There are four fundamental corners in every system, and each must be triangulated: 4 × 3 = 12. Thus there are twelve degrees of freedom, tetrahedrally organized." Amy Edmondson, A Fuller Explanation, Re. R. Buckminster Fuller, vector equilibrium
Relative Volume Table: (tetra volumes) | (cube volumes)
|form||= tv|cv||form||= tv | cv|
|tetra/coupler||= 1 | 1/3||star octa||= 12 | 4|
|double tetra||= 2 | 2/3||star rh dodeca||= 12 | 4|
|duo-tet cube||= 3 | 1||cmpd cubocta ext||= 20 | ~7|
|reg octa||= 4 | 4/3||reg icosa int||:~18 | ~6|
|star tetra||= 5 | 5/3||star icosa ext||= 20 | ~7|
|rhom dodeca||= 6 | 2||cubocta int||= 20 | ~7|
|2F coupler||= 8 | 8/3||2F cube||= 24 | 8|
Icosa = 18.51 tv +
Vector edge Cube = 8.49 tv =
Icosa + Vector edge Cube = 27 tv -
Vector diagonal Cube = 3 tv (2 small duotets) =
2F lattice Cube = 24 tv (2 big duotets) -
Octa = 4 tv =
Star Icosa ext = 20 tv or
Cubocta = 20 tv + 8 ext tetra =
Star Cubocta Cube = 28 tv + 6 ext sq pyramids =
Cmpd Cubocta = 40 tv - 8 ext tetra =
Superocta = 6 octa & 8 tetra = 32 tv
compound of (1st stellation of) cube & octahedron (not star cubocta)
Cumulation: tet, octa, cube, icosa, dodeca can be cumulated 4 times
(Cube & tet can't be stellated but both can be cumulated)
tetra, triakis tetra, cube, 12-faced star deltahedron (star tetra)
cube, tetrakis hexahedron, rhomb dodeca, 24 face star deltahedron (star cube)
octa, sm triakis octa, stella octangula/duotet (star octa)
star tetra, star octa (domain net of cubocta), star icosa
Now excluding the nucleic polyhedron volume:
View of surface tets as skeletal structure, equi length struts
Star Tet: nuc tet void, 4 tets, 6 circ + 12 ext struts = 18
Star Octa: nuc octa void, 8 tets, 12 circ + 24 ext struts = 36
Star Icosa: nuc icosa void, 20 tets, 30 circ + 60 ext struts = 90
I think the star tetra, star octa and star icosa qualify as 'VE', if the central polyhedra (reg tet, reg octa, reg icosa) is considered to be the nucleus. All the connecting vectors are same length AFAICT. Eg. the star icosa has all equal length vectors and has no centroid vectors (which would be a different length). So you need to specify which VE:
VE star tetra
VE star octa
VE star icosa
The star tetra has 12 face planes
Multiplication by division: http://synergeticists.org/synergetics/plates/figs/plate03.html
Noting Kirby's & Alan's relative volume tables here:
A 1F cube has 3 tv. First stellation of a cube is a rhombic dodeca of 6 tv.
First stellation of a rhombic dodeca is Escher's Solid, of 12 tv,
half the volume of the 2F cube, same as the star octa.
The volume of the cuboctahedron is 5/6 of that of the enclosing cube and 5/8 of that of the enclosing octahedron. The cuboctahedron shares its edges and vertex arrangement with two nonconvex uniform polyhedra: the cubohemioctahedron (having the square faces in common) and the octahemioctahedron (having the triangular faces in common).
I suggest adding one more, some star polyhedra with identical regular tetrahedra, the internal central polyhedron is a void*, since the external tetras produce the structure, so no additional struts required:
star tetra: 4 tetras (star tet - tet nucleus)
star octa: 8 tetras (duotet - octa nucleus)
star cubocta: 8 tetras (cube - cubocta nucleus)
star icosa: 20 tetras (4 + 8 + 8) * jitterbug transformation
*These exclude the nuclear polyhedron, since it is a void built by the exoskeleton of structural tets and overlapped struts are unnecessary.
I haven't checked if the orientation of the 20 tets of the star icosa match with those of the others, or if they need to be rotated or jitterbugged to fit the icosa.
compound cubocta & 120 cell
What can fit into an icosa? a tet. an octa. a star tet? a star octa?
rhombic dodeca (cube with pyramidal faces combine coplanar pairs of triangles into rhombi) rhomdodeca from cube+6pyramid (differs from star cube with noncoplanar pairs of triangles)
star rhombic dodeca & star octa:
C60 Carbon buckyball fullerene: 60 vertices divided into 12 pentagon sets; 32 faces of 12 pentas and 20 hexas, 90 edges of 60 pentagonal edges (single bonds)and 30 hexagonal edges (double bonds), each hexagon has alternating single and double bonds. Dual of truncated icosahedron C60 with 32 faces and 60 vertices is the pentakis dodecahedron (triangulated C60) of 60 faces and 32 vertices.
1033.53 The vector equilibrium jitterbug provides the articulative model for demonstrating the always omnisymmetrical, divergently expanding or convergently contracting intertransformability of the entire primitive polyhedral hierarchy, structuring as you go in an omnitriangularly oriented evolution.
1033.54 As we explore the interbonding (valencing) of the evolving structural components, we soon discover that the universal interjointing of systems__and their foldability__permit their angularly hinged convergence into congruence of vertexes (single bonding), vectors (double bonding), faces (triple bonding), and volumetric congruence (quadri-bonding). Each of these multicongruences appears only as one vertex or one edge or one face aspect. The Eulerean topological accounting as presently practiced__innocent of the inherent synergetical hierarchy of intertransformability__accounts each of these multicongruent topological aspects as consisting of only one of such aspects. This misaccounting has prevented the physicists and chemists from conceptual identification of their data with synergetics' disclosure of nature's comprehensively rational, intercoordinate mathematical system.
1033.55 Only the topological analysis of synergetics can account for all the multicongruent__doubled, tripled, fourfolded__topological aspects by accounting for the initial tetravolume inventories of the comprehensive rhombic dodecahedron and vector equilibrium. The comprehensive rhombic dodecahedron has an initial tetravolume of 48; the vector equilibrium has an inherent tetravolume of 20; their respective initial or primitive inventories of vertexes, vectors, and faces are always present__though often imperceptibly so__at all stages in nature's comprehensive convergence transformation.
2 fold: triangle octavalent
3 fold: tetra quadruvalent
4 fold: cube, octa, apparent cubocta, rhom dodeca
5 fold: icosa, penta dodeca
6 fold: actual cubocta in motion when surface squares collapse
7 axes of symmetry, 14 faces of bubbles & cells: 14faces
Carbon C60 buckyball: truncated icosahedron C60
Boron B80 buckyball: truncated rhombic triacontahedron B80
Tetrakaidecahedron: 14 sided polyhedra includes VE, truncated octahedra (space filler) and hexagonal truncated trapezohedron (14 sided soap foam)
The more regular honeycombs dualise neatly:
* The cubic honeycomb is self-dual.
* That of octahedra and tetrahedra is dual to that of rhombic dodecahedra.
Check: quasicrystal matrix composed of icosa & star tetra, where the icosa contain nuclei of star tetra.
Da Vincis' star tetrahedron & icosahedron: star tet
Jitterbug transformation: from 4 fold VE through 5 fold icosa to 3 fold tetra VV
The nucleated cubocta is a unique 3D system in Vector Equilibrium, where all 24 circumferential vectors (edges) are the same length as the 12 nuclear radiating vectors (rays). The non-nucleated VE is not structurally rigid with rubber joints (unlike the tetra, octa and icosa of 60 degrees) due to the 6 collapsable square faces of 90 degrees, but with polarised kinetic energy applied (compressed, extended or torqued at opposing triangle or square faces) will transform as follows:
Compressing polar triangles inward: spins through an incomplete icosa to an octa.
Extending polar triangles outward: unstable convex-concave tri-prism, which if equatorially compressed (at evertices or invertices) or polarly torqued forms a webbed pin-hinged double tetra hourglass.
Torqueing polar triangles laterally: 2f (nuc) triangle, folds to tetra.
Compressing polar squares inward: star square, folds to 1/2-octa to 1f triangle.
Extending polar squares outward produces stretched cube, which if equatorially compressed or polarly torqued forms a pin-hinged double 1/2-octa hourglass, folds to a single 1/2-octa to 1f triangle.
Torqueing polar squares laterally: incomplete 2f tetra, folds to saddle form or hexagon or 2f (nuc) triangle to tetra.
Jitterbug collapse to tetra: http://www.rwgrayprojects.com/synergetics/s04/figs/f6108.html
Jitterbug collapse to octa:
6 loops circumscribe a spherical VE: 6 loop VE
Relative Volumes at Grunch: http://grunch.net/archives/47
Concentric hierarchy: here
Quintet dodeca: http://www.newciv.org/Synergetic_Geometry/Quintet_Dodeca.JPEG
Octet truss IVM virtual space frame allows tourists to see ancient artifacts without disturbing them at Java, Indonesia.
Time & Space Perspective
On spherical gravity "attraction" & EM radial wave "repulsion" relationships
2 round objects have more surface area than 1, so they are "attracted" to each other, collision reduces total surface area. Energy has no surface area so is independent of gravity, (it 'curves' around bodies due to reflection pressure?) it radiates outwardly in periodic (tidal) waves linearly from sphere center.
circumferential vectors & great circles: icosa edge vectors 30, icosa great circles 31; cubocta edge vectors 24, cubocta great circles 25
1052.21 Isaac Newton discovered the celestial gravitation interrelationship and expressed it in terms of the second power of the relative distance between the different masses as determined by reference to the radius of one of the interattracted masses. The gravitational relationship is also synergetically statable in terms of the second power of relative frequency of volumetric quanta concentrations of the respectively interattracted masses. Newton's gravitational constant is a radially (frequency) measured rate of spherical surface contraction, while Einstein's radiational constant is a radial (frequency) rate of spherical expansion. (See Secs. 960.12, 1009.31 and 1052.44.)
1052.30 Gravitational Constant: Excess of One Great Circle over Edge Vectors in Vector Equilibrium and Icosahedron: Pondering on Einstein's last problem of the Unified Field Theory, in which he sought to identify and explain the mathematical differentiations between electromagnetics and gravity__the two prime attractive forces of Universe__and recalling in that connection the conclusion of synergetics that gravity operates in spherical embracement, not by direct radial vectors, and recalling that electromagnetics follows the high-tension convex surfaces, possibly the great-circle trunk system of railroad tracks (see Secs. 452 and 458); led to pondering, in surprise, over the fact that the vector equilibrium, which identifies the gravitational behaviors, discloses 25 great circles for the vector equilibrium in respect to its 24 external vector edges, and the icosahedron, which identifies the electron behaviors of electromagnetics, discloses 31 great circles in respect to its 30 external vector edges.
1052.31 In each case, there is an excess of one great circle over the edge vectors. Recalling that the circumferential vector edges of the vector equilibrium exactly equal the radial explosive/implosive forces, while the icosahedron's 30 external edges are longer and more powerful than its 30 radial vectors [What 30 radial vectors?? A star icosa has external equilength pentalateral-bond tets, a cubocta has equilength quadrilateral-bond & equilength mono-radial vectors.], yet each has an excess of one great circle, which great circles must have two polar axes of spin, we encounter once more the excess two polar vertexes characterizing all topological systems, and witness the excess of embracingly cohering forces in contradistinction to the explosively disintegrative forces of Universe.
A star tetra, a star octa, and a star icosa each contain a nuclear void (empty space tet, octa, icosa) polyhedron surrounded and structured by reg ext tets. A star cubocta does not contain a void nucleus. A star rh dodeca and star non-nucleated cubocta contain void polyhedron nuclei, but lacking full tet surfaces are not structured. Empty stars can be nucleated by structures: star icosa accepts a nuclear star tetra (free spinning? quasicrystal matrix), star cubocta accepts a nuclear star octa (IVM crystal matrix).
domed dive mask
I'd like to get this dive mask, great vision underwater for those who are nearsighted (myopic) like me, and lack sufficient visual accomodation for clear dark adapted acuity.
Vortices in dragonfly flight animation:
Spiral column foldable origami: spiral
Fish bones, fermented fish vs cooked fish, calcium carbonate in sea water, sleep
Fish-bone peptides (FBP) with a high affinity to Ca were isolated using hydroxyapatite affinity chromatography, and FBP II with a high ratio of phosphopeptide was fractionated in the range of molecular weight 5·0–1·0kDa by ultramembrane filtration. In vitro study elucidated that FBP II could inhibit the formation of insoluble Ca salts in neutral pH. In vivo effects of FBP II on Ca bioavailability were further examined in the ovariectomised rat. During the experimental period, Ca retention was increased and loss of bone mineral was decreased by FBP II supplementation in ovariectomised rats. After the low-Ca diet, the FBP II diet, including both normal level of Ca and vitamin D, significantly decreased Ca loss in faeces and increased Ca retention compared with the control diet. The levels of femoral total Ca, bone mineral density, and strength were also significantly increased by the FBP II diet to levels similar to those of the casein phosphopeptide diet group (no difference; P>0·05). In the present study, the results proved the beneficial effects of fish-meal in preventing Ca deficiency due to increased Ca bioavailability by FBP intake
ocean alkalinity & fish
New research reveals the major influence of fish on maintaining the delicate pH balance of our oceans, vital for the health of coral reefs and other marine life.
The discovery, made by a team of scientists from the UK, US and Canada, could help solve a mystery that has puzzled marine chemists for decades. Published 16 January 2009 in Science, the study provides new insights into the marine carbon cycle, which is undergoing rapid change as a result of global CO2 emissions.
Until now, scientists have believed that the oceans' calcium carbonate, which dissolves to make seawater alkaline, came from the external 'skeletons' of microscopic marine plankton. This study estimates that three to 15 per cent of marine calcium carbonate is in fact produced by fish in their intestines and then excreted. This is a conservative estimate and the team believes it has the potential to be three times higher.
Fish are therefore responsible for contributing a major but previously unrecognised portion of the inorganic carbon that maintains the ocean's acidity balance. The researchers predict that future increases in sea temperature and rising CO2 will cause fish to produce even more calcium carbonate.
To reach these results, the team created two independent computer models which for the first time estimated the total mass of fish in the ocean. They found there are between 812 and 2050 million tonnes (between 812 billion and 2050 billion kilos) of bony fish in the ocean. They then used lab research to establish that these fish produce around 110 million tonnes (110 billion kilos) of calcium carbonate per year.
Calcium carbonate is a white, chalky material that helps control the delicate acidity balance, or pH, of sea water. pH balance is vital for the health of marine ecosystems, including coral reefs, and important in controlling how easily the ocean will absorb and buffer future increases in atmospheric CO2.
This calcium carbonate is being produced by bony fish, a group that includes 90% of marine fish species but not sharks or rays. These fish continuously drink seawater to avoid dehydration. This exposes them to an excess of ingested calcium, which they precipitate into calcium carbonate crystals in the gut. The fish then simply excrete these unwanted chalky solids, sometimes called 'gut rocks', in a process that is separate from digestion and production of faeces.
The study reveals that carbonates excreted by fish are chemically quite different from those produced by plankton. This helps explain a phenomenon that has perplexed oceanographers: the sea becomes more alkaline at much shallower depths than expected. The carbonates produced by microscopic plankton should not be responsible for this alkalinity change, because they sink to much deeper depths intact, often becoming locked up in sediments and rocks for millions of years. In contrast, fish excrete more soluble forms of calcium carbonate that are likely to completely dissolve at much shallower depths (e.g. 500 to 1,000 metres).
Lead author Dr Rod Wilson of the University of Exeter (UK) said: "Our most conservative estimates suggest three to 15 per cent of the oceans' carbonates come from fish, but this range could be up to three times higher. We also know that fish carbonates differ considerably from those produced by plankton"
toads, fish, sharks, CaCO3, etc.
Photic sneeze: vestige of past tropical lagoon diving ancestors? Aqua-photic Respiratory Cycle forage divers, fast dark-adapted sunlight exhalations...
Y DNA Haplogroup T: Salt trade, boiling brine to trade via dugout for inland goods?
coastal transit/trade 30ka
Upper Rift seasonal fishing camp 770ka: carp, acorn, olive pit, raisin, bark
carp fishing with acorn bait & crabbing 770ka at Lake Hula
23ka bedding, salt and freshwater springs at Sea of Galillee
Similarities between Hs & Pt: proteins, DNA, chromosome