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Author Topic: 3D Printed Cross Sections of the Menger Sponge  (Read 2855 times)
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tit_toinou
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« on: April 02, 2013, 10:03:15 AM »

Hi all,

I made the cross sections of the menger sponge on Shapeways.com :


╔toile de Menger 8cm




╔toile de Menger 5cm




Triangle de Menger




Losange de Menger
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cKleinhuis
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« Reply #1 on: April 02, 2013, 10:46:46 AM »

that star shapes are just awesome, as i told before, i like that they are just inside the menger when cutting diagonally,
i wonder how smooth the diagonal parts are, because the rasterization may lead to pixel-stepping !?

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divide and conquer - iterate and rule - chaos is No random!
hobold
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« Reply #2 on: April 02, 2013, 07:33:09 PM »

By the way, there is at least one more interesting shape hiding in that Menger sponge.

Starting from the whole cube, you can cut away a total of four "Triangle de Menger" pieces (2nd to last picture). You end up with a regular tetrahedron that has four identical faces, each looking like the equilateral face of of a "Triangle de Menger".
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tit_toinou
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« Reply #3 on: April 02, 2013, 11:02:00 PM »

that star shapes are just awesome, as i told before, i like that they are just inside the menger when cutting diagonally,
i wonder how smooth the diagonal parts are, because the rasterization may lead to pixel-stepping !?
I'm mesmerised too.
There's no pixel-stepping.
I didn't do (for example) what bib does for creating 3d models, i.e. creating little cubes from a DE formula.
I created individually all the little cubes. And I computed the intersections for the cubes crossing the plane.
Computing cross sections of a cube is a pain in the ass, by the way Grin with closed eyes .

By the way, there is at least one more interesting shape hiding in that Menger sponge.

Starting from the whole cube, you can cut away a total of four "Triangle de Menger" pieces (2nd to last picture). You end up with a regular tetrahedron that has four identical faces, each looking like the equilateral face of of a "Triangle de Menger".
How do you "cut" a cube with fours planes and end up with a tetrahedron ?
I can see how to cut it with height planes and end up with an octahedron !
I'll try to create it I think it'll be worth it.

I've also spotted a mix between "Losange de Menger" and "Triangle de Menger" I have to render it.
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hobold
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« Reply #4 on: April 04, 2013, 04:29:42 PM »

How do you "cut" a cube with fours planes and end up with a tetrahedron ?
Consider the eight corners of a cube. Imagine coloring those corners with a checkerboard pattern, that is, pick one corner and color it black. Then color the three adjacent corners white. Now the immediate neighbors of the new white corners can be colored black, and so on. Eventually you will have four black corners and four white corners.

Now you can cut four tetrahedra from the original cube in the following way: start from a black corner and place a cutting plane through three adjacent white corners. This can be done for all four black cube corners.
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tit_toinou
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« Reply #5 on: April 05, 2013, 01:54:24 PM »

@hobold: Ok i'll try that.

I've found a way to get three designs into one object Grin with closed eyes :


Tranches de Menger
.
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cKleinhuis
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« Reply #6 on: April 29, 2013, 03:35:56 PM »

just for completion, haltenny posted this in response the menger contact
https://simonsfoundation.org/multimedia/mathematical-impressions-the-surprising-menger-sponge-slice/
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divide and conquer - iterate and rule - chaos is No random!
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