View source for Menger sponge

{{Short description|Three-dimensional fractal}}
[[Image:Menger-Schwamm-farbig.png|upright=1.4|thumb|An illustration of ''M''<sub>4</sub>, the sponge after four iterations of the construction process]]
In [[mathematics]], the '''Menger sponge''' (also known as the '''Menger cube''', '''Menger universal curve''', '''Sierpinski cube''', or '''Sierpinski sponge''')<ref>{{cite book|last1=Beck|first1=Christian|last2=Schögl|first2=Friedrich|title=Thermodynamics of Chaotic Systems: An Introduction|date=1995|publisher=Cambridge University Press|isbn=9780521484510|pages=97|url=https://books.google.com/books?id=GyPpZ-Lg6KAC&pg=PA97|language=en}}</ref><ref>{{cite book|last1=Bunde|first1=Armin|last2=Havlin|first2=Shlomo|title=Fractals in Science|date=2013|publisher=Springer|isbn=9783642779534|page=7|url=https://books.google.com/books?id=dh7rCAAAQBAJ&pg=PA7|language=en}}</ref><ref>{{cite book|last1=Menger|first1=Karl|title=Reminiscences of the Vienna Circle and the Mathematical Colloquium|date=2013|publisher=Springer Science & Business Media|isbn=9789401111027|pages=11|url=https://books.google.com/books?id=BKIyBwAAQBAJ&pg=PR11|language=en}}</ref> is a [[fractal curve]]. It is a three-dimensional generalization of the one-dimensional [[Cantor set]] and two-dimensional [[Sierpinski carpet]]. It was first described by [[Karl Menger]] in 1926, in his studies of the concept of [[topological dimension]].<ref>{{citation|first= Karl|last= Menger| title= Dimensionstheorie |year =1928| publisher= B.G Teubner |oclc=371071 }}</ref><ref>{{citation|first=Karl |last=Menger|title= Allgemeine Räume und Cartesische Räume. I. |year=1926|journal= Communications to the Amsterdam Academy of Sciences}}. English translation reprinted in {{Citation |editor1-last=Edgar |editor1-first=Gerald A. |title=Classics on fractals |publisher=Westview Press. Advanced Book Program, Boulder, CO |series=Studies in Nonlinearity |isbn=978-0-8133-4153-8 |mr=2049443 |year=2004}}</ref>

==Construction==
[[File:Menger sponge (Level 0-3).jpg|thumb|An illustration of the iterative construction of a Menger sponge up to ''M''<sub>3</sub>, the third iteration]]The construction of a Menger sponge can be described as follows:
# Begin with a cube.
# Divide every face of the cube into nine squares in a similar manner to a [[Rubik's Cube]]. This sub-divides the cube into 27 smaller cubes.
# Remove the smaller cube in the middle of each face, and remove the smaller cube in the center of the larger cube, leaving 20 smaller cubes. This is a level-1 Menger sponge (resembling a void cube).
# Repeat steps two and three for each of the remaining smaller cubes, and continue to iterate ''[[ad infinitum]]''.

The second iteration gives a level-2 sponge, the third iteration gives a level-3 sponge, and so on. The Menger sponge itself is the limit of this process after an infinite number of iterations.

==Properties==
[[File:Menger sponge diagonal section 27.png|thumb|Hexagonal cross-section of a level-4 Menger sponge. (Part of a [[commons:Category:Menger sponge diagonal section, stack|series of cuts]] perpendicular to the space diagonal.)]]
The <math>n</math>th stage of the Menger sponge, <math>M_n</math>, is made up of <math>20^n</math> smaller cubes, each with a side length of [[power of three|(1/3)<sup>''n''</sup>]].  The total volume of <math>M_n</math> is thus <math display="inline">\left(\frac{20}{27}\right)^n</math>.  The total surface area of <math>M_n</math> is given by the expression <math>2(20/9)^n + 4(8/9)^n</math>.<ref>Wolfram Demonstrations Project, ''[http://demonstrations.wolfram.com/VolumeAndSurfaceAreaOfTheMengerSponge/ Volume and Surface Area of the Menger Sponge]''</ref><ref>University of British Columbia Science and Mathematics Education Research Group, ''[http://scienceres-edcp-educ.sites.olt.ubc.ca/files/2012/08/sec_math_geometry_menger.ppt Mathematics Geometry: Menger Sponge]''</ref> Therefore, the construction's volume approaches zero while its surface area increases without bound. Yet any chosen surface in the construction will be thoroughly punctured as the construction continues so that the limit is neither a solid nor a surface; it has a topological dimension of 1 and is accordingly identified as a curve.

Each face of the construction becomes a [[Sierpinski carpet]], and the intersection of the sponge with any diagonal of the cube or any midline of the faces is a [[Cantor set]]. The cross-section of the sponge through its [[centroid]] and perpendicular to a [[space diagonal]] is a regular hexagon punctured with [[hexagram]]s arranged in six-fold symmetry.<ref>{{cite news|url=https://nytimes.com/2011/06/28/science/28math-menger.html|title=The Mystery of the Menger Sponge|first=Kenneth|last=Chang|work=The New York Times |date=27 June 2011|access-date=8 May 2017|via=NYTimes.com}}</ref> The number of these hexagrams, in descending size, is given by the following [[recurrence relation]]:<math>a_n=9a_{n-1}-12a_{n-2}</math>, with <math>a_0=1, \ a_1=6</math>.<ref>{{Cite web|url=https://oeis.org/A299916|title=A299916 - OEIS|website=oeis.org|access-date=2018-08-02}}</ref>

The sponge's [[Hausdorff dimension]] is {{sfrac|log&nbsp;20|log&nbsp;3}} &cong; 2.727.<ref>{{cite book
 | last = Quinn | first = John R.
 | editor1-last = Carfì | editor1-first = David
 | editor2-last = Lapidus | editor2-first = Michel L.
 | editor3-last = Pearse | editor3-first = Erin P. J.
 | editor4-last = van Frankenhuijsen | editor4-first = Machiel
 | contribution = Applications of the contraction mapping principle
 | doi = 10.1090/conm/601/11957
 | isbn = 978-0-8218-9148-3
 | location = Providence, Rhode Island
 | mr = 3203870
 | pages = 345–358
 | publisher = American Mathematical Society
 | series = Contemporary Mathematics
 | title = Fractal geometry and dynamical systems in pure and applied mathematics. II. Fractals in applied mathematics.
 | volume = 601
 | year = 2013}}. See [https://books.google.com/books?id=WbMCAQAAQBAJ&pg=PA351 Example 2, p. 351].</ref> The [[Lebesgue covering dimension]] of the Menger sponge is one, the same as any [[curve]]. Menger showed, in the 1926 construction, that the sponge is a ''[[Urysohn universal space|universal curve]]'', in that every curve is [[homeomorphic]] to a subset of the Menger sponge, where 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. Similarly, 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 [[Planar graph|planar]] and might be embedded in any number of dimensions.

In 2024, Broden, Nazareth, and Voth proved that all knots can also be found within a Menger sponge.<ref>{{Cite web |last=Barber |first=Gregory |date=2024-11-26 |title=Teen Mathematicians Tie Knots Through a Mind-Blowing Fractal |url=https://www.quantamagazine.org/teen-mathematicians-tie-knots-through-a-mind-blowing-fractal-20241126/ |access-date=2024-11-29 |website=Quanta Magazine |language=en}}</ref>

The Menger sponge is a [[closed set]]; since it is also bounded, the [[Heine&ndash;Borel theorem]] implies that it is [[compact set|compact]]. It has [[Lebesgue measure]] 0. Because it contains continuous paths, it is an [[uncountable set]].

Experiments also showed that cubes with a Menger sponge-like structure could dissipate [[Shock wave|shocks]] five times better for the same material than cubes without any pores.<ref name=Shockwave>{{cite journal|last1=Dattelbaum|first1=Dana M.|last2=Ionita|first2=Axinte|last3=Patterson|first3=Brian M.|last4=Branch|first4=Brittany A.|last5=Kuettner|first5=Lindsey|date=2020-07-01|title=Shockwave dissipation by interface-dominated porous structures|journal=AIP Advances|volume=10|issue=7|pages=075016|doi=10.1063/5.0015179| bibcode=2020AIPA...10g5016D |doi-access=free}}</ref>

==Formal definition==
Formally, a Menger sponge can be defined as follows (using [[Intersection (set theory)|set intersection]]):

:<math>M := \bigcap_{n\in\mathbb{N}} M_n</math>

where <math>M_0</math> is the [[unit cube]] and

:<math>M_{n+1} := \left\{
(x,y,z)\in\mathbb{R}^3:
\left( \begin{array}{r}  \exists i,j,k\in\{0,1,2\} \,\, (3x-i,3y-j,3z-k)\in M_n
\\ \text{and at most one of }i,j,k\text{ is equal to 1}\end{array} \right) \right\}.</math>

==MegaMenger==
MegaMenger was a project aiming to build the largest fractal model, pioneered by [[Matt Parker]] of [[Queen Mary University of London]] and [[Laura Taalman]] of [[James Madison University]]. Each small cube is made from six interlocking folded business cards, giving a total of 960&thinsp;000 for a level-four sponge. The outer surfaces are then covered with paper or cardboard panels printed with a Sierpinski carpet design to be more [[Aesthetics|aesthetically]] pleasing.<ref>{{cite web|title=A Million Business Cards Present a Math Challenge|author=Tim Chartier| website=[[HuffPost]] | date=10 November 2014 |author-link=Tim Chartier|url=https://huffingtonpost.com/tim-chartier/a-million-business-cards_b_6128880.html|access-date=2015-04-07}}</ref> In 2014, twenty level-three Menger sponges were constructed, which combined would form a distributed level-four Menger sponge.<ref name="MegaMenger">{{cite web |url=http://www.megamenger.com/ |title=MegaMenger |access-date=2015-02-15}}</ref>

<gallery widths="200" heights="133">
Megamenger Bath.jpg|One of the MegaMengers, at the [[University of Bath]]
cmglee_Cambridge_Science_Festival_2015_Menger_sponge.jpg|A model of a [[Sierpinski tetrahedron|tetrix]] viewed through the centre of the Cambridge Level-3 MegaMenger at the 2015 [[Cambridge Science Festival]]
</gallery>

==Similar fractals==
<!--[[Jerusalem cube]] redirects directly here.-->

===Jerusalem cube===
{{anchor|Jerusalem Cube}}

A ''Jerusalem cube'' is a [[fractal]] object first described by Eric Baird in 2011. It is created by recursively drilling [[Greek cross]]-shaped holes into a cube.<ref>{{cite web|author=Robert Dickau |url=http://www.robertdickau.com/jerusalemcube.html |title=Cross Menger (Jerusalem) Cube Fractal |publisher=Robert Dickau |date=2014-08-31 |access-date=2017-05-08}}</ref><ref>{{cite web |author=Eric Baird |url=http://alt-fractals.blogspot.fr/2011/08/jerusalem-cube.html |title=The Jerusalem Cube |publisher=Alt.Fractals|date=2011-08-18 |access-date=2013-03-13}}, published in 
''Magazine Tangente'' 150, "l'art fractal" (2013), p. 45.</ref> The construction is similar to the Menger sponge but with two different-sized cubes. The name comes from the face of the cube resembling a [[Jerusalem cross]] pattern.<ref name="jerusalem_square">{{cite web |author=Eric Baird |url=http://alt-fractals.blogspot.com/2011/11/jerusalem-square.html |title=The Jerusalem Square |publisher=Alt.Fractals|date=2011-11-30 |access-date=2021-12-09}}</ref>

The construction of the Jerusalem cube can be described as follows:
# Start with a cube.
# Cut a cross through each side of the cube, leaving eight cubes (of rank +1) at the corners of the original cube, as well as twelve smaller cubes (of rank +2) centered on the edges of the original cube between cubes of rank +1.
# Repeat the process on the cubes of ranks 1 and 2.

Iterating an infinite number of times results in the Jerusalem cube.

Since the edge length of a cube of rank N is equal to that of 2 cubes of rank N+1 and a cube of rank N+2, it follows that the scaling factor must satisfy <math>k^2 + 2k = 1</math>, therefore <math>k = \sqrt{2} - 1</math> which means the fractal cannot be constructed using points on a [[Rational number|rational]] [[Lattice (group)|lattice]].

Since a cube of rank N gets subdivided into 8 cubes of rank N+1 and 12 of rank N+2, the Hausdorff dimension must therefore satisfy <math>8k^d + 12(k^2)^d = 1</math>. The exact solution is
:<math>d=\frac{\log\left(\frac{\sqrt{7}}{6}-\frac{1}{3}\right)}{\log\left(\sqrt{2}-1\right)}</math>
which is approximately 2.529

As with the Menger sponge, the faces of a Jerusalem cube are fractals<ref name="jerusalem_square"/> with the same scaling factor. In this case, the Hausdorff dimension must satisfy <math>4k^d + 4(k^2)^d = 1</math>. The exact solution is
:<math>d=\frac{\log\left(\frac{\sqrt{2}-1}{2}\right)}{\log\left(\sqrt{2}-1\right)}</math>
which is approximately 1.786

<gallery widths="200" heights="150">
Cube de Jérusalem, itération 3.png|Third iteration Jerusalem cube
Jerusalem_Cube.jpg|3D-printed model Jerusalem cube
</gallery>

===Others===
[[File:Sierpinskisnowflake.gif|thumb|Sierpinski–Menger snowflake]]

*A [[Mosely snowflake]] is a cube-based fractal with corners recursively removed.<ref>{{cite magazine|url=https://www.wired.com/2012/09/folded-fractal-art-cards|title=Folding Fractal Art from 49,000 Business Cards|first=Lizzie|last=Wade| magazine=Wired |access-date=8 May 2017}}</ref>
*A [[Sierpinski tetrahedron|tetrix]] is a tetrahedron-based fractal made from four smaller copies, arranged in a tetrahedron.<ref>{{cite web|url=http://mathworld.wolfram.com/Tetrix.html|title=Tetrix|first=Weisstein, Eric|last=W.|website=mathworld.wolfram.com|access-date=8 May 2017}}</ref>
*A Sierpinski–Menger snowflake is a cube-based fractal in which eight corner cubes and one central cube are kept each time at the lower and lower recursion steps. This peculiar three-dimensional fractal has the Hausdorff dimension of the natively two-dimensional object like the plane i.e. {{sfrac|log&nbsp;9|log&nbsp;3}}=2

==See also==
*[[Cantor cube]]
*[[Koch snowflake]]
*[[Sierpiński carpet]]
*[[Sierpinski tetrahedron|Sierpiński tetrahedron]]
*[[Sierpinski triangle|Sierpiński triangle]]
*[[List of fractals by Hausdorff dimension]]

==References==
{{reflist|35em}}

==Further reading==
*{{Citation |last1=Iwaniec |first1=Tadeusz |author1-link=Tadeusz Iwaniec |last2=Martin |first2=Gaven |author2-link=Gaven Martin| title=Geometric function theory and non-linear analysis |publisher=The Clarendon Press Oxford University Press |series=Oxford Mathematical Monographs |isbn=978-0-19-850929-5 |mr=1859913 |year=2001}}.
*{{citation|first=Li |last=Zhou |title=Problem 11208: Chromatic numbers of the Menger sponges |journal=[[American Mathematical Monthly]] |year=2007 |volume=114|number=9 |page=842|jstor=27642353}}

==External links==
{{Commons category|Menger sponges}}
*[http://mathworld.wolfram.com/MengerSponge.html Menger sponge at Wolfram MathWorld]
*[http://theiff.org/oexhibits/menger01.html The 'Business Card Menger Sponge' by Dr. Jeannine Mosely – an online exhibit about this giant origami fractal at the Institute For Figuring]
*[http://www.mathematik.com/Menger/Menger2.html An interactive Menger sponge]
*[https://web.archive.org/web/20051217041007/http://www.ibiblio.org/e-notes/3Dapp/Sponge.htm Interactive Java models]
*[http://santisan.free.fr/coco/extras2.htm Puzzle Hunt] — Video explaining Zeno's paradoxes using Menger–Sierpinski sponge
*[https://www.flickr.com/photos/fpsunflower/337024546/ Menger sphere], rendered in [[SunFlow]]
*[https://www.flickr.com/photos/rougeux/sets/72157621702780335/ Post-It Menger Sponge] &ndash; a level-3 Menger sponge being built from Post-its
*[https://www.nytimes.com/2011/06/28/science/28math-menger.html?_r=1 The Mystery of the Menger Sponge.] Sliced diagonally to reveal stars
*{{OEIS el|1=A212596|2=Number of cards required to build a Menger sponge of level n in origami}}
*[http://www.woollythoughts.com/menger.html Woolly Thoughts Level 2 Menger Sponge] by two "Mathekniticians"
*Dickau, R.: [http://www.robertdickau.com/jerusalemcube.html Jerusalem Cube] Further discussion.
*Miller, P.: [https://www.pure-mirage.com/Gallery/MengerSponges/Optimized%20Menger%20Sponges.htm Discussion of explicitly defined Menger sponges for stress testing in 3d display and rendering systems]

{{Fractals}}
{{Fractal software}}

[[Category:Iterated function system fractals]]
[[Category:Topological spaces]]
[[Category:Cubes]]
[[Category:Fractal curves]]
[[Category:Eponymous curves]]