D8 octagon as cuboid 3D fractal of Menger type

[Roger Bagula]

I've very busy on some of these 3D fractals
and improving them. I managed to get two levels of an
dodecahedron 3d fractals
and yesterday, a cube octahedron hollow pyramid.
I appended the EdgeForm[] command in Mathematica
so that at level 3 they aren't mostly black with the edges.

These D8 fractals will tile with Diagonal Menger cubes...
First inkling I've had there is a diagonal Menger cube.
I use relative center coordintes x->{0,6},y->{0,6},z->{0,2}
The three level at 40 cubes ber unit is very slow here:
Top layer : 16 cubes in an octagon ( 8 on edges)
Middle layer: 8 cubes at vertices with gaps
Bottom layer : 16 cubes in an octagon ( 8 on edges)
This method gives an uneven octagon with diagonal
edges longer than the flat edges.
Picture of level 3 at:
http://profile.imeem.com/GUmj0c/photo/yLSNmE-k/TgEyoDeAsF/
Mathematica:
Clear[pieces, menger]
p = {{0, 2, 0}, {0, 3,
0}, {0, 4, 0}, {1, 1, 0}, {1, 5, 0}, {2, 0, 0}, {3, 0, 0}, {4, 0, 0}, {2,
6, 0}, {3, 6, 0}, {4, 6, 0}, {5, 1, 0}, {5, 5, 0}, {6, 2, 0}, {6,
3, 0}, {6, 4, 0}};
p1 = {{0, 2, 2}, {0, 3, 2}, {0, 4, 2}, {1, 1, 2}, {1, 5, 2}, {2, 0,
2}, {3, 0, 2}, {4, 0, 2}, {2, 6, 2}, {3, 6, 2}, {4, 6, 2}, {5, 1,
2}, {5, 5, 2}, {6, 2, 2}, {6, 3, 2}, {6, 4, 2}};
p2 = {{0, 2, 1}, {0,
4, 1}, {2, 0, 1}, {4, 0, 1}, {2, 6, 1}, {4, 6, 1}, {6, 2, 1}, {6, 4, 1}};
pieces = Join[p, p1, p2];
Length[pieces]
40
menger[cornerPt_, sideLen_, n_] :=
menger[cornerPt + #1*(sideLen/3), sideLen/3, n - 1] & /@ pieces;
menger[cornerPt_, sideLen_, 0] :=
{EdgeForm[], Cuboid[cornerPt, cornerPt + sideLen*{1, 1, 1}]};

Show[Graphics3D[Flatten[menger[{0, 0, 0}, 1, 1]]], Boxed -> False]
Show[Graphics3D[Flatten[menger[{0, 0, 0}, 1, 2]]], Boxed -> False]
Show[Graphics3D[Flatten[menger[{0, 0, 0}, 1, 3]]], Boxed -> False]


Respectfully, Roger L. Bagula
11759Waterhill Road, Lakeside,Ca 92040-2905,tel: [phone]
:http://www.geocities.com/rlbagulatftn/Index.html
alternative email: ***@sbcglobal.net

[Roger Bagula]

Picture of the level three outputs:
http://profile.imeem.com/GUmj0c/photo/yLSNmE-k/boadA-WWGt/
Once I figured out the D8 version,
the others were plug in and turn the crank...
I have to do the D5 yet, but I don't think it will be hard.
There aren't "regular" polygons like the group actually requires
and the IFS of these point arrays will be different than these "stacking"
cuboid fractals,
but the generalization of the symmetry seems to be working well.
The finding of the "diagonal" Menger cube was just a side result.
These are actually four new sponges that were never done before as far
as I know.
I did the D3 and so did Paul Bourke but as an equalateral priism.
The Menger Cube is done usually without these saw tooth edges.
I did an D6 IFS that didn't turn out at all well, but it wasn't cuboid.
The Octagon D8 is entirely new ( never done before).

[Roger Bagula]

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 <
***@gmail.com >, \
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] & /@ 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: [phone]
:http://www.geocities.com/rlbagulatftn/Index.html
alternative email: ***@sbcglobal.net

[Roger Bagula]

I made a movie that shows the three lervels I've been able to get:
it's a "viewpoint tumble movie".
It's up at Myspace and Imeem both ( I had doubts about the Imeem upload)

http://vids.myspace.com/index.cfm?fuseaction=vids.individual&VideoID=16266686
http://profile.imeem.com/GUmj0c/video/qcQpr638/d4v_twisted_menger_cube/

Mathematica:
(*D4v like Menger cube by Roger Bagula 20 Aug 2007©*)
(* symmetric isomer of the Menger cube*)
(* patterned from Menger cube code by Szabolcs Horvát <
***@gmail.com >, \
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`
(* remove EdgeForm[] to get black lines on the cubes*)
menger1[cornerPt_, sideLen_, n_] :=
menger1[cornerPt + #1*(sideLen/3), sideLen/3, n - 1] & /@ pieces;
menger1[cornerPt_, sideLen_, 0] :=
{EdgeForm[], Cuboid[cornerPt, cornerPt + sideLen*{1, 1, 1}]};
(* tumble viewpoint*)
a = Delete[Sort[Join[Flatten[Table[If[i == j ==
k == 0, {}, 3*{i, j, k}], {i, -1, 1}, {k, -1, 1}, {j, -1, 1}], 2], {{
0.001, -0.045, 3.383}}, {{-0.064, 3.005, 1.555}}]], 1];
(* change to menger1[{0,
0, 0}, 1, 2] for better output*)
gr = \
Show[Graphics3D[Flatten[menger1[{0, 0, 0}, 1, 3]]], Boxed -> False]
Table[Show[gr, ViewPoint -> a[[n]]], {n, 1, Length[a]}]