In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpiński cube, or Sierpiński sponge) is a fractal curve. It is a three-dimensional generalization of the two-dimensional Sierpinski carpet. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension.
It has similar properties as the Cantor set and the Cantor dust, because the construction requires in both cases the removal of the inner third.
Contents
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 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
The
n
{\displaystyle n}
th stage of the Menger sponge,
M
n
{\displaystyle M_{n}}
, is made up of
20
n
{\displaystyle 20^{n}}
smaller cubes, each with a side length of (1/3)n. The total volume of
M
n
{\displaystyle M_{n}}
is thus
(
20
27
)
n
{\textstyle \left({\frac {20}{27}}\right)^{n}}
. The total surface area of
M
n
{\displaystyle M_{n}}
is given by the expression
2
(
20
/
Formal definition
Formally, a Menger sponge can be defined as follows (using set intersection):
M
:=
⋂
n
∈
N
M
n
{\displaystyle M:=\bigcap _{n\in \mathbb {N} }M_{n}}
where
M
0
{\displaystyle M_{0}}
is the unit cube and
M
n
+
1
:=
{
(
x
,
y
,
z
)
∈
R
3
:
(
∃
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 960,000 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
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. 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.
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
+
2
k
=
1
{\displaystyle k^{2}+2k=1}
, therefore
k
=
2
−
1
{\displaystyle k={\sqrt {2}}-1}
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. log 9/log 3=2