The Sierpinski Curve

Right-click here to download this page and the Java Class File.
The Java applet above draws the Sierpinski curve of order n = 0 to 6.
Each order of the Sierpinski curve consists of 4 copies of the curve of the previous order replicated on a smaller scale and joined at the centre.

order	length of Sierpinski Curve
0	4 (sqrt(2)/4)	1.414
1	4 (3 x sqrt(2)/8) + 4 x 1/4	3.121
2	4 (11 x sqrt(2)/16) + 20 x 1/8	6.389
3	4 (43 x sqrt(2)/32) + 84 x 1/16	12.851
4	4 (171 x sqrt(2)/64) + 340 x 1/32	25.739
5	4 (683 x sqrt(2)/128) + 1364 x 1/64	51.497
6	4 (2731 x sqrt(2)/256) + 5460 x 1/128	103.003
:	:	:
n	4 (2²ⁿ⁺¹ + 1)sqrt(2)/2ⁿ⁺²3 + (2²ⁿ⁺² - 4)/2ⁿ⁺¹3	:

order	area of Sierpinski Curve
0	1/8	0.125
1	1/4 + 4(3/8)/16	0.344
2	5/16 + 4(11/8)/64	0.398
3	21/64 + 4(43/8)/256	0.412
4	85/256 + 4(171/8)/1024	0.4155
5	341/1024 + 4(683/8)/4096	0.4164
6	1365/4096 + 4(2731/8)/16384	0.4166
:	:	:
n	(4ⁿ - 1)/4ⁿ3 + (2²ⁿ⁺¹ + 1)/2²ⁿ⁺³3	:

The length of the Sierpinski curve of order n approaches infinity as n approaches infinity, while the area approaches the value 5/12 = 0.41666... .
As n approaches infinity, we obtain the Sierpinski space-filling curve, a continuous curve passing through every point of the enclosing square.