Kea
I became side-tracked with the discussion of attractors.
I reviewed my old ‘Chaos’ by J Gleick (paperback-1987).
On page 143 is a demonstration of how a torus may become distorted in cross [Poincare] section.
On p271 “Economists analyzing stock market data would try to find attractors of dimension 3.7 or 5.3”
The numbers 3.7 and 5.1 or 5.4 are prominent in the ‘The discovery of the [alpha]-helix and [beta]-sheet, the principal structural features of proteins’ by David Eisenberg in PNAS.
“Why did Pauling delay 3 years in publishing this finding that came to him in only a few hours? He gave the answer in his banquet address at the third symposium of the Protein Society in Seattle in 1989. He was uneasy that the diffraction pattern of -keratin shows as its principal meridional feature a strong reflection at 5.15-Å resolution, whereas the -helix repeat calculated from his models with Corey was at 5.4 Å. As he says in his fourth paper of the PNAS series with Corey: "The 5.15-Å arc seems on first consideration to rule out the alpha-helix, for which the c-axis period must be a multiple of the axis distance per turn..." But then came the paper in 1950 by Bragg, Kendrew, and Perutz enumerating potential protein helices. Pauling told his audience in 1989: "I knew that if they could come up with all of the wrong helices, they would soon come up with the one right one, so I felt the need to publish it."
“The origin of the discrepancy between the repeat of the -helix and the x-ray reflection of alpha-keratin was hit on a year later by Francis Crick (5), then a graduate student with Perutz, and also by Pauling. It is that keratin is a coiled-coil, with alpha-helices winding around each other. The wider excursion of the alpha-helix in the coiled-coil reduces its repeat distance to 5.1 Å. This knack of knowing which contradictory fact to ignore was one of Pauling's great abilities as a creative scientist.‘
http://www.pnas.org/cgi/content/full/100/20/11207
Is the helix itself an attractor?
Is chaos theory related to game theory?
I also looked at ‘attractor’ un MathWorld. The Rössler Attractor may relate to spiral galaxies? and perhaps to the 9_Conclussions p 30-32 of ‘Duality of Orthogonal and Symplectic Matrix Integrals and Quaternionic Feynman Graphs by M. Mulase, A. Waldron. in your 8-11 post.
http://mathworld.wolfram.com/RoesslerAttractor.html
In the ‘NCG predictions: Alain Connes in SciAm’ thread, I think that flight mechanics of animals or objects controlled by animals may be a subset of the NCG torus which may hold for any general n-body interaction.
I suspect this because of the evolution during or shortly after the Cambrian explosion of genus-3-torus semi-circular canals for balance and orientation based on angular momentum in chordates.
Phylum: Chordata
Subphylum: Vertebrata
Superclass: Pisces
Agnathans vs. Gnathostomes:
semicircular canals - agnathans have 1 or 2 - gnathostomes have 3
from ;BIO 342
Comparative Vertebrate Anatomy
Lecture Notes 1 - Chordate Origins & Phylogeny’
http://www.biology.eku.edu/ritchiso/342notes1.htm
‘Building a better semicircular canal: could we balance any better?‘ by Todd Squires (Caltech): “Every vertebrate organism uses fluid-filled semi-circular canals (SCC) to sense angular rotation -- and thus to balance, navigate, and hunt. Whereas the size of most organs typically scales with the size of the organism itself, the SCC are all about the same size--whether in lizards, mice, humans or whales. What is so special about these dimensions? We consider fluid flow in the canals and elastic deformations of a sensory membrane, and isolate physical and physiological constraints required for successful SCC function. We demonstrate that the `parameter space' open to evolution is almost completely constrained; furthermore, the most sensitive possible SCC has dimensions that are remarkably close to those common to all vertebrates”
http://flux.aps.org/meetings/YR03/DFD03/baps/abs/S240011.html
‘Electrical coupling between secondary hair cells in the statocyst of the squid Alloteuthis subulata’ by R Williamson: ‘The cephalopod angular acceleration receptor system has sensory response characteristics similar to those of the vertebrate semicircular canal system and, unusual for an invertebrate, contains secondary receptor hair cells. The experiments reported use intracellular recordings from pairs of hair cells to show that at least one subset of the hair cells is electrically coupled along the entire length of the crista section. The coupling can be reduced by application of heptanol or octanol. Intracellular injection of H+ ions into a hair cell reduces the coupling of cells on the opposite site of the injected hair cell but does not abolish it completely. It is proposed that the coupling is likely to result in an improvement in the signal-to-noise ratio of the receptor system, a reduction in overall frequency response, but an increase in the low frequency sensitivity.
http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=2541872&dopt=Abstract
BACK TO AP BIOLOGY HOME PAGE Chapter 40: Sense Organs by Jerry G. Johnson with the last line “4. Static equilibrium organs called statocysts are found in cnidaria, mollusks, and crustacea.”
http://www.sirinet.net/~jgjohnso/apbio40.html
Statocysts images can be found on Google, but otherwise I know very little about these SCC equivalent organs.
Sorry, but I still have not yet finished the Ghrist book.
