Some people have probably been thinking
D_n for Menger objects:
what is he talking about.
I came from a Chemistry background:
specifically Physical Inorganic spectroscopy using IR and Raman infrared .
We used what are called "Point groups"
In the Cartan D type there are three types in three dimensions :
D_n
D_nh
D_nv
Now the Menger types I have been doing
break down as having an horizontal plane of symmetry : Dnh
What happens when you break that symmetry to get Dnv like symmetry?
It is pretty much as if you took the top and twisted it around by 45
degrees in a cube!
The finding of the "diagonal" Menger contributed to the development of
this
fractal too.
The bottom Diagonal version is bigger by a factor of Sqrt[2] due to that
45 degree rotation.
Although the dimension in Moran similarity terms remains unchanged,
the "roughness" of the external figure increases.
The result is like a steep Mayan pyramid with a platform on the top!
As far as I know no one has taken this approach to Menger sponges before.
Pictures:
http://profile.imeem.com/GUmj0c/photo/yLSNmE-k/ouTiQ9Vr_l/
http://profile.imeem.com/GUmj0c/photo/yLSNmE-k/fQT991izXz/
Mathematica code:
(*D4v like Menger cube by Roger Bagula 16 Aug 2007©*)
(* symmetric isomer of the Menger cube*)
(* patterned from Menger cube code by Szabolcs Horvát <
szhorvat@[EMAIL PROTECTED]
>, \
University of Bergen in
Mathematica newsgroup : Mon, 28 May 2007 09 : 10 : 50*)
Clear[pieces, menger1]
p = {{1, 3, 0}, {2, 2, 0}, {2, 4, 0}, {3, 1, 0}, {3, 5, 0}, {4, 2,
0}, {4, 4, 0}, {5, 3, 0}};
p1 = {{2, 2, 2}, {2, 3, 2}, {2, 4, 2}, {3, 2, 2}, {3, 4, 2}, {4, 2, 2},
{4, 3,
2}, {4, 4, 2}};
p2 = {{1 + 1/2, 3, 1}, {3, 4 + 1/2, 1}, {3, 1 + 1/2, 1}, {4 + 1/2, 3, 1}};
pieces =
Join[p, p1, p2];;
N[Log[Length[pieces]]/Log[3]]
2.7268330278608417`
menger1[cornerPt_, sideLen_, n_] :=
menger1[cornerPt + #1*(sideLen/3), sideLen/3, n - 1] & /@[EMAIL PROTECTED]
pieces;
menger1[cornerPt_, sideLen_, 0] :=
{EdgeForm[], Cuboid[cornerPt, cornerPt + sideLen*{1, 1, 1}]};
gr = Show[Graphics3D[Flatten[menger1[{0, 0, 0}, 1, 3]]], Boxed -> False]
Show[gr, ViewPoint -> {0.001, -0.045, 3.383}]
Show[gr, ViewPoint -> {-0.064, 3.005, 1.555}]
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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