Q:

The following mesh/grid like diagram in tikz

I need to create a picture as shown in the attachment. I know how to create the hexagonal substructure and the code for the same can be found in this question: Hyperbolic polyhedron in tikz
Please help me out as I'm new to tikz.

Edit: The function which maps the subdivided icosahedron to the plane behaves like $z^{6/5}$ at the vertex of degree 5 (hereby denoted by $v$).

A:

You can draw such grids with \foreach loops.
\documentclass[tikz,border=3.14mm]{standalone}
\usetikzlibrary{shapes.geometric,calc}
\begin{document}
\begin{tikzpicture}[web/.style={append after command={foreach \XX in {1,...,#1} 
 {(current.corner \XX)
 -- (current.center)}},regular polygon,regular polygon sides=#1,minimum
  size=1cm,draw,alias=current},
  pics/outerior/.style={code={
  \pgfmathtruncatemacro{\idiv}{pow(2,#1-1)}
  \draw (90:#1*0.5) foreach \XX in {1,...,5}
   {-- (90+72*\XX:#1*0.5) coordinate (P-\XX)};
  \draw (90:#1*0.5+0.5) foreach \XX in {1,...,5}
   {-- coordinate[midway] (M-\XX) (90+72*\XX:#1*0.5+0.5) coordinate (Q-\XX)}; 
  \foreach \XX [remember=\XX as \YY (initially 5)] in {1,...,5} 
  {\draw (P-\XX) -- (Q-\XX);
  \foreach \ZZ in {0,...,#1}
   {\draw ($(Q-\XX)!{(\ZZ+1)/(#1+1)}!(Q-\YY)$) -- ($(P-\XX)!{\ZZ/#1}!(P-\YY)$) --
   ($(Q-\XX)!{\ZZ/(#1+1)}!(Q-\YY)$) ;}
  }
  }}]
 \draw (0,0) node[web=5] (c5) {};
 \clip (-3,-3) rectangle (3,3);
 \draw foreach \X in {1,...,9} {(0,0) pic{outerior=\X}};
\end{tikzpicture}
\end{document}

P.S. Your prescription does not seem to yield the drawn lattice when using regular polygons (and also to introduce polygons that automatically have the internal lines added via append after command.
\documentclass[tikz,border=3.14mm]{standalone}
\usetikzlibrary{shapes.geometric,calc}
\begin{document}
\begin{tikzpicture}[web/.style={append after command={foreach \XX in {1,...,#1} 
 {(current.corner \XX)
 -- (current.center)}},regular polygon,regular polygon sides=#1,minimum
  size=1cm,draw,alias=current}]
 \draw (0,0) node[web=5] (c5) {};
 \foreach \X [remember=\X as \Y (initially 5)] in {1,...,5}
 {\draw 
 let \p1=($(c5.corner \Y)-(c5.corner \X)$),\n1={atan2(\y1,\x1)} 
 in ($(c5.corner \Y)!0.5!(c5.corner \X)$) coordinate (aux) 
    ($(aux)!{-(1/2)*1cm}!90:(c5.corner \X)$) 
    node[web=6,minimum size=6cm/5,rotate=\n1]{};}
\end{tikzpicture}
\end{document}

