July 12, 2010

Potential Geometry in Quantum Physics and Vision

Newscientist

http://www.circular-theory.com/

http://www.google.com/search?hl=en&safe=off&q=E8+Lie+Group+geometry+optics&btnG=Search&aq=f&aqi=&aql=&oq=&gs_rfai=

add ed witten interview

E8 Lie Group

http://www.squidoo.com/lisigarrett

Garret Lisi responds to refutation

The Scientific Promise of Perfect Symmetry

TedTalk – Garret Lisi

Garrett Lisi Through the Wormhole Visualizing Circles

http://www.physorg.com/search/?search=e8+lie+group+geometry+optics

search Scripps Research Institute E8 symmetry DNA

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

“The Golden Ratio, roughly equal to 1.618, was first formally introduced in text by Greek mathematician Pythagoras and later by Euclid in the 5th century BC. In the fourth century BC, Aristotle noted its aesthetic properties.
Aside from interesting mathematical properties, geometric shapes derived from the golden ratio, such as the golden rectangle, the golden triangle, and Kepler’s triangle, were believed to be aesthetically pleasing. As such, many works of ancient art exhibit and incorporate the golden ratio in their design. Various authors can discern the presence of the golden ratio in Egyptian, Sumerian and Greek vases, Chinese pottery, Olmec sculptures, and Cretan and Mycenaean products from as early as the late Bronze Age. The prevalence of this special number in art and architecture even before its formal discovery by Pythagoras is perhaps evidence of an instinctive and primal human cognitive preference for the golden ratio.”

Nature by Numbers

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

It is in De Divina Proportione that the golden ratio is defined as the divine proportion. Pacioli also details the use of the golden ratio as the mathematical definition of beauty when applied to the human face.

“The Ancients, having taken into consideration the rigorous construction of the human body, elaborated all their works, as especially their holy temples, according to these proportions; for they found here the two principal figures without which no project is possible: the perfection of the circle, the principle of all regular bodies, and the equilateral square.” from De Divina Proportione (1509)

http://www.bl.uk/learning/cult/bodies/vitruvius/proportion.html

Complexity & Chaos – Part 13b

http://jersey.uoregon.edu/~js/ast123/lectures/lec17.html

vaccuum

http://www.pnas.org/content/100/20/11216.full.pdf

http://mathworld.wolfram.com/HyperspherePacking.html

http://en.wikipedia.org/wiki/Self-organized_criticality

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

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

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

http://en.wikipedia.org/wiki/First-order_phase_transition

Golden ratio discovered in a quantum world
“When applying a magnetic field at right angles to an aligned spin the magnetic chain will transform into a new state called quantum critical, which can be thought of as a quantum version of a fractal pattern. Prof. Alan Tennant, the leader of the Berlin group, explains “The system reaches a quantum uncertain – or a Schrödinger cat state. This is what we did in our experiments with cobalt niobate. We have tuned the system exactly in order to turn it quantum critical.”

By tuning the system and artificially introducing more quantum uncertainty the researchers observed that the chain of atoms acts like a nanoscale guitar string. Dr. Radu Coldea from Oxford University, who is the principal author of the paper and drove the international project from its inception a decade ago until the present, explains: “Here the tension comes from the interaction between spins causing them to magnetically resonate. For these interactions we found a series (scale) of resonant notes: The first two notes show a perfect relationship with each other. Their frequencies (pitch) are in the ratio of 1.618…, which is the golden ratio famous from art and architecture.” Radu Coldea is convinced that this is no coincidence. “It reflects a beautiful property of the quantum system – a hidden symmetry. Actually quite a special one called E8 by mathematicians, and this is its first observation in a material”, he explains.

The observed resonant states in cobalt niobate are a dramatic laboratory illustration of the way in which mathematical theories developed for particle physics may find application in nanoscale science and ultimately in future technology. Prof. Tennant remarks on the perfect harmony found in quantum uncertainty instead of disorder. “Such discoveries are leading physicists to speculate that the quantum, atomic scale world may have its own underlying order. Similar surprises may await researchers in other materials in the quantum critical state.”

World-Science.net – Golden Ratio hints at atomic symmetry

Isfahan

http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TJ7-45S9JVY-1&_user=10&_origUdi=B6WK3-4DTKDT8-77&_fmt=high&_coverDate=06%2F30%2F2002&_rdoc=1&_orig=article&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=8d025db96d90c39f8f29b3bc89e277a0
onelinkwrong

http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6X1M-45FKPGJ-8&_user=10&_coverDate=12%2F31%2F1980&_rdoc=1&_fmt=high&_orig=search&_sort=d&_docanchor=&view=c&_rerunOrigin=scholar.google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=547877a98733766a0b0526fe09557df6

geometric psychology, google names for further research

“Subjective geometry” is a term coined by Weintraub and Krantz to describe the distortion imposed upon geometric patterns by the visual system itself—so-called optical illusions. The latter are widely regarded as being generated by misplaced “constancy” effects, i.e., they are regarded as stemming from the invariance of an object’s appearance under wide variations in viewing conditions, such as obliquity, rotations, etc. The invariances represented by these constancies—shape constancy, size constancy, etc.—are spatiotemporal invariants of certain Lie subgroups of P4(R) circled plus CO(1, 3) circled plus GL(4, R) that govern Euclidean and non-Euclidean geometry. That Euclidean subgroups describe a Cyclopean visual world; the non-Euclidean, a binocular (bipolar) world of hyperbolic nature, according to the work of Luneburg, Blank, Indow, and others. The visual field of view is itself a geometric object involving not only “figure” and “ground” but also visual contours (orbits of the Lie groups involved), linear perspective, interposition, and contact and symplectic structures. The retina and “cortical retina” are both covered by a family of “circular-surround” cellular response fields (of a “Mexican hat” nature) which constitute an atlas for the visual manifold S. Upon this manifold are defined certain equivariant vector bundles that account for constancy phenomena and certain jet bundles, arising out of the vector bundles by prolongation, that generate the differential invariants characterizing higher form perception. The resultant theory of perceptual-cognitive processing has been termed “geometric psychology,” in analogy to MacLane’s “geometrical mechanics” and Brockett–Hermann–Mayne’s “geometry of systems,” the mathematical structure being very similar in all three instances. Functorial maps from the category GvFB(S) of equivariant fibre bundles to the simplicial category and the category of simplicial objects complete the theory by extending the perceptual system to cognitive phenomena and information-processing psychology.

Constructive Aspect of Visual Perception

Snakes