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In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a
fractal curve A fractal curve is, loosely, a mathematical curve whose shape retains the same general pattern of irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fractal. In general, fractal curves are nowhere recti ...
. It is a three-dimensional generalization of the one-dimensional
Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. Th ...
and two-dimensional Sierpinski carpet. It was first described by
Karl Menger Karl Menger (January 13, 1902 – October 5, 1985) was an Austrian-American mathematician, the son of the economist Carl Menger. In mathematics, Menger studied the theory of algebras and the dimension theory of low- regularity ("rough") curve ...
in 1926, in his studies of the concept of
topological dimension In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically invariant way. Informal discussion For ordinary Euclidean ...
.


Construction

The construction of a Menger sponge can be described as follows: # Begin with a cube. # Divide every face of the cube into nine squares, like
Rubik's Cube The Rubik's Cube is a 3-D combination puzzle originally invented in 1974 by Hungarian sculptor and professor of architecture Ernő Rubik. Originally called the Magic Cube, the puzzle was licensed by Rubik to be sold by Pentangle Puzzles i ...
. 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 more giant cube, leaving 20 smaller cubes. This is a level-1 Menger sponge (resembling a
void cube The Void Cube is a 3-D mechanical puzzle similar to a Rubik's Cube, with the notable difference being that the center pieces are missing, which causes the puzzle to resemble a level 1 Menger sponge. The core used on the Rubik's Cube is also abse ...
). # Repeat steps two and three for each of the remaining smaller cubes, and continue to iterate ''
ad infinitum ''Ad infinitum'' is a Latin phrase meaning "to infinity" or "forevermore". Description In context, it usually means "continue forever, without limit" and this can be used to describe a non-terminating process, a non-terminating ''repeating'' ...
''. 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

The nth stage of the Menger sponge, M_n, is made up of 20^n smaller cubes, each with a side length of (1/3)''n''. The total volume of M_n is thus \left(\frac\right)^n. The total surface area of M_n is given by the expression 2(20/9)^n + 4(8/9)^n. 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 In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. Th ...
. The cross-section of the sponge through its
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...
and perpendicular to a
space diagonal In geometry, a space diagonal (also interior diagonal or body diagonal) of a polyhedron is a line connecting two vertices that are not on the same face. Space diagonals contrast with ''face diagonals'', which connect vertices on the same face (but ...
is a regular hexagon punctured with
hexagram , can be seen as a compound composed of an upwards (blue here) and downwards (pink) facing equilateral triangle, with their intersection as a regular hexagon (in green). A hexagram ( Greek language, Greek) or sexagram (Latin) is a six-pointed ...
s arranged in six-fold symmetry. The number of these hexagrams, in descending size, is given by a_n=9a_-12a_, with a_0=1, \ a_1=6. The sponge's
Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a ...
is ≅ 2.727. The Lebesgue covering dimension of the Menger sponge is one, the same as any
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
. Menger showed, in the 1926 construction, that the sponge is a '' universal curve'', in that every curve is
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorp ...
to a subset of the Menger sponge, where a ''curve'' means any
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
of Lebesgue covering dimension one; this includes
trees In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that ar ...
and
graphs Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties * Graph (topology), a topological space resembling a graph in the sense of discr ...
with an arbitrary
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
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 and might be embedded in any number of dimensions. The Menger sponge is a
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
; since it is also bounded, the Heine– Borel theorem implies that it is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
. It has
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides ...
 0. Because it contains continuous paths, it is an
uncountable set In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
. Experiments also showed that cubes with a Menger sponge structure could dissipate shocks five times better for the same material than cubes without any pores.


Formal definition

Formally, a Menger sponge can be defined as follows: :M := \bigcap_ M_n where M_0 is the unit cube and :M_ := \left\.


MegaMenger

MegaMenger was a project aiming to build the largest fractal model, pioneered by Matt Parker of
Queen Mary University of London Queen Mary University of London (QMUL, or informally QM, and previously Queen Mary and Westfield College) is a public university, public research university in Mile End, East London, England. It is a member institution of the federal University of ...
and Laura Taalman of
James Madison University James Madison University (JMU, Madison, or James Madison) is a public research university in Harrisonburg, Virginia. Founded in 1908 as the State Normal and Industrial School for Women at Harrisonburg, the institution was renamed Madison Col ...
. Each small cube is made from six interlocking folded business cards, giving a total of 960 000 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 aesthetically pleasing. In 2014, twenty level-three Menger sponges were constructed, which combined would form a distributed level-four Menger sponge. Megamenger Bath.jpg, One of the MegaMengers, at the
University of Bath (Virgil, Georgics II) , mottoeng = Learn the culture proper to each after its kind , established = 1886 (Merchant Venturers Technical College) 1960 (Bristol College of Science and Technology) 1966 (Bath University of Technology) 1971 (univ ...
cmglee_Cambridge_Science_Festival_2015_Menger_sponge.jpg, A model of a tetrix viewed through the centre of the Cambridge Level-3 MegaMenger at the 2015 Cambridge Science Festival


Similar fractals


Jerusalem cube

A ''Jerusalem cube'' is a
fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as il ...
object described by Eric Baird in 2011. It is created by recursively drilling
Greek cross The Christian cross, with or without a figure of Christ included, is the main religious symbol of Christianity. A cross with a figure of Christ affixed to it is termed a ''crucifix'' and the figure is often referred to as the ''corpus'' ( ...
-shaped holes into a cube. 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 The Jerusalem cross (also known as "five-fold Cross", or "cross-and-crosslets") is a heraldic cross and Christian cross variant consisting of a large cross potent surrounded by four smaller Greek crosses, one in each quadrant. It was used as t ...
pattern. 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 k^2 + 2k = 1, therefore k = \sqrt - 1 which means the fractal cannot be constructed on a rational grid. 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 8k^d + 12(k^2)^d = 1. The exact solution is :d=\frac which is approximately 2.529 As with the Menger sponge, the faces of a Jerusalem cube are fractals with the same scaling factor. In this case, the Hausdorff dimension must satisfy 4k^d + 4(k^2)^d = 1. The exact solution is :d=\frac which is approximately 1.786 Cube de Jérusalem, itération 3.png, Third iteration Jerusalem cube Jerusalem_Cube.jpg, 3D-printed model Jerusalem cube


Others

*A Mosely snowflake is a cube-based fractal with corners recursively removed. *A tetrix is a tetrahedron-based fractal made from four smaller copies, arranged in a tetrahedron. *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. =2


See also

*
Apollonian gasket In mathematics, an Apollonian gasket or Apollonian net is a fractal generated by starting with a triple of circles, each tangent to the other two, and successively filling in more circles, each tangent to another three. It is named after Greek ma ...
*
Cantor cube In mathematics, a Cantor cube is a topological group of the form ''A'' for some index set ''A''. Its algebraic and topological structures are the group direct product and product topology over the cyclic group of order 2 (which is itself given th ...
*
Koch snowflake The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curv ...
* Sierpiński tetrahedron *
Sierpiński triangle The Sierpiński triangle (sometimes spelled ''Sierpinski''), also called the Sierpiński gasket or Sierpiński sieve, is a fractal attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equi ...
*
List of fractals by Hausdorff dimension According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff dimension, Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." Presented here is a list of fractals, ordered by increasing Hausdorff ...


References


Further reading

*. *


External links


Menger sponge at Wolfram MathWorld
– an online exhibit about this giant origami fractal at the Institute For Figuring]
An interactive Menger sponge
— Video explaining Zeno's paradoxes using Menger–Sierpinski sponge
Menger sphere
rendered in SunFlow
Post-It Menger Sponge
– a level-3 Menger sponge being built from Post-its

Sliced diagonally to reveal stars *

by two "Mathekniticians" *Dickau, R.

Further discussion. {{Fractals, state=expanded Iterated function system fractals Curves Topological spaces Cubes Fractals