Patterns in nature Natural patterns form as wind blows sand in the dunes of the Namib Desert. The crescent shaped dunes and the ripples on their surfaces repeat wherever there are suitable conditions. Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, arrays, cracks and stripes.[1] Early Greek philosophers studied pattern, with Plato, Pythagoras and Empedocles attempting to explain order in nature. The modern understanding of visible patterns developed gradually over time. In the 19th century, Belgian physicist Joseph Plateau examined soap films, leading him to formulate the concept of a minimal surface. Mathematics, physics and chemistry can explain patterns in nature at different levels. History[edit] The American photographer Wilson Bentley (1865–1931) took the first micrograph of a snowflake in 1885.[10]
Trace theory From Wikipedia, the free encyclopedia Theory of trace monoids In mathematics and computer science, trace theory aims to provide a concrete mathematical underpinning for the study of concurrent computation and process calculi. The underpinning is provided by an algebraic definition of the free partially commutative monoid or trace monoid, or equivalently, the history monoid, which provides a concrete algebraic foundation, analogous to the way that the free monoid provides the underpinning for formal languages. The power of trace theory stems from the fact that the algebra of dependency graphs (such as Petri nets) is isomorphic to that of trace monoids, and thus, one can apply both algebraic formal language tools, as well as tools from graph theory.
Songs of Innocence and of Experience Songs of Innocence and of Experience is an illustrated collection of poems by William Blake. It appeared in two phases. A few first copies were printed and illuminated by William Blake himself in 1789; five years later he bound these poems with a set of new poems in a volume titled Songs of Innocence and of Experience Showing the Two Contrary States of the Human Soul. "Innocence" and "Experience" are definitions of consciousness that rethink Milton's existential-mythic states of "Paradise" and the "Fall." Blake's categories are modes of perception that tend to coordinate with a chronology that would become standard in Romanticism: childhood is a state of protected innocence rather than original sin, but not immune to the fallen world and its institutions. Songs of Innocence[edit] Songs of Innocence was originally a complete work first printed in 1789. The poems are each listed below: The Echoing Green The Lamb The Little Black Boy The Blossom The Chimney Sweeper The Little Boy found Infant Joy
Constructal Theory and the Asynsis Principle | ASYNSIS It’s with a sense of relief and vindication that work that I’ve previously termed Dynamical Symmetries or more recently, Asynsis (asymptotic synthesis), regarding optimal information, mass and energy flows in nature has also occurred in parallel, over a similar timeframe of 20-odd years. It’s called Constructal Law or Theory, if you prefer. The author of this research (which comes to similar conclusions), is the renowned Professor Adrian Bejan of the Pratt School of Mechanical Engineering at Duke University, NC, USA. He describes himself as an engineer and thermodynamicist as elaborated on here: The substantial and diverse academic work in the constructal field is to be found here: I also feel that our work is highly complementary in that he approached it as an engineer while I did as an architect. “Both positive and negative feedback loops effect a system.
Essential Math for Games Programmers As the quality of games has improved, more attention has been given to all aspects of a game to increase the feeling of reality during gameplay and distinguish it from its competitors. Mathematics provides much of the groundwork for this improvement in realism. And a large part of this improvement is due to the addition of physical simulation. Creating such a simulation may appear to be a daunting task, but given the right background it is not too difficult, and can add a great deal of realism to animation systems, and interactions between avatars and the world. This tutorial deepens the approach of the previous years' Essential Math for Games Programmers, by spending one day on general math topics, and one day focusing in on the topic of physical simulation. It, like the previous tutorials, provides a toolbox of techniques for programmers, with references and links for those looking for more information. Topics for the various incarnations of this tutorial can be found below. Slides
Rewriting Replacing subterm in a formula with another term In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a formula with other terms. Such methods may be achieved by rewriting systems (also known as rewrite systems, rewrite engines,[1][2] or reduction systems). In their most basic form, they consist of a set of objects, plus relations on how to transform those objects. Rewriting can be non-deterministic. One rule to rewrite a term could be applied in many different ways to that term, or more than one rule could be applicable. Example cases[edit] Logic[edit] In logic, the procedure for obtaining the conjunctive normal form (CNF) of a formula can be implemented as a rewriting system.[6] The rules of an example of such a system would be: (double negation elimination) (De Morgan's laws) (distributivity) [note 1] where the symbol ( Arithmetic[edit] For example, the computation of 2+2 to result in 4 can be duplicated by term rewriting as follows: .
Yggdrasill Da Wikipedia, l'enciclopedia libera. Yggdrasill in un manoscritto islandese del XVII secolo. Yggdrasill [ˈygˌdrasilː], nella mitologia norrena, è l'albero cosmico, l'albero del mondo. Natura, struttura, funzione[modifica | modifica sorgente] Secondo Völuspá è un frassino (norreno askr); secondo Rodolfo di Fulda, monaco benedettino del IX secolo, che lo denomina come Irminsul[1] è invece un tasso o una quercia, (alberi comunque sacri presso i popoli del Nord Europa); il suo nome significa con ogni probabilità "cavallo di Yggr", dove "cavallo" è metafora per "forca", "patibolo", mentre Yggr è uno dei tanti nomi di Óðinn. Immenso, Yggdrasill sprofonda sin nel regno infero, mentre i suoi rami sostengono l'intera volta celeste. L'albero Yggdrasill è il luogo dell'assemblea (Thing) quotidiana degli Dèi che vi giungono cavalcando il ponte di Bifröst (l'Arcobaleno), vigilato dal dio Heimdallr. Altro nome dell'albero cosmico è Mímameiðr ("albero di Mími"). Note[modifica | modifica sorgente] Irminsul
Emergence In philosophy, systems theory, science, and art, emergence is a process whereby larger entities, patterns, and regularities arise through interactions among smaller or simpler entities that themselves do not exhibit such properties. Emergence is central in theories of integrative levels and of complex systems. For instance, the phenomenon life as studied in biology is commonly perceived as an emergent property of interacting molecules as studied in chemistry, whose phenomena reflect interactions among elementary particles, modeled in particle physics, that at such higher mass—via substantial conglomeration—exhibit motion as modeled in gravitational physics. In philosophy, emergence typically refers to emergentism. In philosophy[edit] Main article: Emergentism In philosophy, emergence is often understood to be a claim about the etiology of a system's properties. Definitions[edit] This idea of emergence has been around since at least the time of Aristotle. Strong and weak emergence[edit]