What is the Zero Density Cube?

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  • Thread starter Thread starter chaotixmonjuish
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    Cube Density Zero
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Discussion Overview

The discussion revolves around the concept of a geometric figure known as the Zero Density Cube, which appears to relate to fractal structures. Participants explore the properties and construction methods of these figures, particularly focusing on the Sierpinski sponge and the Menger sponge, and their implications regarding density and volume.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant describes a figure where small cubes are cut out from each face of a larger cube, suggesting it has a density of 1 or 0.
  • Another participant proposes that this figure resembles a "Sierpinski sponge," which involves dividing a cube into smaller cubes and removing the middle cube, resulting in a fractal dimension.
  • A third participant introduces the "Menger sponge," noting that it is constructed similarly but involves removing the center cube, leading to 20 smaller cubes and resulting in infinite surface area and zero volume.
  • A later reply acknowledges the similarity between the Sierpinski sponge and the Menger sponge, indicating a connection between the two concepts.

Areas of Agreement / Disagreement

Participants express differing views on the specific figure being discussed, with some suggesting it may be the Sierpinski sponge while others argue it aligns more closely with the Menger sponge. The discussion remains unresolved regarding the exact identity and properties of the Zero Density Cube.

Contextual Notes

Participants do not clarify the definitions of density in this context, nor do they resolve the mathematical implications of the infinite iterations involved in constructing these fractal figures.

chaotixmonjuish
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I may be posting this in the wrong section:

A while ago I was reading a collection of articles on Wikipedia and I stumbled upon this article that was pretty interesting. It was about a cube, on each face of the cube a small cube is cut out. It is done many more times. The next thing I recall was that the figure had a density of 1 or 0. I'm not sure what this is called. Does anyone have any idea about this?
 
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Sounds to me like your are talking about a "Sierpinski sponge". Divide a cube into 27 small cubes by cutting each face with two planes and remove the middle cube. Now do the same with each of the remaining cubes. It has fractal dimension.
 
I might be describing the same thing as HofI, but the Menger sponge is constructed in much the same way except that you also remove the centre cube, so that you're left with 20 smaller cubes. Iterating this infinitely produces the Menger sponge, which has infinite surface area and zero volume.
 
It was the first one, but the second one seems pretty similar
 

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