[math-fun] fractal sponges
Some of you may know that I built a large origami model of Menger's sponge a few years ago out of 66,048 business cards. Recently I have been playing around with a different fractal sponge and I'm wondering if anyone else has studied it or named it. I have tried some web searches, but if this other sponge is out there I cannot find it among the thousands of sites that mention Menger's sponge. To make Menger's sponge, take a cube and divide it into 3x3x3 smaller cubes, then throw away the cubes in the centers of the faces and in the center of the body, keeping only the corner and edge cubes, then repeat this process with the remaining cubes an infinite number of times. To make my sponge, at each iteration throw away the corners and the body center, keeping the edge and face cubes. Some people protest that there is no point in throwing away the body center cube because you cannot tell that it is missing. But this is only true at the first iteration -- after that the missing corners allow you to see into the holes. I have modeled it in Mathematica and it looks pretty cool. When viewed along one of the original cube's long diagonals, this fractal is riddled with holes that have a cross section shaped like a snowflake curve. Because of this I have been calling it a Snowflake Sponge. If it already has a name I'd like to know. You can create other fractals by keeping different subsets of the sub-cubes. I wrote my Mathematica program to take as arguments a list of numbers from 0 to 3, indicating whether or not to keep the corner, edge, face or body cubes. If you want it to be connected then obviously the numbers have to be consecutive. Menger's sponge is {0,1}, my sponge is {1,2}. {2,3} makes an interesting fractal but it's more a tree than a sponge. -- Jeannine Mosely
On Tue, Oct 27, 2009 at 11:35 PM, Jeannine Mosely <j9mosely@gmail.com>wrote:
Some of you may know that I built a large origami model of Menger's sponge a few years ago out of 66,048 business cards. Recently I have been playing around with a different fractal sponge and I'm wondering if anyone else has studied it or named it. I have tried some web searches, but if this other sponge is out there I cannot find it among the thousands of sites that mention Menger's sponge.
Nice idea, Jeannine! Have you put your picture somewhere we can look at? I tried web searches using the fact that its fractal dimension is log_3(18), approx. 2.63092975, and couldn't find any mention that way. --Michael
To make Menger's sponge, take a cube and divide it into 3x3x3 smaller cubes, then throw away the cubes in the centers of the faces and in the center of the body, keeping only the corner and edge cubes, then repeat this process with the remaining cubes an infinite number of times. To make my sponge, at each iteration throw away the corners and the body center, keeping the edge and face cubes.
Some people protest that there is no point in throwing away the body center cube because you cannot tell that it is missing. But this is only true at the first iteration -- after that the missing corners allow you to see into the holes.
I have modeled it in Mathematica and it looks pretty cool. When viewed along one of the original cube's long diagonals, this fractal is riddled with holes that have a cross section shaped like a snowflake curve. Because of this I have been calling it a Snowflake Sponge. If it already has a name I'd like to know.
You can create other fractals by keeping different subsets of the sub-cubes. I wrote my Mathematica program to take as arguments a list of numbers from 0 to 3, indicating whether or not to keep the corner, edge, face or body cubes. If you want it to be connected then obviously the numbers have to be consecutive. Menger's sponge is {0,1}, my sponge is {1,2}. {2,3} makes an interesting fractal but it's more a tree than a sponge.
-- Jeannine Mosely _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush.
Jeannine, There is Mathematica code here to generate many related fractal polyhedra of that general nature: (See Section 8 of the paper. It is from Bridges 2008) http://www.georgehart.com/ProceduralGeneration/Bridges08-Hart10pages.pdf http://www.georgehart.com/ProceduralGeneration/ProceduralGenerationOf3DForms... I don't think any of them have standard names other than "Sierpinski tetrahedron" and "Menger sponge". You can often cheat and make these physically on rapid prototyping machines even if they are not mathematically connected, by enlarging the lowest-level polyhedra slightly so they overlap. The tetrahedral example is shown here: http://www.georgehart.com/rp/rp.html Also, some close cousins in the 3x3x3 cube family that don't keep octahedral symmetry are explored here: http://www.i.h.kyoto-u.ac.jp/~tsuiki/sakuhin-e.html George http://www.georgehart.com/ Jeannine Mosely wrote:
Some of you may know that I built a large origami model of Menger's sponge a few years ago out of 66,048 business cards. Recently I have been playing around with a different fractal sponge and I'm wondering if anyone else has studied it or named it. I have tried some web searches, but if this other sponge is out there I cannot find it among the thousands of sites that mention Menger's sponge.
To make Menger's sponge, take a cube and divide it into 3x3x3 smaller cubes, then throw away the cubes in the centers of the faces and in the center of the body, keeping only the corner and edge cubes, then repeat this process with the remaining cubes an infinite number of times. To make my sponge, at each iteration throw away the corners and the body center, keeping the edge and face cubes.
Some people protest that there is no point in throwing away the body center cube because you cannot tell that it is missing. But this is only true at the first iteration -- after that the missing corners allow you to see into the holes.
I have modeled it in Mathematica and it looks pretty cool. When viewed along one of the original cube's long diagonals, this fractal is riddled with holes that have a cross section shaped like a snowflake curve. Because of this I have been calling it a Snowflake Sponge. If it already has a name I'd like to know.
You can create other fractals by keeping different subsets of the sub-cubes. I wrote my Mathematica program to take as arguments a list of numbers from 0 to 3, indicating whether or not to keep the corner, edge, face or body cubes. If you want it to be connected then obviously the numbers have to be consecutive. Menger's sponge is {0,1}, my sponge is {1,2}. {2,3} makes an interesting fractal but it's more a tree than a sponge.
-- Jeannine Mosely _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Nice idea! from the side it kind of looks like a cross crosslet, so you might also call it a Cross Sponge, or Cross Crosslet Sponge: http://images.google.com/images?q=Cross+crosslet Here is a quick render of the outside: http://www.flickr.com/photos/sbprzd/4052575863/ And here is a view from the inside (requires Flash, double click to view full screen and use your mouse to move the view) http://theseblog.free.fr/fsppa.php?q=4104357940534061060aaf598b44 Cheers, Seb On Wed, Oct 28, 2009 at 04:35, Jeannine Mosely <j9mosely@gmail.com> wrote:
To make my sponge, at each iteration throw away the corners and the body center, keeping the edge and face cubes.
participants (4)
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George W. Hart -
Jeannine Mosely -
Michael Kleber -
Seb Perez-D