Existence of a special pentagonal tiling

Question

Is there a pentagonal tiling composed of only one shape of pentagon so that each pentagon touches exactly 5 other pentagons? Two pentagons are in touch if they share at least one common point.

Few examples:

Here each pentagon touches exactly 6 other pentagons:

Here each pentagon touches exactly 7 other pentagons:

Here each pentagon touches exactly 8 other pentagons:

Here each yellow pentagon touches 5 other pentagons, each pink pentagon touches 6 other pentagons and each blue pentagon touches 7 other pentagons:

I just can not find any tiling that there are only 5 touches per pentagon.

Maybe it is trivial question but I am not very familiar with tiling theory.

Answer 1

If such a tiling existed, it would necessarily have to be edge-to-edge with $3$ pentagons at every vertex. The tiling's skeleton (the graph formed by the vertices and edges of contact) would therefore be a $3$-regular infinite planar graph with each face being of size $5$. But by Euler's formula, this should be a dodecahedron, which is not a tiling. Thus no such tiling exists.