symbols. There are several interesting families of pictures both natural as well as artificial which are cycle languages. Outlines of chromosome shapes regarded as sequences of basic shape elements form a natural family of cycles. These were among the earliest examples for which grammars were given. Several of the fractals including some classical space-filling curves provide us with interesting examples of picture cycles [9,20]. Map-generating systems [5,8,18] are discussed for the formulation of developmental mechanisms producing cell division patterns in meristematic cell layers [6] and for computer implementations. In the case of edge production rules with single labels, the wall division rules may be based on circular edge sequences and these again provide us with natural examples of cycles. Certain families of kolam patterns viz the kambi kolam literally meaning wire decoration which can be represented as a single strand provide us with illustrative examples of picture cycle languages. Each kambi begins and ends at the same point i.e. each kambi is an unending line or a closed curve with or without loops[3].

Rosenfeld's cycle grammars and kolam

Infinite arrays and infinite computations

In 1968 Lindenmayer introduced a mathematical model of developmental systems [28]. This involves parallel rewriting in which each letter in a string is rewritten using production rules and there is no distinction between terminal and nonterminal symbols reflecting the simultaneous growth of each cell during different stages of development. Since then L-systems have been studied extensively, resulting in the number of research papers growing exponentially [19,29,54,56]. An interesting comment from a referee is a pointer to the extent to which the theory had developed within a span of five years. In 1973 we submitted a paper entitled Parallel O-Lindenmayer Languages which we abbreviated as POL [69]. Then pat came the referee’s comment: “P already stands for ’propagating’ and almost all letters of the alphabet have been used up. You may try PaOL”.

Array Languages and Lindenmayer Systems —A Survey

The u-v theorem for context-free languages is extended to prove an intercalation theorem for the family of context-free matrix languages. A row-wise iteration factor theorem is proved for the families of regular and context-free matrix languages. Characterizations of regular and context-free matrix languages are given in terms of vertical regular sequences and simple operations on vertical regular sequences. Closure of regular and context-free matrix languages under array nondeterministic finite state transducer mappings is established and an image theorem proved. This is used to give another characterization of regular matrix languages. Further it is shown that the family of regular matrix languages is a principal abstract family of matrices (AFM). The effect of string control and array control on these families are examined.

Rani Siromoney

Papers

Radial Grammars and Radial L-Systems

Rosenfeld's cycle grammars and kolam

Infinite arrays and infinite computations

Array Languages and Lindenmayer Systems —A Survey

Characterizations of regular and context-free matrices