Menger sponge 208811 224311580 2008-07-08T08:07:13Z Ghewgill 10757 [[WP:UNDO|Undid]] revision 224154662 by [[Special:Contributions/82.73.59.206|82.73.59.206]] ([[User talk:82.73.59.206|talk]]) - seriously, we don't need more of this {{Otheruses|Sponge (disambiguation)}} [[Image:Menger sponge (IFS).jpg|right|thumb|Menger sponge]] In [[mathematics]], the '''Menger sponge''' is a [[fractal]] curve. It is the '''universal curve''', in that it has [[topological dimension]] one, and any other [[curve]] or [[graph (mathematics)|graph]] is [[homeomorphic]] to some subset of the Menger sponge. It is sometimes called the ''Menger-Sierpinski sponge'' or the ''Sierpinski sponge''. It is a three-dimensional extension of the [[Cantor set]] and [[Sierpinski carpet]]. It was first described by [[Austria]]n [[mathematician]] [[Karl Menger]] in 1926 while exploring the concept of topological dimension. ==Construction == Construction of a Menger sponge can be visualized as follows: # Begin with a cube. (''first image'') # Divide every face of the cube into 9 squares. This will sub-divide the cube into 27 smaller cubes, like a [[Rubik's Cube]] # Remove the cube at the middle of every face, and remove the cube in the center, leaving 20 cubes (''second image''). This is a Level 1 Menger sponge. # Repeat steps 1-3 for each of the remaining smaller cubes. The second repetition will give you a Level 2 sponge (''third image''), the third a Level 3 sponge (''fourth image''), and so on. The Menger sponge itself is the limit of this process after an infinite number of iterations. [[Image:Menger sponge (Level 1-4).jpg|600px|center|'''Menger sponge''', first four levels of the construction.]] The number of cubes increases by <math>20^n</math>, with <math>n</math> being the number of iterations performed on the first cube: {| width="228px" | width="20%"|'''Iters''' | width="40%"|'''Cubes''' | width="40%"|'''Sum''' |- | align="center"| 0 | align="left"| 1 | align="left"| 1 |- | align="center"| 1 | align="left"| 20 | align="left"| 21 |- | align="center"| 2 | align="left"| 400 | align="left"| 421 |- | align="center"| 3 | align="left"| 8,000 | align="left"| 8,421 |- | align="center"| 4 | align="left"| 160,000 | align="left"| 168,421 |- | align="center"| 5 | align="left"| 3,200,000 | align="left"| 3,368,421 |- | align="center"| 6 | align="left"| 64,000,000 | align="left"| 67,368,421 |} At the first level, no iterations are performed, (20<sup>0</sup> = 1). ==Properties == [[Image:Menger.png|thumb|right|An illustration of ''M<sub>4</sub>'', the fourth iteration of the construction process]] Each face of the Menger sponge is a [[Sierpinski carpet]]; furthermore, any intersection of the Menger sponge with a diagonal or medium of the initial cube ''M<sub>0</sub>'' is a [[Cantor set]]. The Menger sponge is a [[closed set]]; since it is also bounded, the [[Heine-Borel theorem]] implies that it is [[compact set|compact]]. Furthermore, the Menger sponge is [[uncountable set|uncountable]] and has [[Lebesgue measure]] 0. The [[topological dimension]] of the Menger sponge is one, the same as any [[curve]]. Menger showed, in the 1926 construction, that the sponge is a '''universal curve''', in that any possible one-dimensional curve is [[homeomorphic]] to a subset of the Menger sponge, where here a curve means any [[compact space| compact]] [[metric space]] of [[Lebesgue covering dimension]] one; this includes [[tree (graph theory)|trees]] and [[graph theory|graphs]] with an arbitrary [[countable]] number of edges, vertices and closed loops, connected in arbitrary ways. In a similar way, the [[Sierpinski carpet]] is a universal curve for all curves that can be drawn on the two-dimensional plane. The Menger sponge constructed in three dimensions extends this idea to graphs that are not flat, and might be embedded in any number of dimensions. Thus any geometry of [[quantum loop gravity]] can be embedded in a Menger sponge. Interestingly, the Menger sponge simultaneously exhibits an infinite surface area and encloses zero volume. The sponge has a [[Hausdorff dimension]] of (ln 20) / (ln 3) (approx. 2.726833). <br clear=all> ==Formal definition == Formally, a Menger sponge can be defined as follows: :<math>M := \bigcap_{n\in\mathbb{N}} M_n</math> where ''M<sub>0</sub>'' is the [[unit cube]] and :<math>M_{n+1} := \left\{\begin{matrix} (x,y,z)\in\mathbb{R}^3: & \begin{matrix}\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}\end{matrix} \end{matrix}\right\}.</math> == See also == * [[List of fractals by Hausdorff dimension]] * [[Sierpinski triangle]] * [[Sierpinski triangle#Analogs in higher dimension|Sierpinski tetrahedron]] * [[Koch snowflake]] ==References== * Karl Menger, ''General Spaces and Cartesian Spaces'', (1926) Communications to the Amsterdam Academy of Sciences. English translation reprinted in ''Classics on Fractals'', Gerald A.Edgar, editor, Addison-Wesley (1993) ISBN 0-201-58701-7 * Karl Menger, ''Dimensionstheorie'', (1928) B.G Teubner Publishers, Leipzig. == External links == {{Commons|Menger sponge}} * [http://www.mathematik.com/Menger/Menger2.html An interactive Menger sponge] * [http://ibiblio.org/e-notes/VRML/Poly/Poly.htm Fractal polyhedra] (VRML) and [http://ibiblio.org/e-notes/3Dapp/Sponge.htm interactive Java models] * [http://santisan.free.fr/coco/extras.htm Puzzle Hunt] — Video explaining Zeno's paradoxes using Menger-Sierpinski sponge * [http://theiff.org/oexhibits/menger01.html The Business Card Menger Sponge Project] * [http://www.flickr.com/photos/84445194@N00/sets/72157594256801666/ Menger Sponge Assembly] Construction of a Level-3 Menger Sponge from business cards * [http://www.mengermania.com Mengermania] — Construction of a Level 4 Menger Sponge from index cards with detailed status updates * [http://www.pure-mirage.com/html/Optimized%20Menger%20Sponges.htm Menger Sponge Animations] — Menger Sponge Animations up to Level 9, discussion of optimization for 3d. *[http://www.cornellcollege.edu/mathematics/ L3 Menger Sponge with business cards 2006] - An L3 Menger Sponge by students at Cornell College built in 2006 * [http://www.msstate.edu/web/phototemplate.php?Id=1737 Level 3 Menger Sponge made of Business Cards] - A Level 3 Menger Sponge built by students at Mississippi State University out of 48,000 folded business cards. 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