Menger sponge

An illustration of M₄, the sponge after four iterations of the construction process.

In mathematics, the Menger sponge (also known as the Menger universal curve) is a fractal curve. It is a three-dimensional generalization of the Cantor set and Sierpinski carpet. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension.

The Menger sponge simultaneously exhibits an infinite surface area and zero volume.

Construction

A sculptural representation of iterations 0 (bottom) to 3 (top).

The construction of a Menger sponge can be described as follows:

1. Begin with a cube (first image).
2. Divide every face of the cube into 9 squares, like a Rubik's Cube. This will sub-divide the cube into 27 smaller cubes.
3. Remove the smaller cube in the middle of each face, and remove the smaller cube in the very center of the larger cube, leaving 20 smaller cubes (second image). This is a level-1 Menger sponge (resembling a Void Cube).
4. Repeat steps 2 and 3 for each of the remaining smaller cubes, and continue to iterate ad infinitum.

The second iteration will give you a level-2 sponge (third image), the third iteration gives 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.

An illustration of the iterative construction of a Menger sponge up to M₃, the third iteration.

Menger sponge animation through (4) recursion steps.

The following table lists the properties of sponges M_n where n is the number of iterations performed on the first cube of unit side length.

n	Number of cubes N = 20n	Side of a cube, s = (1/3)n	Relative sizes of s	Total volume, V = Ns3 = (20/27)n	Relative sizes of V	Surface area, A = 2(20/9)n + 4(8/9)n	Relative sizes of A
0	1	1 (exact)	1	1 (exact)	1	6 (exact)	6
1	20	0.3333333333	0.3333333333	0.74074	0.74074	8 (exact)	8
2	400	0.1111111111	0.1111111111	0.54870	0.5487	13.04	13.04
3	8,000	0.0370370370	0.037037037	0.40644	0.40644	24.76	24.76
4	1.6×105	0.0123456790	0.012345679	0.30107	0.30107	51.27	51.27
5	3.2×106	0.0041152263	0.0041152263	0.22301	0.22301	110.60	110.6
6	6.4×107	0.0013717421	0.0013717421	0.16520	0.1652	242.83	242.83
7	1.28×109	0.0004572474	0.0004572474	0.12237	0.12237	536.99	536.99
8	2.56×1010	0.0001524158	0.0001524158	0.09064	0.09064	1,190.96	1190.96
9	5.12×1011	0.0000508053	5.08053E-5	0.06714	0.06714	2,644.51	2644.51
10	1.024×1013	0.0000169351	1.69351E-5	0.04974	0.04974	5,874.84	5874.84
11	2.048×1014	0.0000056450	5.645E-6	0.03684	0.03684	13,053.55	13053.55
12	4.096×1015	0.0000018817	1.8817E-6	0.02729	0.02729	29,006.44	29006.44
13	8.192×1016	0.0000006272	6.272E-7	0.02021	0.02021	64,457.45	64457.45
14	1.6384×1018	0.0000002091	2.091E-7	0.01497	0.01497	143,237.63	143237.63
15	3.2768×1019	0.0000000697	6.97E-8	0.01109	0.01109	318,304.82	318304.82
16	6.5536×1020	0.0000000232	2.32E-8	0.00822	0.00822	707,343.14	707343.14
17	1.31072×1022	0.0000000077	7.7E-9	0.00609	0.00609	1,571,872.84	1571872.84
18	2.62144×1023	0.0000000026	2.6E-9	0.00451	0.00451	3,493,050.04	3493050.04
19	5.24288×1024	0.0000000009	9.0E-10	0.00334	0.00334	7,762,332.79	7762332.79
20	1.048576×1026	0.0000000003	3.0E-10	0.00247	0.00247	17,249,627.84	17249627.84

Properties

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The cross-section of a Menger sponge through its centroid and perpendicular to a space diagonal has sixfold symmetry containing hexagrams.

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 M₀ 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. It has Lebesgue measure 0. It is an uncountable set.

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 universal curve, in that every curve is homeomorphic to a subset of the Menger sponge, where a curve means any compact metric space of Lebesgue covering dimension one; this includes trees and 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 planar, and might be embedded in any number of dimensions.

The Menger sponge has infinite surface area but zero volume.

The sponge has a Hausdorff dimension of log 20/log 3 (approximately 2.726833).

Formal definition

Formally, a Menger sponge can be defined as follows:

$M := \bigcap_{n\in\mathbb{N}} M_n$

where M₀ is the unit cube and

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

MegaMenger

A model of a tetrix viewed through the centre of the Cambridge Level-3 MegaMenger at the 2015 Cambridge Science Festival

One of the MegaMengers, at the University of Bath

MegaMenger is 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 6 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.

Similar fractals

Sierpinski-Menger snowflake. Eight corner cubes and the 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. log 9/log 3=2

• A Jerusalem Cube is a cube-based fractal with a Greek cross recursively removed from each face.
• A Mosely snowflake is a cube-based fractal with corners recursively removed.
• A tetrix is a tetrahedron-based fractal with tetrahedrons in its middle recursively removed.

See also

• Apollonian gasket
• Cantor cube
• Koch snowflake
• List of fractals by Hausdorff dimension
• Sierpiński tetrahedron
• Sierpiński triangle
• Sierpiński sponge

Notes and references

• .

External links

• Menger sponge at Wolfram MathWorld
• The 'Business Card Menger Sponge' by Dr. Jeannine Mosely – an online exhibit about this giant origami fractal at the Institute For Figuring
• An interactive Menger sponge
• Interactive Java models
• Puzzle Hunt — Video explaining Zeno's paradoxes using Menger–Sierpinski sponge
• Menger Sponge Animations — Menger sponge animations up to level 9, discussion of optimization for 3d.
• Menger sphere, rendered in SunFlow
• Post-It Menger Sponge – a level-3 Menger sponge being built from Post-its
• The Mystery of the Menger Sponge. Sliced diagonally to reveal stars
• Number of cards required to build a Menger sponge of level n in origami
• Woolly Thoughts Level 2 Menger Sponge by two "Mathekniticians"