Novel structure incorporating 3 simplest structural polyhedral forms:
" the ARC in =4D= " the tail as a rotating/fused/swiveling structure
I found 6 correlations. [Update: a 7th found, the Proxigean extreme spring tide occurs every 31 years, see bottom of post.] The last one was hard, the ictet has 31 sides, but what was the 12? I knew there must be a twelve. Then I remembered Bucky's 12 radial spokes supporting a tensional wheel. Just as a rotating tail (but not a swiveling tail) such as a propeller or a clock must balance outwards to avoid offset erosion of the prop axle, the wheel must have 12 contacts in balance, as the sphere in a matrix must have 12 contact points. The result is extraordinary. (I did not originally see the link of 31 GC & vertebral nerve prs and 12 cranial nerves, Rybo at Synergeo group did, a copy of his graphic is at the bottom of the ARC blog. Ken at AAT group noted the 31 equal temperment of an octave.)
1 growth31 vert neural pairs, 12 cranial nerve pairs, pentameric
2 form31 great circles in icosa, 12 vertices, pentagonal
3 energy31 equal temperment in octave, 12 tones, pentatonic
4 time31~ day/night cycles in month, 12 month, heptapent
5 triax31=sum of 1st 7 factors, of 12 factors of 60, hexapent
6 ictet31 triangular sides, 12 deg. of freedom of tail pentabase
7 orbit31 year cycle extreme proxigean spring tide, 12 pt ellipsis
Bio video of helical propelled bacteria: Rhodobacter sphaeroides
http://www.rowland. harvard.edu/ labs/bacteria/ showmovie. php?mov=rsphe_ f_swim_1
http://tinyurl. com/qzdb4r
Good graphical explanation of flagella, cilia, rod:
http://lecturer. ukdw.ac.id/ dhira/BacterialS tructure/ Flagella. html
Note: (Flower petals/leaves often split as 2, 3, 5, which are prime)
The number 60, a highly composite number, has twelve factors:
1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
1 + 2 + 3 + 4 + 5 + 6 + 10 = 31 (sum of 1st 7 factors)
In spherically arranged tightfit ball packing layers:
12, 42, 92, 162, 252, 362 (shell layer balls in VE, Icosa)
10, 40, 90, 160, 250, 360 (subtract 2)
1, 4, 9, 16, 25, 36 (divide by 10)
f = 1, 2, 3, 4, 5, 6 (sq root)
form: 1 tet = 4 sides (tetrahedron)
form: 1 octa = 8 sides (octahedron)
form: 1 octet = 11 sides (2 merge into 1 double pane = tube)
form: 1 icosa = 20 sides (icosahedron)
form: 1 ictet = 31 sides (icosa w/ pinhinged octet "tail")
Ictet = icosa + tail, polar moment, monoaxis
So now we have a composite structure of all 3 structural elements, the overall shape is an icosaspheroid with an octet tail which can rotate in any direction (or if reversely docked, swinghinge in an arc), answering the question, from whence did the flagella and sperm arise. Also explains the persistence of the tail in all motile organisms. Also explains why humans have 31 vertebral neural pairs, 12 cranial neural pairs and no tail. At least it would seem so.
AFAIK no one has noted the significance of the combined 'Ictet' structure as a universal jointed spheric mobile entity, and its numerical parallel of 31 sides with the 31 equal temperment of the octave, the 31 vertebral neural pairs in humans, the 31 lunar days and 31 great circles of the icosahedron, the 31 sum of 1st 7 factors of 60.
http://www.tabletoptelephone.com/~hopspage/Fuller.html
http://en.wikipedia.org/wiki/Icosahedron
http://www.buckminster.info/Pics/Tetrahedra/TetQuarkMite.gif
At the single open vertex of the tetrahedron (the far left blue point) of the octet is the universal 'pin joint' which is attached to one of the 12 pentagonal sutures (pore/window) of the (truncated/regular) icosahedron or hexapent buckyball, the octet tail allowed to swivel radially or horizontally on a favorable plane.
(The picture includes the internal electrostatic bonding forces referred to as a mite or quark, whose form is an irregular tetrahedron.)
Note that the icosa meets the octet at a pentagon window, the tet meets the octa with each of the tet base corners having 5 vectors. So there is a continuum of pentagonal adhesion/cohesion, aka pentabase. Synergetic.
DDeden, August 18, 2009
(ictet not abbreviation for interpenetrated icosatet, rather an abbreviation for attached icosaoctatet, in the manner of R.B. Fuller term octet to name attached octatet form in patent.)
(More information on ICTET 3112 available at AAT & Synergeo Yahoo groups.)

background addendum:
The term harmony derives from the Greek (harmonÃa), meaning "joint", re. "to fit
together, to join". The harmonious major triad is composed of three tones in a
simple whole number ratio of 6:5:4. The major triad chord in music theory
consists of a root, a major third, and a perfect fifth.
In structural theory, the simplest closed structures, the Platonic triad, are
the tetrahedron, the octahedron and the icosahedron. It would seem to me that
these 2 triads or chords are the same relationship expressed in different form,
(sound) energy and physical matter. [Compare to diving/backfloating partners
with infant]
The relevance to the 'ictet' above is that the chordal form triad is
preserved, the icosa, tet and oct are securely joined, yet the pin joint allows
vibrational energy propulsion via propeller rotation equivalent to a wheel of 12 spokes tensionally held or 12 degrees of freedom. A dual pin hinge would produce lateral swiveling.
An alternative form, not considered here but perhaps relevant to echinoforms or
spongiforms, is a central icosa with an octet flagella (or spine) docked at each
of the 12 surface pentagons, resembling a stellated icosahedron.
Tetrahedron: triangular corner (3 lines converge)
Octahedron: square corner (4 lines converge)
Icosahedron: pentagon corner (5 lines converge)
Potentially supporting information as to the antiquity of the ictet tail structure, this time as a fused external structure: fused structural segments produce 'primary cilia' (flagella, filament) different from motile cilia and found in animal skin and brain cells. Not impossibly this 'primary cilia' became the source of the notochord & tail in pre/vertebrates.

http://www.eurekalert.org/pub_releases/200908/uocssc082109.php
"Unlike the more familiar motile cilia, primary cilia do not move, and only one pokes out of each cell. They have recently been discovered to play an essential role in assuring normal embryological development. May protect against skin cancer"
http://www.eurekalert.org/pub_releases/200908/uocsbt082109.php
"UCSF scientists have discovered that a tiny filament extending from cells, until recently regarded as a remnant of evolution, may play a role in the most common malignant brain tumor in children. "In the last few years, primary cilia have been shown to be essential for the cellsignaling that drives both human development, including the differentiation of stem cells into neurons, and some diseases, including polycystic kidney disease. The fact that the two UCSF studies implicate primary cilia in two totally different tissues suggests the finding is likely to be very general."
Note: A skew icosahedron is inscribed within an octahedron in a 4 frequency tetrahedron.
http://groups.yahoo.com/group/synergeo/message/54332
"In other words, you can't make a regular icosahedron out of regular tetrahedrons. In
http://groups.yahoo.com/group/synergeo/message/10326
I once said that, "The dihedral angle between the faces of a regular tetrahedron is 1.230959418 radians or 70.52877938 degrees. This is also the angle between alternating faces of a regular octahedron. Notice that it is not quite 72 degrees, so you can't put exactly five
tetrahedra around a common edge or five octahedra around a common vertex, as was attempted in the gapball or octaball Photos." AM
http://home.usit.net/~rybo6/rybo/id2.html

The number 31 pops up again, this time in regard to tidal cycles.
"During the last 400 years, there have been 39 instances or 'Extreme Proxigean Spring Tides' There were, in fact cases of extreme tidal flooding recorded during these particular spring tides which occur once every 31 years."
1800 yr ocean tidal cycle
http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=18099
The gravitational force of the moon is one tenmillionth that of earth, but when you combine other forces such as the earth's centrifugal force created by its spin, you get tides.
When the moon is full or new, the gravitational pull of the moon and sun are combined. At these times, the high tides are very high and the low tides are very low. This is known as a spring tide. The net result of this is that the Earth gets deformed into a slightly squashed, ellipsoidal shape due to these tidal forces.
The tidal bulge of the Moon follows along the path on the earth's surface which intersects with the orbital plane of the Moon. This plane is tilted about 23 degrees with respect to the equatorial plane of the earth. The result is that near the equator, the difference between high tide and low tide is actually rather small, compared to other latitudes.
The Proxigean Spring Tide is a rare, unusually high tide. This very high tide occurs when the moon is both unusually close to the Earth (at its closest perigee, called the proxigee) and in the New Moon phase (when the Moon is between the Sun and the Earth). The proxigean spring tide occurs at most once every 1.5 years.
During the last 400 years, there have been 39 instances or 'Extreme Proxigean Spring Tides' where the tideproducing severity has been near the theoretical maximum. The last one of these was on March 7 1995 at 22:00 hours Greenwich Civil Time during a lunar Full Moon. There were, in fact cases of extreme tidal flooding recorded during these particular spring tides which occur once every 31 years."
If you see earths' orbit as a circle (slightly lopsided, ellipse) around the sun, imagine the sun as the center of a sunflower, and earth orbit as the edge of that center, then the moon could be seen as a set of 366 flower petals around it. In the last 400 years, the extreme proxigean spring tide occurred 39 times. Maybe at 366 it skipped the extreme tide.
Also note Rybo's picture at bottom of page showing the 31 spinal nerve pairs and 12 cranial nerve pairs.
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