\documentclass[12pt]{article}
%\usepackage {times}
%\usepackage {bm}
\usepackage{tikz}
\usepackage{pgf}
\usetikzlibrary{snakes}
%\usetikzlibrary{patterns}
\usepackage[active,pdftex,tightpage]{preview}
\PreviewEnvironment[]{tikzpicture}
%\usepackage{comment}
%\usepackage[active]{srcltx}
%\usepackage{paralist}

%\pagestyle{plain}
\begin{document}
        \def\s{0.028}
        \def\sh{1}
	\def\stwo{1.85}
	\def\sthree{3.11}
	\def\top{2.5}
        \begin{tikzpicture}[scale=4,>=stealth]
	  \begin{scope}
	    \draw[snake=coil,segment length=2mm,segment amplitude=.4mm,segment aspect=0.5,thick] (-0.05,-\top+1) -- (1.05,-\top+1);%(0,0) ellipse (1 and 2);
	    \draw[snake=coil,segment length=2mm,segment amplitude=.4mm,segment aspect=0.5,thick]  (1.05,\top) -- (-.05,\top);%(0,0) ellipse (1 and 2);
	    \clip[snake=coil,segment length=2mm,segment amplitude=.4mm,segment aspect=0.5] (-0.05,-\top+1) -- (1.05,-\top+1) -- (1.05,\top) -- (-0.05,\top) -- cycle;%(0,0) ellipse (1 and 2);
	
	%1
		\foreach \x in {0,7,8}		{
			\foreach \y in {0,2,7,8}		{
				\ifnum \y>\x
					\draw[fill=gray] ({1+(\x-\y)*1/9},{3*\sh-1+(\x+\y)*1/9}) -- +(1/9,1/9) -- +(0,2/9);
				\else
%					\draw[fill=gray] ({0+(\x-\y)*1/9},{2*\sh-1+(\x+\y)*1/9}) -- +(1/9,1/9) -- +(0,2/9);
				\fi
				\ifnum \x>\y
%					\draw[fill=gray] ({0+(\x-\y)*1/9},{2*\sh-1+(\x+\y)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
				\else
					\draw[fill=gray] ({1+(\x-\y)*1/9},{3*\sh-1+(\x+\y)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
				\fi
			}
		}
	%2	
		\foreach \x in {0,1,2,7,8}		{
			\foreach \y in {0,1,2,4,5}		{
				\ifnum \y>\x
				\else
					\draw[fill=gray] ({0+(\x-\y)*1/9},{2*\sh-1+(\x+\y)*1/9}) -- +(1/9,1/9) -- +(0,2/9);
				\fi
				\ifnum \x>\y
					\draw[fill=gray] ({0+(\x-\y)*1/9},{2*\sh-1+(\x+\y)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
				\fi

			}
		}
	%3
		\foreach \x in {0,1,2,7,8}		{
			\foreach \y in {0,1,2,4,5}		{
				\ifnum \y>\x
					\draw[fill=gray] ({1+(\x-\y)*1/9},{\sh-1+(\x+\y)*1/9}) -- +(1/9,1/9) -- +(0,2/9);
				\else
%					\draw[fill=gray] ({0+(\x-\y)*1/9},{-1+(\x+\y)*1/9}) -- +(1/9,1/9) -- +(0,2/9);
				\fi
				\ifnum \x>\y
%					\draw[fill=gray] ({0+(\x-\y)*1/9},{-1+(\x+\y)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
				\else
					\draw[fill=gray] ({1+(\x-\y)*1/9},{\sh-1+(\x+\y)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
				\fi
			}
		}
	%4
		\foreach \x in {0,2,3,4,5}		{
			\foreach \y in {0,1,2,4,5}		{
				\ifnum \y>\x
%					\draw[fill=gray] ({1+(\x-\y)*1/9},{\sh-1+(\x+\y)*1/9}) -- +(1/9,1/9) -- +(0,2/9);
				\else
					\draw[fill=gray] ({0+(\x-\y)*1/9},{-1+(\x+\y)*1/9}) -- +(1/9,1/9) -- +(0,2/9);
				\fi
				\ifnum \x>\y
					\draw[fill=gray] ({0+(\x-\y)*1/9},{-1+(\x+\y)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
				\else
%					\draw[fill=gray] ({1+(\x-\y)*1/9},{\sh-1+(\x+\y)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
				\fi
			}
		}
	%5
		\foreach \x in {0,2,3,4,5}		{
			\foreach \y in {3,5,7,8}		{
				\ifnum \y>\x
					\draw[fill=gray] ({1+(\x-\y)*1/9},{-\sh-1+(\x+\y)*1/9}) -- +(1/9,1/9) -- +(0,2/9);
				\fi
				\ifnum \x>\y
				\else
					\draw[fill=gray] ({1+(\x-\y)*1/9},{-\sh-1+(\x+\y)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
				\fi
			}
		}
	%6
		\foreach \x in {0,1,8}		{
			\foreach \y in {3,5,7,8}		{
				\ifnum \y>\x
%					\draw[fill=gray] ({1+(\x-\y)*1/9},{-\sh-1+(\x+\y)*1/9}) -- +(1/9,1/9) -- +(0,2/9);
				\else
					\draw[fill=gray] ({0+(\x-\y)*1/9},{-2*\sh-1+(\x+\y)*1/9}) -- +(1/9,1/9) -- +(0,2/9);
				\fi
				\ifnum \x>\y
					\draw[fill=gray] ({0+(\x-\y)*1/9},{-2*\sh-1+(\x+\y)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
				\else
%					\draw[fill=gray] ({1+(\x-\y)*1/9},{-\sh-1+(\x+\y)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
				\fi
			}
		}
                
		\draw (0,-2.1) -- (0,2.1+\sh);
		\draw[thick] (1/3,-2.1) -- (1/3,2.1+\sh);
		\draw (2/3,-2.1) -- (2/3,2.1+\sh);
		\draw (1,-2.1) -- (1,2.1+\sh);

		\draw[thin,dashed] (1/9,-1.95) -- (1/9,1.95+\sh);
		\draw[thin,dashed] (2/9,-1.95) -- (2/9,1.95+\sh);
		\draw[thick,dashed] (4/9,-1.95) -- (4/9,1.95+\sh);
		\draw[thin,dashed] (5/9,-1.95) -- (5/9,1.95+\sh);
		\draw[thin,dashed] (7/9,-1.95) -- (7/9,1.95+\sh);
		\draw[thin,dashed] (8/9,-1.95) -- (8/9,1.95+\sh);

		\draw[ultra thick] (0,-1) -- (1,0) -- (0,1) -- cycle;
                \draw[ultra thick] (0,0+\sh) -- (1,-1+\sh) -- (1,1+\sh) -- cycle;
		\draw (0,-1+2*\sh) -- (1,0+2*\sh) -- (0,1+2*\sh) -- cycle;
                \draw (0,0-\sh) -- (1,-1-\sh) -- (1,1-\sh) -- cycle;
	\end{scope}
		\draw[thick,<-] (7/18,\top +0.08 ) -- (7/18,\top+0.17);
		\draw (7/18,\top+0.25) node {$C(i_1,\cdots,i_{k\cdot N-1})$};
		\draw[snake=brace,very thick] (6/18,\top+0.03) -- (8/18,\top+0.03);
		\draw (-0.03,0) node[left] {\LARGE${R_j^{(k-1)}}$};
		\draw (1.03,0+\sh) node[right] {\LARGE${L_j^{(k-1)}}$};
		\draw[snake=brace,very thick] (1,-\top+1-0.1) -- (0,-\top+1-0.1);
		\draw (1/2,-\top+1-0.2) node {$C(i_1,\cdots,i_{(k-1)\cdot N})$};

	\begin{scope}[xshift=\stwo{}cm]

		  \begin{scope}
	   	 \draw[snake=coil,segment length=2mm,segment amplitude=.4mm,segment aspect=0.5,thick] (-0.05,-\top+1) -- (1.05,-\top+1);%(0,0) ellipse (1 and 2);
	   	 \draw[snake=coil,segment length=2mm,segment amplitude=.4mm,segment aspect=0.5,thick]  (1.05,\top) -- (-.05,\top);%(0,0) ellipse (1 and 2);
	  	  \clip[snake=coil,segment length=2mm,segment amplitude=.4mm,segment aspect=0.5] (-0.05,-\top+1) -- (1.05,-\top+1) -- (1.05,\top) -- (-0.05,\top) -- cycle;%(0,0) ellipse (1 and 2);
		
	%1
			\foreach \x in {0,7,8}		{
				\foreach \y in {0,2,7,8}		{
					\ifnum \y>\x
						\draw[fill=black!25] ({1+(\x-\y)*1/9},{3*\sh-1+(\x+\y)*1/9}) -- +(1/9,1/9) -- +(0,2/9);
					\else
	%					\draw[fill=black!25] ({0+(\x-\y)*1/9},{2*\sh-1+(\x+\y)*1/9}) -- +(1/9,1/9) -- +(0,2/9);
					\fi
					\ifnum \x>\y
	%					\draw[fill=black!25] ({0+(\x-\y)*1/9},{2*\sh-1+(\x+\y)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
					\else
						\draw[fill=black!25] ({1+(\x-\y)*1/9},{3*\sh-1+(\x+\y)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
					\fi
				}
			}
		%2	
			\foreach \x in {0,1,2,7,8}		{
				\foreach \y in {0,1,2,4,5}		{
					\ifnum \y>\x
					\else
						\draw[fill=black!25] ({0+(\x-\y)*1/9},{2*\sh-1+(\x+\y)*1/9}) -- +(1/9,1/9) -- +(0,2/9);
					\fi
					\ifnum \x>\y
						\draw[fill=black!25] ({0+(\x-\y)*1/9},{2*\sh-1+(\x+\y)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
					\fi
	
				}
			}
		%3
			\foreach \x in {0,1,2,7,8}		{
				\foreach \y in {0,1,2,4,5}		{
					\ifnum \y>\x
						\draw[fill=black!25] ({1+(\x-\y)*1/9},{\sh-1+(\x+\y)*1/9}) -- +(1/9,1/9) -- +(0,2/9);
					\else
	%					\draw[fill=black!25] ({0+(\x-\y)*1/9},{-1+(\x+\y)*1/9}) -- +(1/9,1/9) -- +(0,2/9);
					\fi
					\ifnum \x>\y
	%					\draw[fill=black!25] ({0+(\x-\y)*1/9},{-1+(\x+\y)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
					\else
						\draw[fill=black!25] ({1+(\x-\y)*1/9},{\sh-1+(\x+\y)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
					\fi
				}
			}
			
		%4
			\foreach \x in {0,2,3,4,5}		{
				\foreach \y in {0,1,2,4,5}		{
					\ifnum \y>\x
	%					\draw[fill=black!25] ({1+(\x-\y)*1/9},{\sh-1+(\x+\y)*1/9}) -- +(1/9,1/9) -- +(0,2/9);
					\else
						\draw[fill=black!25] ({0+(\x-\y)*1/9},{-1+(\x+\y)*1/9}) -- +(1/9,1/9) -- +(0,2/9);
					\fi
					\ifnum \x>\y
						\draw[fill=black!25] ({0+(\x-\y)*1/9},{-1+(\x+\y)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
					\else
	%					\draw[fill=black!25] ({1+(\x-\y)*1/9},{\sh-1+(\x+\y)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
					\fi
				}
			}
		%5
			\foreach \x in {0,2,3,4,5}		{
				\foreach \y in {3,5,7,8}		{
					\ifnum \y>\x
						\draw[fill=black!25] ({1+(\x-\y)*1/9},{-\sh-1+(\x+\y)*1/9}) -- +(1/9,1/9) -- +(0,2/9);
					\fi
					\ifnum \x>\y
					\else
						\draw[fill=black!25] ({1+(\x-\y)*1/9},{-\sh-1+(\x+\y)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
					\fi
				}
			}
		%6
			\foreach \x in {0,1,8}		{
				\foreach \y in {3,5,7,8}		{
					\ifnum \y>\x
	%					\draw[fill=black!25] ({1+(\x-\y)*1/9},{-\sh-1+(\x+\y)*1/9}) -- +(1/9,1/9) -- +(0,2/9);
					\else
						\draw[fill=black!25] ({0+(\x-\y)*1/9},{-2*\sh-1+(\x+\y)*1/9}) -- +(1/9,1/9) -- +(0,2/9);
					\fi
					\ifnum \x>\y
						\draw[fill=black!25] ({0+(\x-\y)*1/9},{-2*\sh-1+(\x+\y)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
					\else
	%					\draw[fill=black!25] ({1+(\x-\y)*1/9},{-\sh-1+(\x+\y)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
					\fi
				}
			}
        	        
			\draw (0,-2.1) -- (0,2.1+\sh);
			\draw[thick] (1/3,-2.1) -- (1/3,2.1+\sh);
			\draw (2/3,-2.1) -- (2/3,2.1+\sh);
			\draw (1,-2.1) -- (1,2.1+\sh);
	
			\draw[thin,dashed] (1/9,-1.95) -- (1/9,1.95+\sh);
			\draw[thin,dashed] (2/9,-1.95) -- (2/9,1.95+\sh);
			\draw[thick,dashed] (4/9,-1.95) -- (4/9,1.95+\sh);
			\draw[thin,dashed] (5/9,-1.95) -- (5/9,1.95+\sh);
			\draw[thin,dashed] (7/9,-1.95) -- (7/9,1.95+\sh);
			\draw[thin,dashed] (8/9,-1.95) -- (8/9,1.95+\sh);
	
			\draw[ultra thick] (0,-1) -- (1,0) -- (0,1) -- cycle;
	                \draw[ultra thick] (0,0+\sh) -- (1,-1+\sh) -- (1,1+\sh) -- cycle;
			\draw (0,-1+2*\sh) -- (1,0+2*\sh) -- (0,1+2*\sh) -- cycle;
	                \draw (0,0-\sh) -- (1,-1-\sh) -- (1,1-\sh) -- cycle;
		\end{scope}
			\draw[thick,<-] (7/18,\top +0.08 ) -- (7/18,\top+0.17);
			\draw (7/18,\top+0.25) node {$C(i_1,\cdots,i_{k N-1})$};
			\draw[snake=brace,very thick] (6/18,\top+0.03) -- (8/18,\top+0.03);
%			\draw (-0.05,0) node[left] {\Large$\mathbf{R_j^{(k-1)}}$};
%			\draw (1.05,0+\sh) node[right] {\Large$\mathbf{L_j^{(k-1)}}$};
			\draw[snake=brace,very thick] (1,-\top+1-0.1) -- (0,-\top+1-0.1);
			\draw (1/2,-\top+1-0.2) node {$C(i_1,\cdots,i_{(k-1) N})$};
	
			\draw[fill=black!80] ({1+(0-5)*1/9},{\sh-1+(0+5)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
			\draw[<-,very thick]  ({1+(0-5)*1/9-1/9-0.01},{\sh-1+(0+5)*1/9+1/9}) -- (-0.15,{\sh-1+(0+5)*1/9+1/9}) node[left] {$\Delta_1^j$};
			\draw[fill=black!80] ({0+(5-1)*1/9},{-1+(5+1)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
			\draw[<-,very thick] ({0+(5-1)*1/9-1/9-0.01},{-1+(5+1)*1/9+1/9}) -- (-0.15,{-1+(5+1)*1/9+1/9}) node[left]{$\Delta_2^j$};
			\draw[fill=black!80] ({0+(4-0)*1/9},{-1+(4+0)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
			\draw[<-,very thick] ({0+(4-0)*1/9-1/9-0.01},{-1+(4+0)*1/9+1/9}) -- (-0.15,{-1+(4+0)*1/9+1/9}) node[left]{$\Delta_3^j$};
	 \end{scope}

	\edef\tone{2.8}
	\edef\ttwo{0.3}
	\edef\tthree{-2.2}
	\begin{scope}[xshift=\sthree{}cm,scale=0.5]
	%1
			\foreach \x in {0,1}		{
				\foreach \y in {0,2}		{
					\ifnum \y>\x
						\draw[fill=black!25] ({1+(\x-\y)*1/3},{\tone-1+(\x+\y)*1/3}) -- +(1/3,1/3) -- +(0,2/3);
					\else
	%					\draw[fill=black!25] ({0+(\x-\y)*1/9},{2*\sh-1+(\x+\y)*1/9}) -- +(1/9,1/9) -- +(0,2/9);
					\fi
					\ifnum \x>\y
	%					\draw[fill=black!25] ({0+(\x-\y)*1/9},{2*\sh-1+(\x+\y)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
					\else
						\draw[fill=black!25] ({1+(\x-\y)*1/3},{\tone-1+(\x+\y)*1/3}) -- +(-1/3,1/3) -- +(0,2/3);
					\fi
				}
			}
%			\draw[pattern=grid] ({1+(1-2)*1/3},{\tone-1+(1+2)*1/3}) -- +(1/3,1/3) -- +(0,2/3);
%			\draw[pattern=grid] ({1+(0-0)*1/3},{\tone-1+(0+0)*1/3}) -- +(-1/3,1/3) -- +(0,2/3);
			\begin{scope}
				\clip  ({1+(1-2)*1/3},{\tone-1+(1+2)*1/3}) -- +(1/3,1/3) -- +(0,2/3);
				\draw[ystep=0.07,xstep=10] ({1+(1-2)*1/3},{\tone-1+(1+2)*1/3}) grid ({1+(1-2)*1/3+1/3},{\tone-1+(1+2)*1/3+2/3});
		      	\end{scope}
			\draw[very thick,->] ({1+(1-2)*1/3+1/6+1/3},{\tone-1+(1+2)*1/3+1/3+0.3}) node[right=-3pt] {$A$} -- +(-1/3,-0.3) ;
			\begin{scope}
				\clip  ({1+(0-0)*1/3},{\tone-1+(0+0)*1/3}) -- +(-1/3,1/3) -- +(0,2/3);
				\draw[ystep=0.07,xstep=10] ({1+(0-0)*1/3-1/3},{\tone-1+(0+0)*1/3}) grid ({1+(0-0)*1/3},{\tone-1+(0+0)*1/3+2/3});
		      	\end{scope}
			\draw[very thick,->] ({1+(0-0)*1/3+1/6},{\tone-1+(0+0)*1/3+1/3-0.3}) node[right=-3pt] {$B$} -- +(-1/3+0.05,+0.3) ;
			\draw[very thick]  ({1+(1-2)*1/3},{\tone-1+(1+2)*1/3}) -- +(1/3,1/3) -- +(0,2/3) -- cycle;
			\draw[very thick]  ({1+(0-0)*1/3},{\tone-1+(0+0)*1/3}) -- +(-1/3,1/3) -- +(0,2/3) -- cycle;

			\draw[very thick] (0,\tone) -- +(1,1) -- +(1,-1) -- cycle;
			\draw[thick] (1/3,-1/3+\tone) -- (1/3,1/3+\tone);
			\draw[thick] (2/3,-2/3+\tone) -- (2/3,2/3+\tone);
	%2
			\foreach \x in {0,1,2}		{
				\foreach \y in {1,2}		{
					\ifnum \y>\x
						\draw[fill=black!25] ({1+(\x-\y)*1/3},{\ttwo-1+(\x+\y)*1/3}) -- +(1/3,1/3) -- +(0,2/3);
					\else
	%					\draw[fill=black!25] ({0+(\x-\y)*1/9},{2*\sh-1+(\x+\y)*1/9}) -- +(1/9,1/9) -- +(0,2/9);
					\fi
					\ifnum \x>\y
	%					\draw[fill=black!25] ({0+(\x-\y)*1/9},{2*\sh-1+(\x+\y)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
					\else
						\draw[fill=black!25] ({1+(\x-\y)*1/3},{\ttwo-1+(\x+\y)*1/3}) -- +(-1/3,1/3) -- +(0,2/3);
					\fi
				}
			}
%			\draw[pattern=dots] ({1+(2-2)*1/3},{\ttwo-1+(2+2)*1/3}) -- +(-1/3,1/3) -- +(0,2/3);
%			\draw[pattern=dots] ({1+(1-2)*1/3},{\ttwo-1+(1+2)*1/3}) -- +(+1/3,1/3) -- +(0,2/3);
			\begin{scope}
				\clip ({1+(2-2)*1/3},{\ttwo-1+(2+2)*1/3}) -- +(-1/3,1/3) -- +(0,2/3);
				\draw[xstep=0.07,ystep=10] ({1+(2-2)*1/3-1/3},{\ttwo-1+(2+2)*1/3}) grid ({1+(2-2)*1/3},{\ttwo-1+(2+2)*1/3+2/3});
		      	\end{scope}
			\draw[very thick,->] ({1+(2-2)*1/3+1/6},{\ttwo-1+(2+2)*1/3+1/3+0.3}) node[right=-3pt] {$C$} -- +(-1/3,-0.3) ;
			\begin{scope}
				\clip ({1+(1-2)*1/3},{\ttwo-1+(1+2)*1/3}) -- +(+1/3,1/3) -- +(0,2/3);
				\draw[xstep=0.07,ystep=10] ({1+(1-2)*1/3},{\ttwo-1+(1+2)*1/3}) grid ({1+(1-2)*1/3+1/3},{\ttwo-1+(1+2)*1/3+2/3});
		      	\end{scope}
			\draw[very thick,->] ({1+(1-2)*1/3+1/6+1/3-0.05},{\ttwo-1+(1+2)*1/3+1/3+0.2}) node[right=-3pt] {$D$} -- +(-1/3+0.05,-0.2) ;
			\draw[very thick] ({1+(2-2)*1/3},{\ttwo-1+(2+2)*1/3}) -- +(-1/3,1/3) -- +(0,2/3) -- cycle;
			\draw[very thick] ({1+(1-2)*1/3},{\ttwo-1+(1+2)*1/3}) -- +(+1/3,1/3) -- +(0,2/3) -- cycle;
			\draw[very thick] (0,\ttwo) -- +(1,1) -- +(1,-1) -- cycle;
			\draw[thick] (1/3,-1/3+\ttwo) -- (1/3,1/3+\ttwo);
			\draw[thick] (2/3,-2/3+\ttwo) -- (2/3,2/3+\ttwo);
	%3	
			\foreach \x in {1}		{
				\foreach \y in {0,1}		{
					\ifnum \y>\x
						\draw[fill=black!25] ({1+(\x-\y)*1/3},{\tthree-1+(\x+\y)*1/3}) -- +(1/3,1/3) -- +(0,2/3);
					\else
	%					\draw[fill=black!25] ({0+(\x-\y)*1/9},{2*\sh-1+(\x+\y)*1/9}) -- +(1/9,1/9) -- +(0,2/9);
					\fi
					\ifnum \x>\y
	%					\draw[fill=black!25] ({0+(\x-\y)*1/9},{2*\sh-1+(\x+\y)*1/9}) -- +(-1/9,1/9) -- +(0,2/9);
					\else
						\draw[fill=black!25] ({1+(\x-\y)*1/3},{\tthree-1+(\x+\y)*1/3}) -- +(-1/3,1/3) -- +(0,2/3);
					\fi
				}
			}
			\draw[very thick] (0,\tthree) -- +(1,1) -- +(1,-1) -- cycle;
			\draw[thick] (1/3,-1/3+\tthree) -- (1/3,1/3+\tthree);
			\draw[thick] (2/3,-2/3+\tthree) -- (2/3,2/3+\tthree);
        	        
	%
			\draw[thick] (2/3,\tone+1.1) -- (2/3,\tthree-1.1);
			\draw[thick] (1,\tone+1.1) -- (1,\tthree-1.1);


			\draw[thick,<-] (5/6,\tone +1.26 ) -- (5/6,\tone+1.44);
			\draw (5/6,\tone+1.6) node {$C(i_1,\cdots,i_{k N})$};
			\draw[snake=brace,very thick] (2/3,\tone+1.16) -- (1,\tone+1.16);

	 \end{scope}
	 \draw[very thick,->,bend angle=45,bend right] (1.1,1.5) .. controls ({(-1.1+\stwo-0.1)/4+1.1},1.7) and ({3*(-1.1+\stwo-0.1)/4+1.1},1.7) .. (\stwo-0.1,1.5) node[midway,above,text width=4cm,text centered] {We keep only the left triangles in $L_j^{(k-1)}\cup R_j^{(k-1)}$ since $U(i_{k N})=L$.};
%	 \draw[very thick,->,bend angle=45,bend right] (1.2+\stwo,1.5) .. controls ({(-1.2-\stwo-0.2+\sthree)/4+1.2+\stwo},1.7) and ({3*(-1.2-\stwo-0.2+\sthree)/4+1.2+\stwo},1.7) .. (\sthree-0.2,1.5) node[midway,above,text width=4cm,text centered] {Enlarge $\Delta_1^j$, $\Delta_1^j$, $\Delta_3^j$ triangles};

		\draw (\sthree+1/2+0.01,\tone/2) node[right] {\LARGE${\Delta_1^j}$};
		\draw (\sthree+1/2+0.01,\ttwo/2) node[right] {\LARGE${\Delta_2^j}$};
		\draw (\sthree+1/2+0.01,\tthree/2) node[right] {\LARGE${\Delta_3^j}$};
		\draw (\sthree+0.03,{(\ttwo+\tone)/4+0.18}) node[text width=2cm,text centered] {Realization of $\Lambda^{k N}$};
		\draw[->,thick] (\sthree+0.02,{(\ttwo+\tone)/4+0.13+0.18}) -- (\sthree+1/12,\tone/2);
		\draw[->,thick] (\sthree+0.08,{(\ttwo+\tone)/4-0.13+0.18}) -- (\sthree+3/12+0.02,\ttwo/2+0.17);
		\draw ({(1+\stwo)/2},\top+0.1) node[text width=2cm,text centered] {Realization of $\Lambda^{k N-1}$};
		\draw[<-,thick] (15/18-0.01,\top-0.17) -- (1.2,\top+0.1);
		\draw[<-,thick] (\stwo+2/18+0.08,\top-0.08) -- (\stwo-0.2,\top+0.1);
%		\draw[->,very thick]  ({\stwo+1+(0-5)*1/9+0.02},{\sh-1+(0+5)*1/9+1/9}) -- (\sthree,\tone/2);
%		\draw[->,very thick] ({\stwo+0+(5-1)*1/9+0.02},{-1+(5+1)*1/9+1/9}) -- (\sthree,\ttwo/2);
%		\draw[->,very thick] ({\stwo+0+(4-0)*1/9+0.02},{-1+(4+0)*1/9+1/9}) -- (\sthree,\tthree/2);
		\draw[->,very thick,dashed]  ({\stwo+1+(0-5)*1/9},{\sh-1+(0+5)*1/9+2/9}) -- (\sthree+1/2,\tone/2+1/2);
		\draw[->,very thick,dashed]  ({\stwo+1+(0-5)*1/9},{\sh-1+(0+5)*1/9+0/9}) -- (\sthree+1/2,\tone/2-1/2);
		\draw[->,very thick,dashed] ({\stwo+0+(5-1)*1/9},{-1+(5+1)*1/9+2/9}) -- (\sthree+1/2,\ttwo/2+1/2);
		\draw[->,very thick,dashed] ({\stwo+0+(5-1)*1/9},{-1+(5+1)*1/9+0/9}) -- (\sthree+1/2,\ttwo/2-1/2);
		\draw[->,very thick,dashed] ({\stwo+0+(4-0)*1/9},{-1+(4+0)*1/9+2/9}) -- (\sthree+1/2,\tthree/2+1/2);
		\draw[->,very thick,dashed] ({\stwo+0+(4-0)*1/9},{-1+(4+0)*1/9+0/9}) -- (\sthree+1/2,\tthree/2-1/2);
        \end{tikzpicture}

\end{document}

