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Graftal
An L-system or Lindenmayer system is a parallel rewriting system and a type of formal grammar. An L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules that expand each symbol into some larger string of symbols, an initial " axiom" string from which to begin construction, and a mechanism for translating the generated strings into geometric structures. L-systems were introduced and developed in 1968 by Aristid Lindenmayer, a Hungarian theoretical biologist and botanist at the University of Utrecht. Lindenmayer used L-systems to describe the behaviour of plant cells and to model the growth processes of plant development. L-systems have also been used to model the morphology of a variety of organisms and can be used to generate self-similar fractals. Origins As a biologist, Lindenmayer worked with yeast and filamentous fungi and studied the growth patterns of various types of bacteria, such as the cyanobacteria '' Anabae ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal Weeds
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called Affine geometry, affine self-similar. Fractal geometry lies within the mathematical branch of measure theory. One way that fractals are different from finite geometric figures is how they Scaling (geometry), scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory. One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if the radius of a filled sphere i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dragon Trees
A dragon is a reptilian legendary creature that appears in the folklore of many cultures worldwide. Beliefs about dragons vary considerably through regions, but dragons in western cultures since the High Middle Ages have often been depicted as winged, horned, and capable of breathing fire. Dragons in eastern cultures are usually depicted as wingless, four-legged, serpentine creatures with above-average intelligence. Commonalities between dragons' traits are often a hybridization of feline, reptilian and avian features. Scholars believe huge extinct or migrating crocodiles bear the closest resemblance, especially when encountered in forested or swampy areas, and are most likely the template of modern Oriental dragon imagery. Etymology The word ''dragon'' entered the English language in the early 13th century from Old French ''dragon'', which in turn comes from la, draconem (nominative ) meaning "huge serpent, dragon", from Ancient Greek , (genitive , ) "serpent, giant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anabaena
''Anabaena'' is a genus of filamentous cyanobacteria that exist as plankton. They are known for nitrogen-fixing abilities, and they form symbiotic relationships with certain plants, such as the mosquito fern. They are one of four genera of cyanobacteria that produce neurotoxins, which are harmful to local wildlife, as well as farm animals and pets. Production of these neurotoxins is assumed to be an input into its symbiotic relationships, protecting the plant from grazing pressure. A DNA sequencing project was undertaken in 1999, which mapped the complete genome of ''Anabaena'', which is 7.2 million base pairs long. The study focused on heterocysts, which convert nitrogen into ammonia. Certain species of ''Anabaena'' have been used on rice paddy fields, proving to be an effective natural fertilizer. Nitrogen fixation by ''Anabaena'' Under nitrogen-limiting conditions, vegetative cells differentiate into heterocysts at semiregular intervals along the filaments. Heterocyst c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turtle Graphics
In computer graphics, turtle graphics are vector graphics using a relative cursor (the "turtle") upon a Cartesian plane (x and y axis). Turtle graphics is a key feature of the Logo programming language. Overview The turtle has three attributes: a location, an orientation (or direction), and a pen. The pen, too, has attributes: color, width, and on/off state (also called ''down'' and ''up''). The turtle moves with commands that are relative to its own position, such as "move forward 10 spaces" and "turn left 90 degrees". The pen carried by the turtle can also be controlled, by enabling it, setting its color, or setting its width. A student could understand (and predict and reason about) the turtle's motion by imagining what they would do if they were the turtle. Seymour Papert called this "body syntonic" reasoning. A full turtle graphics system requires control flow, procedures, and recursion: many turtle drawing programs fall short. From these building blocks one can build ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractint
Fractint is a freeware computer program to render and display many kinds of fractals. The program originated on MS-DOS, then ported to the Atari ST, Linux, and Macintosh. During the early 1990s, Fractint was the definitive fractal generating program for personal computers. The name is a portmanteau of ''fractal'' and ''integer'', since the first versions of Fractint used only integer arithmetic (also known as fixed-point arithmetic), for faster rendering on computers without math coprocessors. Since then, floating-point arithmetic and arbitrary-precision arithmetic modes have been added. Features FractInt can draw most kinds of fractals that have appeared in the literature. It also has a few "fractal types" that are not strictly speaking fractals, but may be more accurately described as display hacks. These include cellular automata. History Fractint originally appeared in 1988 as FRACT386, a computer program for rendering fractals very quickly on the Intel 80386 process ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
D0L System
In mathematics and computer science, a morphic word or substitutive word is an infinite sequence of symbols which is constructed from a particular class of endomorphism of a free monoid. Every automatic sequence is morphic. Definition Let ''f'' be an endomorphism of the free monoid ''A''∗ on an alphabet ''A'' with the property that there is a letter ''a'' such that ''f''(''a'') = ''as'' for a non-empty string ''s'': we say that ''f'' is prolongable at ''a''. The word : a s f(s) f(f(s)) \cdots f^(s) \cdots \ is a pure morphic or pure substitutive word. Note that it is the limit of the sequence ''a'', ''f''(''a''), ''f''(''f''(''a'')), ''f''(''f''(''f''(''a''))), ... It is clearly a fixed point of the endomorphism ''f'': the unique such sequence beginning with the letter ''a''.Lothaire (2011) p. 10Honkala (2010) p.505 In general, a morphic word is the image of a pure morphic word under a coding, that is, a morphism that maps letter to letter. If a morphic word is c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Context-free Grammar
In formal language theory, a context-free grammar (CFG) is a formal grammar whose production rules are of the form :A\ \to\ \alpha with A a ''single'' nonterminal symbol, and \alpha a string of terminals and/or nonterminals (\alpha can be empty). A formal grammar is "context-free" if its production rules can be applied regardless of the context of a nonterminal. No matter which symbols surround it, the single nonterminal on the left hand side can always be replaced by the right hand side. This is what distinguishes it from a context-sensitive grammar. A formal grammar is essentially a set of production rules that describe all possible strings in a given formal language. Production rules are simple replacements. For example, the first rule in the picture, :\langle\text\rangle \to \langle\text\rangle = \langle\text\rangle ; replaces \langle\text\rangle with \langle\text\rangle = \langle\text\rangle ;. There can be multiple replacement rules for a given nonterminal symbol. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Language
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of symbols, letters, or tokens that concatenate into strings of the language. Each string concatenated from symbols of this alphabet is called a word, and the words that belong to a particular formal language are sometimes called ''well-formed words'' or ''well-formed formulas''. A formal language is often defined by means of a formal grammar such as a regular grammar or context-free grammar, which consists of its formation rules. In computer science, formal languages are used among others as the basis for defining the grammar of programming languages and formalized versions of subsets of natural languages in which the words of the language represent concepts that are associated with particular meanings or semantics. In computational complexity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Identity
In logic, the law of identity states that each thing is identical with itself. It is the first of the historical three laws of thought, along with the law of noncontradiction, and the law of excluded middle. However, few systems of logic are built on just these laws. History Ancient philosophy The earliest recorded use of the law appears to occur in Plato's dialogue '' Theaetetus'' (185a), wherein Socrates attempts to establish that what we call "sounds" and "colours" are two different classes of thing: It is used explicitly only once in Aristotle, in a proof in the ''Prior Analytics'': Medieval philosophy Aristotle believed the law of non-contradiction to be the most fundamental law. Both Thomas Aquinas (''Met.'' IV, lect. 6) and Duns Scotus (''Quaest. sup. Met.'' IV, Q. 3) follow Aristotle in this respect. Antonius Andreas, the Spanish disciple of Scotus (d. 1320), argues that the first place should belong to the law "Every Being is a Being" (''Omne Ens est Ens'', Qq. in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defined inductively using the construction of an ordered pair. Mathematicians usually write tuples by listing the elements within parentheses "" and separated by a comma and a space; for example, denotes a 5-tuple. Sometimes other symbols are used to surround the elements, such as square brackets "nbsp; or angle brackets "⟨ ⟩". Braces "" are used to specify arrays in some programming languages but not in mathematical expressions, as they are the standard notation for sets. The term ''tuple'' can often occur when discussing other mathematical objects, such as vectors. In computer science, tuples come in many forms. Most typed functional programming languages implement tuples directly as product types, tightly associated with algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chomsky Hierarchy
In formal language theory, computer science and linguistics, the Chomsky hierarchy (also referred to as the Chomsky–Schützenberger hierarchy) is a containment hierarchy of classes of formal grammars. This hierarchy of grammars was described by Noam Chomsky in 1956. It is also named after Marcel-Paul Schützenberger, who played a crucial role in the development of the theory of formal languages. Formal grammars A formal grammar of this type consists of a finite set of '' production rules'' (''left-hand side'' → ''right-hand side''), where each side consists of a finite sequence of the following symbols: * a finite set of ''nonterminal symbols'' (indicating that some production rule can yet be applied) * a finite set of ''terminal symbols'' (indicating that no production rule can be applied) * a ''start symbol'' (a distinguished nonterminal symbol) A formal grammar provides an axiom schema for (or ''generates'') a ''formal language'', which is a (usually infinite) s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
