The Sierpinski Sponge |

The Sierpinski Sponge is a fractal. Small parts look like big parts, that look like even bigger parts.

It is build by repeatedly glueing 20 smaller Sierpinski Sponges together like this:

Written properly, in the mathematical way, the Sierpinski Sponge can be defined like this:

sierpinski 0
= cube where "cube" has the corners (±1,±1,±1) sierpinski n = (move (+2/3,+2/3,+2/3) )º(mul 1/3) (sierpinski n-1) ∪ (move ( 0 ,+2/3,+2/3) )º(mul 1/3) (sierpinski n-1) ∪ (move (-2/3,+2/3,+2/3) )º(mul 1/3) (sierpinski n-1) ∪ (move (+2/3, 0 ,+2/3) )º(mul 1/3) (sierpinski n-1) ∪ (move (-2/3, 0 ,+2/3) )º(mul 1/3) (sierpinski n-1) ∪ (move (+2/3,-2/3,+2/3) )º(mul 1/3) (sierpinski n-1) ∪ (move ( 0 ,-2/3,+2/3) )º(mul 1/3) (sierpinski n-1) ∪ (move (-2/3,-2/3,+2/3) )º(mul 1/3) (sierpinski n-1) ∪ (move (+2/3,+2/3, 0 ) )º(mul 1/3) (sierpinski n-1) ∪ (move (-2/3,+2/3, 0 ) )º(mul 1/3) (sierpinski n-1) ∪ (move (+2/3,-2/3, 0 ) )º(mul 1/3) (sierpinski n-1) ∪ (move (-2/3,-2/3, 0 ) )º(mul 1/3) (sierpinski n-1) ∪ (move (+2/3,+2/3,-2/3) )º(mul 1/3) (sierpinski n-1) ∪ (move ( 0 ,+2/3,-2/3) )º(mul 1/3) (sierpinski n-1) ∪ (move (-2/3,+2/3,-2/3) )º(mul 1/3) (sierpinski n-1) ∪ (move (+2/3, 0 ,-2/3) )º(mul 1/3) (sierpinski n-1) ∪ (move (-2/3, 0 ,-2/3) )º(mul 1/3) (sierpinski n-1) ∪ (move (+2/3,-2/3,-2/3) )º(mul 1/3) (sierpinski n-1) ∪ (move ( 0 ,-2/3,-2/3) )º(mul 1/3) (sierpinski n-1) ∪ (move (-2/3,-2/3,-2/3) )º(mul 1/3) (sierpinski n-1) |

When you say *the* Sierpinski Sponge, you are really refering to "sierpinski ∞". Unfortunately it is not really possible to draw this, but "sierpinski 4" gives a good impression of what it must look like.

There is a 2-dimensional version of the Sierpinski Sponge, called the Sierpinski Carpet.

The Sierpinski Carpet is the "footprint" that the Sierpinski Sponge makes. It can be constructed by repeatedly glueing 8 smaller Sierpinski Carpets together.

It is possible to calculate the area (A) and perimeter (P) of the Sierpinski Carpet.

A 0 = 2^{2} = 4

A n = 8·(1/3)^{2}·(A n-1) = (8/9)·(A n-1)

It says that "sierpcarpet 0" has area 4, and that the area of "sierpcarpet n" (A n) is the same as all the (8) small ((1/3)^{2}) Sierpinski Carpets area (A n-1). From this you can derive that:

A n = (8/9)^{n}·4

The perimeter is found in a similar way:

P 0 = 4·2 = 8

P n = 8·(1/3)·(P n-1) - 8·(1/3)·2·2

= (8/3)·(P n-1) - (32/3)

The perimeter (P n) is the same as all the (8) small (1/3) Sierpinski Carpets perimeters (P n-1), apart from the length of all the (8) small (1/3) sides (2) where they meet, and they are two (2) sides meeting. From this you can likewise derive that:

P n = (8/5)·(8/3)^{n} + (32/5)

You can calculate the volume (V) and area (A) for the Sierpinski Sponge. It is done in the same way as for the Sierpinski Carpet. I omit comments, and write only the hard facts...

V 0 = 2^{3} = 8

V n = 20·(1/3)^{3}·(V n-1)

which gives

V n = (20/27)^{n}·8

and

A 0 = 6·2^{2} = 24

A n = 20·(1/3)^{2}·(A n-1) - 24·(1/3)^{2}·(A_{t} n-1)·2

= (20/9)·(A n-1) - (64/3)·(8/9)^{n-1}

(A_{t} is the Sierpinski Carpet's area)

which gives

A n = (8/9)^{n}·16 + (20/9)^{n}·8

What influence does this have for *the* Sierpinski Sponge, "sierpinski ∞"? Well, as n → ∞, so will V n → 0, while A n → ∞. Stated a bit simplistically "sierpinski ∞" doesn't have any volume, but has an infinite surface area!

The part you have to cut out of a solid cube to get a Sierpinski Sponge, looks like this:

While the Sierpinski Sponge has a volume of 0, the Sierpinski Luft has a volume of 8 (when n → ∞). The surface area of the Sierpinski Luft → ∞, just like that of the Sierpinski Sponge.

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last changed 27.mar.2005 © 1997-2005 Bjørn Hee, mailto:webmaster@h33.dk |