Tuesday, November 24, 2009

Relative volumes & concentric hierarchy

tetra tripod table, balloonage

Icosa alloys mimic elements:

Sphere/tet/cube packing, entropy, tets in quasicrystal disks, density in box container
quasi-crystal tet packing
Researchers packed tetrahedra into a cubic box more densely than ever before: 85.03
(Cubes have a 100 percent packing fraction in a cubic box, while spheres pack at only 74 percent.) The tetrahedron was for decades conjectured to be the only solid that packs less densely than spheres, until just last year when U-M mathematics graduate student Elizabeth Chen found an arrangement of 77% that proved that speculation wrong.

the more significant finding is that the tetrahedrons can unexpectedly organize into intricate quasicrystals at a point in the computer simulation when they take up roughly half the space in the theoretical box.

In this computer experiment, many thousands of tetrahedrons organized into dodecagonal, or 12-fold, quasicrystals made of parallel stacks of rings around pentagonal dipyramids. A pentagonal dipyramid contains five tetrahedrons arranged into a disk. The researchers discovered that this motif plays a key role in the overall packing. In the simulation, the tetrahedrons organized into a quasicrystal and settled on a packing that, when compressed further, used up 83 percent of the space. Engel then reorganized the shapes into a "quasicrystalline approximate," which is a periodic crystal closely resembling the quasicrystal. He found an arrangement that filled more than 85 percent of the space.

This is the first result showing such a complicated self-arrangement of hard particles without help from attractive interactions such as chemical bonds, Glotzer said.

Thermos vacuum allows IR radiation outwards through glass, photonic crystals reduce IR heat loss better than pure vacuum
In a typical thermos, a vacuum is used to reduce heat transfer. Scientists have found that layers of photonic crystals in a vacuum can reduce the thermal conductance to about half that of a pure vacuum. Basically, heat can be transferred from one material to another in three main ways: convection, conduction, and radiation. Conduction and convection both require some kind of material medium for heat to pass through; therefore, the lack of material in a pure vacuum greatly minimizes the effectiveness of these two processes. However, heat can also be transferred through infrared radiation, a form of light that is invisible but can be felt as heat. In the example of the thermos, infrared radiation can travel through the vacuum to the thermos' outer wall; when absorbed by the outer wall, the radiation causes the molecules in the outer wall to vibrate and release heat. Significantly, photonic crystals can have band gaps that forbid propagation of certain frequency ranges of light. In this case, they could be used to block infrared radiation.

The scientists found that a 100-micron-thick structure made of a stack of 10 photonic crystal layers, each 1 μm thick and separated by 90-μm gaps of vacuum, could reduce the thermal conductance to about half that of a pure vacuum. In a more recent study, Fan and his colleagues calculated the fraction of all frequencies that the photonic crystal allows through. They were somewhat surprised to find that the thermal conductance doesn't depend on the thickness of the layers but only on how fast light travels through the material, or its index of refraction.

Circles & spheres, flakes and crystals, water & ice
super cold
Take a regular tetrahedron and set the distance from vertex to tet center as unit 1. Then an
edge of the tet is 1.632993.. . So this number is directly related to the ubiquitous Maraldi angle, 109.47.. degrees (the caltrop angle -- the vertex-center-vertex angle in a regular tet).
CircumsphereRadius/Edge = (1/4)*SQRT(6)
Edge/CircumsphereRadius = 4/SQRT(6)
Vertex-Vertex central Maraldi Angle = arc cos -1/3 ( or ) 2*ACOS(SQRT(1/3))
Giacomo_F._Maraldi: "In math known for obtaining experimentally the angle in the
rhombic dodecahedron shape in 1712, which is still called Maraldi angle."

"Water is a network-forming matter. You can imagine the structure of the network as a kitchen sponge, Matsumoto continues. The sponge structure is originally a kind of foam but membranes are lost, and only the beams - bonds - remain. In both network of water and kitchen sponge, four bonds meet at a point, or node, to form a three dimensionally connected random network. As Plateau pointed out in 19th century, four beams of a foam crosses at a node with regular tetrahedral angle - Maraldi's angle - similar to the waters hydrogen bond network. Matsumoto used computer simulation to look at three ways to change the volume of the foam cells: extension of the bonds, a change in the containing angle between the bonds, and a change in network topology. By discriminating the three contributions, the mechanism became very clear. One contributes to thermal expansion, another one contributes to thermal contraction, and the last one does not. Density maximum is a result of these competing contributions, he explains. "

Caltrops: reg tet 'land mines' The simple design paradigm says that the most elegant, efficient, iconic inventions are necessarily the simplest; like the elastic band, the brick, and the pizza. A caltrop is a simple piece of shaped metal (concrete tank killers); a spiky tetrahedron which, when liberally scattered on the ground, causes a great deal of annoyance to any passing dudes or ponies. And the brilliant thing is that however you drop them, they always land spiky point up.

Biological size, volume, area, fluid flow
size & form

Interaction on Bucky Fuller's tetrahedra as unit volume in synergetic sequence, with input from Allan, Kirby, myself:

Shape: Volume (notes)
MITE: 1/8 (AAB mods)
Tetra: 1 (24 A mods)
Coupler: 1 (8 MITE's)
Stella Octangula: 1.5 (Little octahedron of .5 volume + little tets totalling volume of 1. Notice that the Stella Octangula occupies half the volume of its enclosing cube.)
Cubocta: 2.5 (1/2 ƒ)
Octa: 4 (dual of cube)
Cube: 3 (dual of octa)
Cube-Octahedron Compound: 4.5 (first stellation of cub-octahedron)*
Rh-Dodeca: 6 (12 half-couplers, other ways)
Escher's Solid: 12 tetras (or 12 couplers [see above], other ways. Notice that the Escher's Solid occupies half the volume of its enclosing cube.)
Cubocta: 20 (volume = 8*2.5)
Cube: 24 (2ƒ)
Stella Octangula: 12 (Octa volume of 4 + 8 tets. Notice that the Stella Octangula occupies half the volume of its enclosing cube.)
Octahedron: 32 (2ƒ)
Cube-Octahedron Compound: 36 (larger version)

*The (small) cube-octahedron compound has a volume of 4.5, since the cube corners are
half the heights of the regular tetrahedrons.

Relative Volume Sequence: (mine, preliminary, w/ response from Allan)

tet 1 (reg or irreg)
reg tet volume, of course!
coupler 1 (irreg octa)

tet duo 2 (vertex/edge/face bonded)
two tetrahedrons

duotet cube 3
The volume of the cube is 3 tetrahedrons.

octa 4
The volume of the octahedron is 4 tetrahedrons.

The volume of the cube-octahedron compound or first stellation of cub-octahedron is 4.5 tetrahedra. If the octants are elevated even more to regular tetrahedra, then you will get the cuboctahedral star with a volume of 5. (see below)

star tet 5 [tetra-star]
The star tetrahedron is the unfolded net of the pentachoron, which is bounded by 5 tetrahedra.
rhombic triacontahedron 5
reg tet stell cubocta 5 [cubocta star w/ reg tet ecto]

rh dodeca 6
Rhombic dodecahedron has twice the volume of the cube. The volume of the stellated rhombic dodecahedron is 12, which is half the volume of the enclosing 2ƒ cube of volume 24. (below)

star octa 12 [octa star]
Stellated octahedron has half the volume of the enclosing 2ƒ cube of volume 24.
stellated rh dodeca 12

icosa int 18.51 endo
If you think that the icosahedron has volume 20, then you're obviously from another (fourth) dimension! The icosahedron, as we know in our flat Euclidean three-dimensional realm, has a endo- volume of about 18.51 tetrahedra.

cubocta 20 endo
This is the jitterbug, fully extended.

star icosa ext 20 [icosa star reg tet ecto]
But that's assuming that each face has a REGULAR tetrahedron on it.

star cubocta 40 [cubocta star reg tet endo ecto]
The 2ƒ version of the cube-octahedron compound or stellated cuboctahedron has a volume of 36 tetrahedra. For the star cuboctahedron to have a volume of 40, each face would have to be stellated with equilateral triangles (i.e. half-octahedra and regular tetrahedra). (see above)

> Icosa (Fuller explanation, Amy Edmondson, paraphrased)
> Its endo-volume of approximately 18.51 does not fit rationally into
> the cosmic hierarchy with whole numbers nor click-stop when
> jitterbugging.

> I found this interesting, a clue perhaps re. 18.51 1/electron

> "The icosahedron contracts to a radius less than the radii of the
> vector equilibrium from which it derived. There is a sphere that is
> tangent to the other 12 spheres at the center of an icosahedron, but
> that sphere is inherently smaller. Its radius is less than the
> spheres in tangency which generate the 12 vertexes of the vector
> equilibrium or icosahedron. Since it is no longer the same-size
> sphere, it is not in the same frequency or in the same energetic
> dimensioning. The two structures are so intimate, but they do not
> have the same amount of energy. For instance, in relation to the
> tetrahedron as unity, the [endo]volume of the icosahedron is 18.51 in
> respect to the vector equilibrium's [endo]volume of 20 [and also the
> star icosa's ectovolume of 20]. The ratio is tantalizing because the
> mass of the electron in respect to the mass of the neutron is one
> over 18.51. That there should be such an important kind of seemingly
> irrational number provides a strong contrast to all the other
> rational data of the tetrahedron as unity, the octahedron as four,
> the vector equilibrium as 20, and the rhombic dodecahedron as six:
> beautiful whole rational numbers". Syn 400.00 system

> "When the volume of a tetrahedron is specified as one unit, other
> ordered polyhedra are found to have precise whole-number volume
> ratios, as opposed to the cumbersome and often irrational quantities
> generated by employing the cube as the unit of volume. Furthermore,
> the tetrahedron has the most surface area per unit of volume".
> (sphere has least) A Fuller Explanation

> http://www.grunch.net/synergetics/ivm.html
> shows IVM in a tet
> from previous:

> Relative volumes: (endo = interior, ecto = exterior shell)

> Tet: (@IVM), vol 1
> Oct: (@IVM), vol 4
> Star Tet: center tet endovol 1 + ectovol 4 (ext tets), vol 5
> Star Oct: center oct endovol 4 + ectovol 8 (@IVM), vol 12
> Star Icosa: center icosa endovol 18.51 + ectovol 20, vol 38.51
> Star Cubocta: endovol 20 + ectovol 20, vol 40

> DD
> On Nov 23, 2009, at 8:00 AM, rybo6 wrote:

> > "Topologically, lines are composed of points."

> > I don't know how it is defined elsewhere, but to me, a line is a
> > point with depth (not = zero), and a point is a line with depth = 0.
> > [Where 'depth' is any direction.] This does not conflict with the
> > definition of a point being a line crossing (which is the same as 2
> > or more vectors meeting at a vertex).
> > The irreg tets that make an icosa, do their struts meet at the
> > center, or do each reach to the opposite face? I thought each face
> > triangle had a tet apex at the icosa center (not modelable with
> > toothpicks), but maybe it goes to the opposite side at a point. Or
> > are there no irreg tets in an icosa?

> > Here (@ link bottom) see the regular icosa and star icosa:
> > http://en.wikipedia.org/wiki/Small_triambic_icosahedron

> > Here see that 2 halves of duotet = cubocta & 2 halves of cubocta =
> > duotet. Do the relative volumes equate?
> > http://groups.yahoo.com/group/synergeo/attachments/folder/1932305706/item/652172707/view

> > http://www.rwgrayprojects.com/synergetics/s09/figs/f5031.html
> > http://www.rwgrayprojects.com/synergetics/s10/figs/f0632.html
> > Relative volumes: (endo = interior, ecto = exterior)

> > Tet: vol 1 (@IVM)
> > Oct: vol 4 (@IVM)
> > Star Tet: center tet endovol 1 + ectovol 4 (ext tets)
> > Star Oct: center oct endovol 4 + ectovol 8 (@IVM)
> > Star Icosa: center icosa endovol 18.51 + ectovol 20
> > Star Cubocta: endovol 20 + ectovol 20

> > Icosa (Fuller explanation, Amy Edmondson, paraphrased)
> > Its endo-volume of approximately 18.51 does not fit rationally into
> > the cosmic hierarchy with whole numbers nor click-stop when
> > jitterbugging. The icosahedron is a phase in between octahedron and
> > vector equilibrium, rather than a definitive stopping point in the
> > flow. The icosahedron is thus restricted to single-layer
> > construction, able to contract/collapse to rigidity, its radius too
> > small to permit having same-size nuclear sphere. (461.05)
> > You could not have two adjacent layers of vector equilibria and then
> > have them collapse to become the icosahedron, it has to be an outside
> > layer, remote from other layers... . It may have as high a frequency
> > as nature may require. The center is vacant. (456.20-1).

> > If the center of an icosa is vacant, should its structural volume be
> > zero? Consider the volume of a donut/torus, is the donut hole volume
> > included in the donut's volume? The donut hole does not contribute to
> > the structure of the donut, it is vacant, but it is part of the donut
> > definition. That is sort of what I think the inner volume of an icosa
> > is, a sort of donut hole, a vestige, therefore not a whole number,
> > perhaps like the interstitial spaces in ball packing.

> > DD

Cubocta = cube & octahedron dual intercept, VE when complete
Octet = octahedron & tetrahedron 3D lattice (CCP), cubocta or star oct IVM
Cuboctet = IVM with cubocta voids (vector flexor) expand-contract jitterbug
shrinks from 20 tetvol hollow cubocta with single bonds to 4 tetvol octet with double bonds

Relativity on earth
Bucky's design science goal was "To make the world work for 100% of humanity in the shortest possible time through spontaneous cooperation without ecological offense or the disadvantage of anyone." Tim Tyler: How can you tell if the world is 'working' for someone? What % of humanity is the world 'working' for today? It seems rather unrealistic to expect nobody to be disadvantaged. Advantages in nature are relative - "evolution is driven by relative fitnesses, not absolute fitnesses. Santayana's aphorism, ``It is not enough to succeed; others must fail.''
At any time, an organism's chances of surviving depend not on how fit it is, but on how fit it is relative to its competition."

Neutron stars do really exist. Long after the protons and electrons have long given up the struggle to maintain their identity against the force of gravity, all that is left is neutrons, pressed together into one big atomic nucleus a few kilometres across.

Stars are big balls of gases. Their size is determined by the balance between two opposing forces: gravity pulling the gas inwards, and pressure pushing it outwards. Just like the pressure of air in a balloon, pressure reflects the fact that it's hard to push things together. The pressure depends on how many things you're trying to push together (density), but it also depends on how hard they are to push together. At higher temperatures, the air molecules have more energy, so it takes more effort to keep them from bouncing off each other. There's a relation, then, between the amount of matter and the pressure it exerts in a given setting, which is called the equation of state. We have a fairly good idea of this relationship for the interior of stars like our own sun. Eventually, as our sun radiates energy away, the internal pressure will fall and the gravitational force will increase the density until the point at which electrons are forced together (or, more precisely into degenerate states) forming a white dwarf, and for these conditions we also have a fairly good idea of the equation of state. But for more massive stars, the collapse keeps going past this point and is only halted when the remaining neutrons are forced too 'close' together. And at this point we reach some uncertainty (at least according to 1980 era graduate texts) in what the equation of state is. So there is some real scientific uncertainty in the mass of the neutron teaspoon.

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