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](https://tex.stackexchange.com/q/485019/173540)

Please help me out as I'm new to tikz.

[![enter image description here](https://i.stack.imgur.com/YriHo.png)](https://i.stack.imgur.com/YriHo.png)

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 $v$).