Approx. Theory ~ its Appl. 16:2, 2000, 45- -51

HAUSDORFF MEASURE OF FRACTAL--

SIERPINSKI CARPET

Chen Fangyue ~

(Zhefiang Normal Uniersity , Peking University, China)

and

Xie Tingfan

(China Institute o f Meterogy , China)

Received Mar. 15,2000

Abstract

In this paper we present a )~eneral conclusion of looking for the exact value o f Hausdorf f measure of

Sierpinski carpet and construct a special partial cover o f the carpet. And then we obtained an upper bound

of the value, which is the least one as we know. A conjecture for the measure is proposed at last.

1. Introuduction

Sierpinski carpet, one of the most well-known self-similar set, is a typical fractal, its

Hausdorff dimension is easlily calculable, but its Hausdorff measure, like many other frac-

tals, is very difficult to be computed. No satisfying result has been obtained up to now.

In this paper we present a general conclusion of looking for the exact value of the mea-

sure, and constuet a special partial cover of the set. Then we obtain an upper bound of the

value9 which is near 1. It is the least one as far as we know. At last, a conjecture about

the measure is proposed.

2 Construction and Lemmas

Visting scholar of Peking University

46 Approx. Theory ~ its Appl. 16: 2, 2000

The construction of the classical Sierpinski carpet which was raised by Sierpinski in

1916 m is as follows :

Take a unit square So in Eculid plane W and divide it into 9 little squares equally. Re-

move the interior of the middle one, the remaining set is denoted by $1. Do the same pro-

cedure for every little square in S,, and get a set Sz, '" . Repeat the procedure infinitely,

we get a squence of sets (as Fig. 1)

So ~ Sl ~ S~ D '" D S~...

So

~ "~

S, $2

Fig 1

It can be easily seen that S, consistes of 8 + squares ,each denoted by ["73'. We call [-]3*

k order fundamental square. The edge length of [--13' is 1 . 6- Let

then S is called Sierpinski carpet ~].

S = ~S, , J--1 It is clear that S can be formed by 8 + little Sierpinski carpets, each is similar to S with

s -~-SI --~-S : S N similaity ratio 3"" The little SJerpinski carpet which is denoted by satisfies r-]~ k (k= 1,2, . . . ) . We can know that S is a self-similar set which satisfies open set condi-

tions [2]. Thus, the Hausdorff dimension of S in [ I ] .

and

s = dimu(S) = 1og38,

where H' (S) ( * ) denotes the dimension of Hausdorff measure. In this note we always as-

sume H'(S):yaO and H' (S)~ +oo.

Lemma 1. For every ~>0, we have

H' (S) = H's(S).

Chert Fangyue et al : Hausdorff Measure of Fractal 47

proof. Acoording to the definition of Hausdorff measure [23 ,for any r there is a 5 -`

covering a = {Ui}~+7 of S, such that

H$(S) ~ ~,IU, I'~ 1 - ~. (2)

Otherwise

H'(S n U,) iU~[, < 1 - e, i = 1,2...,N,

48 Approx. Theory ~ its Appl. 16:2, 2000

then we have

N

~,H'

Chen Fangyue et al ~ Hausdorff Measure of Fractal 49

are neibouring with each other (as Fig 1).

Let the center of G be the coordlinate origin and take a closed-real disk G" in G (as

Fig 2)

U" = { (x,y) ]xZ + yz

50 Approx. Theory ~ its Appl. 16: 2, 2000

for k=5,Ms=376,H ' (S)

Chen Fangyue et al, Hausdorff Measure of Fractal 51

References

[1"] Mandelbrot, B. B. , The Fractal Geometry of Nature, W.H. Freeman and Company, New

York(1983).

['2] Faloner, K. J . , Fractal Geometry-Mathematical Foundations and Application, Baffines Lane,

Chiehester ~ John Weiley and Sone Ltd(1990).

[3] Zhou Z. L. , The Hausdorff Measures of the self-similr sets-the Koch Curve, Science in Chi-

na. , 28(1998) No. 2, 103--107.

Chen Fangyue

Department of Mathematics

Zhejiang Normal University

Jinhua 321004

PRC

Xie Tingfan ~

China Institute of Meterogy

Hangzhou, 310034

PRC