# Talk:Menger sponge

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## Math block

This math block:

${\displaystyle M_{n+1}:=\left\{(x,y,z)\in \mathbb {R} :\exists i,j,k\in \{0,1,2\}:((3x-i,3y-j,3z-k)\in M_{n}){\mbox{ and at most one of }}i,j,k{\mbox{ is equal to 1}}\right\}}$

is too wide to fit on mid to small size screens (1024px or smaller). Could someone familiar with the notation find a good way to break it into two or more lines?

I'd suggest breaking it at the colon (before the 3x-i part) though I can't figure out how to get that to work. That's the point where it switches from describing the set it's a part of to giving the formula (basically). 68.47.64.155

## Incorrect?

[[Image:Gasket14.png|thumb|3D [[Sierpinski carpet]]]] I was editing the page Cantor dust, (adding a image of the 3D version of the set). There I found, the text talks about a 3D version of the Cantor dust and names it "the Menger sponge". But this page shows a 3D Sierpinski carpet. The images to the right shows the difference. Does anybody what's correct? // Solkoll 22:59, 29 Dec 2004 (UTC)

Both are correct, just different generalizations of the Cantor set in three dimensions. Both the Cantor dust and Menger sponge have the Cantor set within them, but also space around the Cantor set that is not directly described by the set. This space around the set is described by extending the set into multidimensions, but there are multiple ways to interpret the pattern of the Cantor set, as it does not directly define its construction in more dimensions. Pengwy 02:33, 22 May 2006 (UTC)

## Sponge?=

It might be good to explain why this is a "sponge" and not a cube or something else.

## Universality

I mentioned this at Lebesgue covering dimension too, "any object of Lebesgue covering dimension one" doesn't embed in the Menger sponge if "object" means "topological space". I think the right version needs "compact" and "metrizable". I'm going to put a "disputed" note but the article is misleadingA Geek Tragedy 15:06, 11 February 2007 (UTC)

## Guisi aka SILK

In case anyone cares, there is a 2006 Taiwanese film called Silk that uses the Menger Sponge as a plot device. The scientist in the film uses the menger Sponge construction to try and come up with an Anti-Gracity device but winds up instead capturing a form of energy we would call a ghost. There's a fair amount of visual representation of the sponge. Neat film but it's a horror film with science fiction overtones. It's IMDB entry is here:

GUISI - Lisapollison 07:22, 27 February 2007 (UTC)

## Why the "sum" column?

I do not understand the relevance of the "sum" column, where the total numbers of different size cubes in all steps up to the present one is summed. I'd like someone to explain the reason for it; else, I'm minded to remove it. JoergenB (talk) 15:59, 1 January 2008 (UTC)

yea i dont get that either 84.216.44.247 (talk) 17:29, 25 November 2008 (UTC)

## "Lebesgue measure 0"

User:Michael C Price has twice replaced the statement that the Menger sponge has Lebesgue measure 0 with the statement that its volume has Lebesgue measure 0. Sorry, Michael, but this makes no sense. It's the set itself has Lebesgue measure 0. Perhaps what you're trying to say is something like (1) "its interior has Lebesgue measure 0" (true, but not very interesting since its interior is empty) or (2) "its 3-dimensional Hausdorff measure, as opposed to some lower-dimensional measure, is 0" (true but unnecessary to state here, since it's an immediate consequence of the fact that the sponge has Hausdorff dimension less than 3) or (3) "its Lebesgue measure, as opposed to some not-3-dimensional meausure applied to its boundary, is 0" (true, but "its Lebesgue measure" already *means* that; unless someone's in danger of thinking it means something else, why belabour the point?). None of those things, in any case, is correctly stated as "its volume has Lebesgue measure 0". Clearly you are dissatisfied with the statement that the Menger sponge has Lebesgue measure 0; could you please explain why, so that we can avoid gratuitous repeated reversions? Gareth McCaughan (talk) 20:22, 21 March 2010 (UTC)

## stars

The proposed Museum of Mathematics in New York has an M3 which has been sliced diagonally to reveal 6 pointed stars inside (instead of the squares that the typical person would expect) http://www.nytimes.com/2011/06/28/science/28math-menger.html 06:41, 29 June 2011 (UTC)

Cool! -- cheers, Michael C. Price talk 07:54, 29 June 2011 (UTC)

## Mosely Snowflake

Is anyone interested in creating a related article for the Mosely snowflake? I found this article [1]] enlightening. 70.247.162.84 (talk) 18:42, 16 September 2012 (UTC)