# icosa crystal structure

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

1. ### brian a m stucklessGuest

\$ 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"):
EXAMPLEs:
Vol / surface = m^3 / m^2 = Amp / (->H) = diameter / 6.
WHERE:
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 >.!!

#######editing..
/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