Define $$ A = (0,0,0), \ B=\bigg(\frac{1}{n},0,0\bigg), \ C= \bigg(\frac{1}{n},\frac{1}{n},0\bigg) ,\ D= \bigg( 0,\frac{1}{n},0\bigg),\ E= \bigg(\frac{1}{2n},\frac{1}{2n}, \frac{a}{n}\bigg) $$ for some $a>0$.
Consider $\Sigma=\Delta ABE \bigcup\Delta BCE\bigcup\Delta CDE\bigcup\Delta DAE $. When $T_{ij}$ is a translation map on $\mathbb{R}^3$ by $T_{ij}(x,y,z) =( x+ \frac{i}{n},y+\frac{j}{n},z)$, then define $$ M_n =\bigcup_{0\leq i,\ j\leq n-1}\ T_{ij}(\Sigma)$$
When $d_n$ is intrinsic metric on $M_n$, then $d_n((0,0,0),(1,1,0)) > \sqrt{2}$ are equal and areas of $M_n$ are equal. Here does the Gromov-Hausdorff limit of $ M_n$ exists ?
