Angel "Java" Lopez en Blog

Publicado el 31 de Agosto, 2012, 13:24

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Geometría es un tema que en este siglo me lo encuentro cada vez más, tanto relacionado con la teoría de grupos, como con teorías físicas. Una primera lista de los enlaces que he visitado. Vean en el fragmento de la Wikipedia, cómo con Descartes la geometría va fusionándose con el álgebra. Gran parte de la búsqueda de simetría en las teorías físicas, es una lucha por volver a lo geométrico, sin depender de sistema de coordenadas o marco de referencia. Visiten también lo que hay sobre el grupo E8.

http://en.wikipedia.org/wiki/Geometry

Geometry (Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of a formal mathematical science emerging in the West as early as Thales (6th Century BC). By the 3rd century BC geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow.[1] Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. The field of astronomy, especially mapping the positions of the stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia. Both geometry and astronomy were considered in the classical world to be part of the Quadrivium, a subset of the seven liberal arts considered essential for a free citizen to master.

The introduction of coordinates by René Descartes and the concurrent developments of algebra marked a new stage for geometry, since geometric figures, such as plane curves, could now be represented analytically, i.e., with functions and equations. This played a key role in the emergence of infinitesimal calculus in the 17th century. Furthermore, the theory of perspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin ofprojective geometry. The subject of geometry was further enriched by the study of intrinsic structure of geometric objects that originated with Euler and Gauss and led to the creation of topology and differential geometry.

In Euclid's time there was no clear distinction between physical space and geometrical space. Since the 19th-century discovery of non-Euclidean geometry, the concept of space has undergone a radical transformation, and the question arose: which geometrical space best fits physical space? With the rise of formal mathematics in the 20th century, also 'space' (and 'point', 'line', 'plane') lost its intuitive contents, so today we have to distinguish between physical space, geometrical spaces (in which 'space', 'point' etc. still have their intuitive meaning) and abstract spaces. Contemporary geometry considers manifolds, spaces that are considerably more abstract than the familiar Euclidean space, which they only approximately resemble at small scales. These spaces may be endowed with additional structure, allowing one to speak about length. Modern geometry has multiple strong bonds with physics, exemplified by the ties between pseudo-Riemannian geometry and general relativity. One of the youngest physical theories, string theory, is also very geometric in flavour.

Geometric visual hallucinations
http://rstb.royalsocietypublishing.org/content/356/1407/299.full.pdf

The Geomblog: Geometry @ Barriers
http://geomblog.blogspot.com/2010/09/geometry-barriers.html

Geometric algorithms
The Universe of Discourse : A new proof that the square root of 2 is irrational
http://blog.plover.com/math/sqrt-2-new.html

Euclides.org
http://www.euclides.org

George W. Hart
http://www.georgehart.com/
Geometric sculptures and puzzles

The Museum of Mathematics
http://momath.org/

Triángulos disjuntos (Puzzle)
http://gaussianos.com/triangulos-disjuntos/

Touch Trigonometry
http://www.touchtrigonometry.org/

An introduction to the ancient and modern geometry of conics
http://openlibrary.org/works/OL7563147W/An_introduction_to_the_ancient_and_modern_geometry_of_conics

Superformula
http://en.wikipedia.org/wiki/Superformula

Pappus chain
http://en.wikipedia.org/wiki/Pappus_chain

Physics intuitions: Forwards multiplying, backwards dividing
http://commonsensequantum.blogspot.com/2011/04/forwards-multiplying-backwards-dividing.html

Physics intuitions: Morley triangle derived from the tripling of an angle
http://commonsensequantum.blogspot.com/2011/03/morley-triangle-derived-from-tripling.html

Physics intuitions: Archimedes angle trisection or tripling?
http://commonsensequantum.blogspot.com/2011/03/archimedes-angle-trisection-or-tripling.html

Angle trisection
http://en.wikipedia.org/wiki/Angle_trisection#With_a_marked_ruler

Physics intuitions: Playing with angles
http://commonsensequantum.blogspot.com/2011/03/playing-with-angles.html

Albrecht Dürer"s ruler and compass constructions
http://divisbyzero.com/2011/03/22/albrecht-durers-ruler-and-compass-constructions/

PLaSM
http://plasm.net/
functional language for computing with geometry

Manifolds « The Unapologetic Mathematician
http://unapologetic.wordpress.com/2011/02/22/manifolds/

Heron's Formula
http://golem.ph.utexas.edu/category/2011/02/herons_formula.html

La construccion del dodecaedro en los elementos de Euclides
http://gaussianos.com/la-construccion-del-dodecaedro-en-los-elementos-de-euclides/

Introduction to Clifford Algebra
http://www.av8n.com/physics/clifford-intro.htm

Clifford Algebras
http://math.ucr.edu/home/baez/octonions/node6.html
Cli ord Algebras, Cli ord Groups,
and a Generalization of the Quaternions:

The Pin and Spin Groups
http://www.cis.upenn.edu/~cis610/clifford.pdf

AIM math: Representations of E8
http://www.aimath.org/E8/

Dirac belt trick
http://gregegan.customer.netspace.net.au/APPLETS/21/21.html

Three geometric theorems
http://divisbyzero.com/2010/12/24/three-geometric-theorems/

What is E8?
http://www.aimath.org/E8/e8.html

Three geometric theorems « Division by Zero
http://divisbyzero.com/2010/12/24/three-geometric-theorems/

A Geometric Theory Of Everything
http://www.science20.com/quantum_diaries_survivor/geometric_theory_everything

A Geometric Theory of Everything: Scientific American
http://www.scientificamerican.com/article.cfm?id=a-geometric-theory-of-everything

A Geometric Theory of Everything « Not Even Wrong
http://www.math.columbia.edu/~woit/wordpress/?p=3292

E8 (mathematics) - Wikipedia, the free encyclopedia
http://en.wikipedia.org/wiki/E8_(mathematics)

Cayley graphs and the geometry of groups « What"s new
http://terrytao.wordpress.com/2010/07/10/cayley-graphs-and-the-geometry-of-groups/

On growth and form : Thompson, D'Arcy Wentworth, 1860-1948
http://www.archive.org/details/ongrowthform00thom

The Geometry of the MRB constant
http://math-blog.com/2010/11/21/the-geometry-of-the-mrb-constant/

La línea de Simson | Gaussianos
http://gaussianos.com/la-linea-de-simson/

Pictures of Modular Curves (III) | The n-Category Café
http://golem.ph.utexas.edu/category/2010/11/pictures_of_modular_curves_iii.html

Reflections and Rotations
http://www.youtube.com/watch?v=gJMJBc-f9IQ&feature=related

Trigonometric functions and rational multiples of pi
http://divisbyzero.com/2010/10/28/trigonometric-functions-and-rational-multiples-of-pi/

Finding Haystacks (and Similar Structures) in Geometry
http://valis.cs.uiuc.edu/%7Esariel/papers/10/barriers/barriers.pdf

Los centros del triangulo: el punto de Lemoine
http://gaussianos.com/los-centros-del-triangulo-el-punto-de-lemoine/

A Geometric Paradox | Futility Closet
http://www.futilitycloset.com/2010/09/04/a-geometric-paradox/

Physics intuitions: Alternative Pythagorean quadruples and other extensions to Pythagoras theorem
http://commonsensequantum.blogspot.com/2010/08/alternative-pythagorean-quadruples-and.html

Physics intuitions: A Pythagorean relation for any triangle?
http://commonsensequantum.blogspot.com/2010/08/pythagorean-relation-for-any-triangle.html

Los centros del triángulo: el centro de la circunferencia de los nueve puntos | Gaussianos
http://gaussianos.com/los-centros-del-triangulo-el-centro-de-la-circunferencia-de-los-nueve-puntos/

Physics intuitions: Lost theorem about angular proportions
http://commonsensequantum.blogspot.com/2010/06/lost-theorem-about-angular-proportions.html

Variations on dividing circular area into equal parts
http://commonsensequantum.blogspot.com/2010/03/variations-on-dividing-circular-area.html

Bill Kerr: 40 maths shapes challenges
http://billkerr2.blogspot.com/2009/08/40-maths-shapes-challenges.html

GeoGebra
http://www.geogebra.org/cms/
Free mathematics software for learning and teaching

Quantum mechanics and geometry
http://sbseminar.wordpress.com/2009/11/16/quantum-mechanics-and-geometry/

Rhombus tilings and an over-constrained recurrence
http://sbseminar.wordpress.com/2009/10/21/rhombus-tilings-and-an-over-constrained-recurrence/

Historia de la Geometria
http://www.euclides.org/menu/articles/historiadelageometria.htm

Mis Enlaces
http://www.delicious.com/ajlopez/geometry

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Angel "Java" Lopez
http://www.ajlopez.com
http://twitter.com/ajlopez