icosa crystal structure

Discussion in 'Numerical Analysis' started by brian a m stuckless, Sep 30, 2005.

  1. $ icosa crystal structure
    NOT "tied to a definition" of the meter;
    But, rather "accountability" to a meter.

    The TYPiCAL crystallographical topology
    CAN'T uniformly describe crystal groups
    in terms of a STANDARD volume and edge.

    Unity isn't restricted to edge-lengths.

    NOTE: iSOTOPiC FiNE STRUCTURE Mass & SHAPEfactor 666.!!
    "COMMON & GENERAL structure" has to do with (about) Euclidean "SHAPE".
    And so, therefore, STRUCTURE PRE-defines ALL of the iNTEGER NUMBERs.
    [Sqrt5 = diagonal of double squares; Sqrt2 = diagonal of square.]

    A "VOLUME to SURFACE area" quotient can QUANTiFY a particular "SHAPE"
    (but a shape QUANTiTY WiTHOUT having PARTiCULARly-known "STRUCTURE"):
    Vol / surface = m^3 / m^2 = Amp / (->H) = diameter / 6.
    1. Diameter / 6 implies a Euclidean sphere.
    2. Edgelength / 6 implies a cube.
    3. Triakis-icosahedron tip to tip / 6 , an iCOSA CORE symmetry.

    What this means is that if either of these has a diameter, edgelength
    or tip to tip of six (6) units (of ANY Unit System}, that STRUCTURE
    has a very simple 'Volume to Surface Area' QUOTiENT = ONE (1) unit.
    [Made memorable SiMPLY as 6,6,6; (SPHERE, CUBE, TriAKiS-iCOSAHEDRON).]
    Note THESE are the ONLY three STRUCTUREs which CORRESPOND in THiS way.

    Go see: < a SPACE.wpd > attached to < info-itsy-bitsy bytes it >.!!

    /6 /6 /6
    A sphere, cube, and golden rhombic triakisicosahedron
    share a common equation of volume to surface area ratio
    of diameter / 6, edgelength / 6, and tip-to-tip / 6,
    ..respectively, all = scaling factor / 6.

    The volume to surface area ratio of each will be unity
    if the sphere has a diameter of six units, the cube has
    an edgelength of six units, and the golden rhombic
    triakisicosahedron has a golden rhombic lattice long
    diagonal of six units, as well.

    This gives a sphere volume of 36*pi=113.0973 cubic
    units, a cube volume of exactly 216 cubic units, and
    a golden rhombic triakisicosahedron volume of exactly
    667.4767 cubic units, each with a volume to surface
    area ratio of one ..unity.

    A golden rhombic triakisicosahedron is an icosahedral
    crystal core, with 20 golden parallelipiped unit-cells.
    This initial nucleus of twenty golden rhombohedral cells
    is a golden rhombic triakisicosahedron ..see the attached.

    Golden rhombic triakisicosahedra ..20-fold RADiAL symmetry.

    Golden rhombic parallelipiped unit-cells, with pointed-end
    angles of arctan 2 degrees, are the all-space-filling and
    can be stacked around the icosahedral cyrstal core-cluster
    of 20 unit-cells, maintaining the radial symmetry.

    An icosahedral crystal is a composite of 20 golden rhombic
    parallelipiped sectors.

    It has central angles of arctan 2 degrees, which is the
    central angles of a regular icosahedron & golden rhombic
    triakisicosahedron, --the apex angles of golden rhombii,
    triacontahedra and the Great Pyramid of Egypt.

    A triacontahedron is a six dimensional cube and twelve
    will exactly close-pack all space around a golden rhombic
    triakisicosahedron. It has 30 golden rhombus faces.

    The long to short diagonal ratio of a golden rhombus is
    phi --the golden ‘diamond' diagonal ratio, in this case.

    The area of one side of a golden rhombus is the short
    diagonal multiplied by the long diagonal, divided by 2.

    Go to:
    < http://www.georgehart.com/virtual-polyhedra/vp.html >

    Select: ‘Stellations of the Rhombic Triacontahedron'

    Choose: ‘list of models'
    See: #26 (U) (5 colors)
    --for the golden rhombic triakisicosahedron
    See: #1 core (5 colors)
    --for the golden rhombic triacontahedron
    --right click on the objects to select stick-view, and spin.

    Choose: ‘background' --to view stills of both the
    triacontahedron and triakisicosahedron on one page.

    Both of these are 5-cube composites in a circumsphere.
    The 30 golden rhombic faces of a triacontahedron are
    uniform portions of 30 faces of 5 composite cubes. The
    20 tips of the golden rhombic triakisicosahedron mark
    the 40 corners of 5 composite cubes.

    Together, these two are all-space-filling.

    With triacontahedra as hubs and the triakisicosahedra
    as struts, these two also can form all of the golden
    rhombic isozonohedra.

    Yours truly, `````arcsign`````
    VERY sincerely u c, ```Brian
    brian a m stuckless, Sep 30, 2005
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