1

Determine the fractal dimension of the fractal generated by successively removing the open middle one-thirds of the closed intervals at the previous level as indicated by the cell below.

Input := 

Clear[level]
level[0]=Show[
	Graphics[Line[{{0,0},{1,0}}]],
	Epilog->{
		{PointSize[.025],Red,Point[{0,0}]},
		{PointSize[.025],Red,Point[{1,0}]},
		{Text["0",{0,-.1}]},	
		{Text["1",{1,-.1}]}
		},	
	DisplayFunction->Identity];
level[1]=Show[
	Graphics[{Line[{{0,0},{1/3,0}}],
		Line[{{2/3,0},{1,0}}]}],
	Epilog->{
		{PointSize[.025],Red,Point[{0,0}]},
		{PointSize[.025],Red,Point[{1/3,0}]},
		{PointSize[.025],Red,Point[{2/3,0}]},
		{PointSize[.025],Red,Point[{1,0}]},
		{Text["0",{0,-.1}]},	
		{Text["1/3",{1/3,-.1}]},
		{Text["2/3",{2/3,-.1}]},	
		{Text["1",{1,-.1}]}
		},	
	DisplayFunction->Identity];
level[2]=Show[
	Graphics[{
		Line[{{0,0},{1/9,0}}],
		Line[{{2/9,0},{1/3,0}}],
		Line[{{2/3,0},{7/9,0}}],
		Line[{{8/9,0},{1,0}}]}],
	Epilog->{
		{PointSize[.025],Red,Point[{0,0}]},
		{PointSize[.025],Red,Point[{1/9,0}]},
		{PointSize[.025],Red,Point[{2/9,0}]},
		{PointSize[.025],Red,Point[{1/3,0}]},
		{PointSize[.025],Red,Point[{2/3,0}]},
		{PointSize[.025],Red,Point[{7/9,0}]},
		{PointSize[.025],Red,Point[{8/9,0}]},
		{PointSize[.025],Red,Point[{1,0}]},
		{Text["0",{0,-.1}]},	
		{Text["1/9",{1/9,.1}]},
		{Text["2/9",{2/9,-.1}]},
		{Text["1/3",{1/3,.1}]},
		{Text["2/3",{2/3,-.1}]},	
		{Text["7/9",{7/9,.1}]},	
		{Text["8/9",{8/9,-.1}]},
		{Text["1",{1,.1}]}
		},	
	DisplayFunction->Identity];
Show[GraphicsArray[{{level[0]},{level[1]},
	{level[2]}}]]

Output =

-GraphicsArray-

Answer and Discussion