Sierpinski Curve

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The Sierpinski curve was discovered by the Polish Mathematician Waclaw Sierpinski.
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 ((√2)/4)	1.414
1	4 (3(√2)/8) + 4 x 1/4	3.121
2	4 (11(√2)/16) + 20 x 1/8	6.389
3	4 (43(√2)/32) + 84 x 1/16	12.851
4	4 (171(√2)/64) + 340 x 1/32	25.739
:	:	:
n	(2ⁿ⁺¹ + 1/2ⁿ)(√2)/3 + (2ⁿ⁺¹ - 1/2^n-1)/3	:

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	0.416
:	:	:
n	(4ⁿ - 1)/4ⁿ3 + (2²ⁿ⁺¹ + 1)/2²ⁿ⁺³3 = (1 - 1/4ⁿ)/3 + (1/4 + 1/2²ⁿ⁺³)/3	:

The length of the Sierpinski curve of order n approaches infinity as n approaches infinity, while the area approaches the value 5/12 = 0.4167 .
As n approaches infinity, we obtain the Sierpinski space-filling curve, a continuous curve passing through every point of the enclosing square.