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Yoga of Grothendieck

adding articles and wiki

Infinite Dimensionality in Physical Representation

Link to post on Infinity – Dangerous Knowledge doc
Link to Circle is absolute intelligence
Link to Science and Art posts
Link to Collective Unconscious
Link to Expansion of Space
Potential Geometry in Quantum Physics and Vision

Collect Harmony of the Spheres quotes

embed pics of mobius geometry

Golden Ratio Discovered in Quantum World

Spacetime May Have Fractal Properties on a Quantum Scale

The New New Math of String Theory

Scalar Gravitation and Extra Dimensions

(In The Character of Physical Law)Richard Feynman:

“It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of space/time is going to do?”

“So I have often made the hypothesis that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed, and the laws will turn out to be simple, like the checker board with all its apparent complexities. But this speculation is of the same nature as those other people make—’I like it,’ ‘I don’t like it’—and it is not good to be prejudiced about these things.”

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

Georg Cantor:
“Accordingly I distinguish an eternal uncreated infinity or absolutum which is due to God and his attributes, and a created infinity or transfinitum, which has to be used wherever in the created nature an actual infinity has to be noticed, for example, with respect to, according to my firm conviction, the actually infinite number of created individuals, in the universe as well as on our earth and, most probably, even in every arbitrarily small extended piece of space.”

“Georg Cantor’s grand meta-narrative, Set Theory, created by him almost singlehandedly in the span of about fifteen years, resembles a piece of high art more than a scientific theory.” -Y. Manin

“Thus, exquisite minimalism of expressive means is used by Cantor to achieve a sublime goal: understanding infinity, or rather infinity of infinities.” -Y. Manin

Georg Cantor – Dangerous Knowledge – Infinity of Infinities

The Story of Maths – To Infinity and Beyond

BigThink.com Michio Kaku: “In short, the quantum theory allows us to understand the world of the very small and the fundamental properties of matter.

Our deepest understanding of the atomic world comes from the advent of the quantum theory. Having this deep understanding of the various elements of the theory allows us to do much more than just move atoms around or know exactly why things behave the way they do. The theory itself underlies the entire architecture of the world we see today and beyond. It has ultimately allowed us to develop the most advanced technologies to make our lives easier. The marvels of science that we see and use every single day including the Internet, your cell phone, GPS, your email, HD television—all of it—comes from our deep understanding of this theory. This theory offers a very different way to view the world they we live in—one where the simple laws of conventional physics simply don’t apply at all. Quantum theory is so eccentric and peculiar that even Einstein himself couldn’t wrap his head around it. The great physicist, Richard Feynman once stated that “It is impossible, absolutely impossible to explain it in any classical way”.

Some of what quantum theory predicts and states is almost like something out of science fiction. Matter can essentially be in an infinite number of places at any given time; it is possible that there are many worlds or a multiverse; things disappear and reappear somewhere else; you cannot simultaneously know the exact position and momentum of an object; and even quantum entanglement (Einstein referred to it as spooky action at a distance) where it’s possible for two quantum particles to link together effectively making them part of the same entity or entangled. Even if these particles are separated, a change in one is ultimately and instantly reflected in it’s counterpart. At the end of the day, the world of entanglement caused physicists like Einstein to both dislike the predictions and feel nothing more as if their were serious errors in the calculations. As Einstein once wrote: “I find the idea quite intolerable that an electron exposed to radiation should choose of its own free will, not only its moment to jump off, but also its direction. In that case, I would rather be a cobbler, or even an employee in a gaming house, than a physicist”.”
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“Quantum Computing, which is making direct use of the quantum mechanical phenomena, such as superposition and entanglement to perform operations on data. In contrast with a classical computer which has memory made of bits where each bit represents a one or a zero (binary code), a quantum computer will operate on what is called “qubits.” According to Wikipedia, a single qubit can represent a one, a zero, or, crucially, any quantum superposition of these; moreover, a pair of qubits can be in any quantum superposition of 4 states, and three qubits in any superposition of 8 and so on. Superposition refers to the quantum mechanical property which states that all particles exist in not one state but all possible states at once. In short, a quantum computer will essentially be able to crack any algorithm, solve mathematical problems much more quickly and ultimately operate millions of times faster than conventional computers.”
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“The list goes on an on an on: Quantum Dots; Quantum Wires or Carbon Nanotubes; Metamaterials; Invisibility; Quantum Optics; Teleportation; Communication; Space Elevators; Limitless Quantum Energy; Room Temperature Superconductors; Personal Fabicators; Nanotechnology and even Time Travel. Other applications that will strive are advances in battery technology; solar panels; stealth applications; and even advances in biotechnology and medicine. Needless to say, we have only scratched the surface of some of these technologies and time will perfect them. We’ve got a very interesting future ahead of us….”

Information in the Holographic Universe by Jacob D. Bekenstein [July 14,2003] Scientific American
Jakob Bekenstein:
“The proliferation of variations on the holographic motif makes it clear that the subject has not yet reached the status of physical law. But although the holographic way of thinking is not yet fully understood, it seems to be here to stay. And with it comes a realization that the fundamental belief, prevalent for 50 years, that field theory is the ultimate language of physics must give way. Fields, such as the electromagnetic field, vary continuously from point to point, and they thereby describe an infinity of degrees of freedom. Superstring theory also embraces an infinite number of degrees of freedom. Holography restricts the number of degrees of freedom that can be present inside a bounding surface to a finite number; field theory with its infinity cannot be the final story. Furthermore, even if the infinity is tamed, the mysterious dependence of information on surface area must be somehow accommodated.”

Complexity & Chaos – Part 3b (Strange Geometry)

Jakob Bekenstein:
“Thermodynamic entropy is popularly described as the disorder in a physical system. In 1877 Austrian physicist Ludwig Boltzmann characterized it more precisely in terms of the number of distinct microscopic states that the particles composing a chunk of matter could be in while still looking like the same macroscopic chunk of matter. For example, for the air in the room around you, one would count all the ways that the individual gas molecules could be distributed in the room and all the ways they could be moving.”

Singularities and Infinity
Indefinability as ‘failure’ in physics

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

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

Mathematical

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

http://en.wikipedia.org/wiki/Zero-energy_Universe

http://physicsworld.com/cws/article/print/23009

http://icosmos.co.uk/

http://www.flowresearch.com/circular.html

Circular Theory Website

http://en.wikipedia.org/wiki/Lie_sphere_geometry
“The main idea which leads to Lie sphere geometry is that lines (or planes) should be regarded as circles (or spheres) of infinite radius and that points in the plane (or space) should be regarded as circles (or spheres) of zero radius.”

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

quotes

http://en.wikipedia.org/wiki/Category:Infinity

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

http://en.wikipedia.org/wiki/Space_(mathematics)

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

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

http://www.sciencemuseum.org.uk/images/I046/10314748.aspx

http://en.wikipedia.org/wiki/Functional_analysis
In the modern view, functional analysis is seen as the study of vector spaces endowed with a topology, in particular infinite dimensional spaces. In contrast, linear algebra deals mostly with finite dimensional spaces, or does not use topology. An important part of functional analysis is the extension of the theory of measure, integration, and probability to infinite dimensional spaces, also known as infinite dimensional analysis.

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

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

“Real numbers can be thought of as points on an infinitely long number line.”

http://en.wikipedia.org/wiki/Fifth_dimension_(geometry)
“In 1993 the physicist Gerard ‘t Hooft put forward the holographic principle, which explains that the information about an extra dimension is visible as a curvature in a spacetime with one fewer dimensions. For example, holograms are three-dimensional pictures placed on a two-dimensional surface, which gives the image a curvature when the observer moves. Similarly, in general relativity, the fourth dimension is manifested in observable three dimensions as the curvature of path of a moving infinitesimal (test) particle. Hooft has speculated that the fifth dimension is really the spacetime fabric.”

Space-Time-Matter

http://astro.uwaterloo.ca/~wesson/

http://www.archive.org/details/spacetimematter00weyluoft

http://www.popmath.org.uk/sculpmath/pagesm/fibundle.html

http://en.wikipedia.org/wiki/Circle_bundle
In physics, circle bundles are the natural geometric setting for electromagnetism. A circle bundle is a special case of a sphere bundle.

http://en.wikipedia.org/wiki/Kaluza_Klein
In 1926, Oskar Klein proposed that the fourth spatial dimension is curled up in a circle of very small radius, so that a particle moving a short distance along that axis would return to where it began. The distance a particle can travel before reaching its initial position is said to be the size of the dimension. This extra dimension is a compact set, and the phenomenon of having a space-time with compact dimensions is referred to as compactification.

In modern geometry, the extra fifth dimension can be understood to be the circle group U(1), as electromagnetism can essentially be formulated as a gauge theory on a fiber bundle, the circle bundle, with gauge group U(1). Once this geometrical interpretation is understood, it is relatively straightforward to replace U(1) by a general Lie group.

http://faculty.washington.edu/smcohen/320/GrainySpace.html

http://en.wikipedia.org/wiki/Infinite-dimensional_space

http://en.wikipedia.org/wiki/Foundational_crisis_of_mathematics#Foundational_crisis

Mathematics vs. Metamathematics

http://en.wikipedia.org/wiki/Hilbert%27s_program

http://en.wikipedia.org/wiki/M%C3%B6bius_transformation

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

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

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

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

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

quotes, in music, etc

http://www.music.princeton.edu/~dmitri/sciencearticle.html

http://www.time.com/time/magazine/article/0,9171,1582330,00.html

http://harvardmagazine.com/2007/01/mapping-music.html

http://www.brainmusic.org/EducationalActivitiesFolder/Tymoczko_chords2006.pdf

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

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

Local Geometry

http://en.wikipedia.org/wiki/Point_at_infinity
“The point at infinity, also called ideal point, is a point which when added to the real number line yields a closed curve called the real projective line. The real projective line is not equivalent to the extended real number line, which has two different points at infinity.
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This construction can be generalized to an arbitrary topological space. The space so obtained is called the one-point compactification or Alexandroff compactification of the original space. Thus the circle is the one-point compactification of the line, and the sphere is the one-point compactification of the plane.
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In hyperbolic geometry, the ideal point is also called the omega point. Given a line l and a point P not on l, right- and left-limiting parallels to l through P meet at a point on the boundary circle of the Poincaré disk model and the Klein model called the omega point. Pasch’s axiom and the exterior angle theorem still hold for an omega triangle, defined by two points in hyperbolic space and an omega point.”

http://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model

http://www.mcescher.com/Gallery/recogn-bmp/LW436.jpg

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

http://en.wikipedia.org/wiki/Real_projective_line
The symbol \infty represents the point at infinity, an idealized point that bridges the two “ends” of the real line.

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

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

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

http://en.wikipedia.org/wiki/Riemann_sphere
add quotes

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

http://en.wikipedia.org/wiki/N-dimensional_space

“Sometimes it is convenient in science to describe the state of an object with n degrees of freedom as if it were a point in some n-dimensional space. For example, classical mechanics describes the three-dimensional position and momentum of a point particle as a point in 6-dimensional phase space.”

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

http://en.wikipedia.org/wiki/Differential_geometry_of_surfaces
Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss (1825-1827), who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.
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“An important role in their study has been played by Lie groups (in the spirit of the Erlangen program), namely the symmetry groups of the Euclidean plane, the sphere and the hyperbolic plane. These Lie groups can be used to describe surfaces of constant Gaussian curvature; they also provide an essential ingredient in the modern approach to intrinsic differential geometry through connections.”

http://en.wikipedia.org/wiki/Connection_(mathematics)

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

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

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

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

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

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

http://www.livescience.com/strangenews/091216-reappearing-particle-trio.html

http://en.wikipedia.org/wiki/Contact_geometry#Legendrian_submanifolds_and_knots

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

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

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

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

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

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

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

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

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

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

http://mdigest.jrc.ec.europa.eu/soille/soille-rivest96.pdf

http://www.pbs.org/wnet/hawking/universes/html/bound.html
“A proposal first advanced by Stephen Hawking and Jim Hartle, the no-boundary universe is one in which the universe does not start with a singularity. It uses American physicist Richard Feynman’s proposal to treat quantum mechanics as a “sum over histories,” meaning that a particle does not have one history in space-time but instead follows every possible path to reach its current state. By summing these histories—a difficult process that must be done by treating time as imaginary—you can find the probability that the particle passes through a particular point.”

http://webcache.googleusercontent.com/search?q=cache:cQGLK4Vo8hYJ:www.neuroquantology.com/journal/index.php/nq/article/viewFile/397/444+10500+compactified&cd=2&hl=en&ct=clnk&gl=us