Hypothetical fractal process in the chemistry of ionic crystals:
Cubic crystals are well known in nature.
Sodium Chloride is a very good example.
The idea is to make up a statistical analog crystal to different types
on Menger sponges
using substitution of lithium or potassium.
Example:
mix:
7 moles of LiCl ( or KCl)
with
20 moles of NaCl
so that the metal ion
Moran dimension in sodium is
Log[20]/Log[3]
Random Menger cubes with 7 of one and 20 other would result
in the crystal that results overall from the solution.
( differential solubilities may flaw the result further: experiments
would have to be done)
There is little hope that a Menger structure would result,
but the result should be statistically Menger in properties.
The fracturing of the crystals of this type would depend on the relative
amounts of Lithium to Sodium.
Comparison with pure Sodium Chloride fractures to
different integer fractional ( rational) mole amounts of impurities of
LiCl or KCl
would give an experimental method of gaging fractal
statistical dimension properties so that a model
for crystal could be formulated.
A program in Mathematica to make up a random Menger cube is possible
with a slight alteration of the original program to put in 7 voids in a
27 cube
matrix of cubes.
The Random von Koch in Falconer's "Fractal Geometry"
and percolation fractals are very much like this process.
Picture Link:
http://profile.imeem.com/GUmj0c/photo/Qh05u0X9I5/
Mathematica:
Clear[a, menger]
(* Random Menger cubes : by Roger Bagula 26 Aug 2007© : Similarity
Dimension \
2.771243749161422*)
(* random voids in a Menger cube statistic of 27 cubes and 7 random
voids*)
a := Complement[
Flatten[Table[{i, j, k}, {i, 0, 2}, {j, 0, 2}, {k, 0, 2}],
2], Table[{Random[Integer, {0, 2}], Random[Integer, {0, 2}], \
Random[Integer, {0, 2}]}, {n, 1, 7}]]
N[Log[Length[a]]/Log[3]]
2.771243749161422`
menger[cornerPt_, sideLen_, n_] :=
menger[cornerPt + #1*(sideLen/3), sideLen/3, n - 1] & /@[EMAIL PROTECTED]
a;
menger[cornerPt_, sideLen_, 0] :=
{EdgeForm[], Cuboid[cornerPt, cornerPt + sideLen*{1, 1, 1}]};
Show[Graphics3D[Flatten[menger[{0, 0, 0}, 1, 3]]], Boxed -> False]
Respectfully, Roger L. Bagula
11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814
:http://www.geocities.com/rlbagulatftn/Index.html
alternative email: rlbagula@[EMAIL PROTECTED]
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