Pentomino Tessellation Explanation

Question

I need to explain why this pentomino tessellates in a mathematically coherent way.

Here is the pentomino and the tessellation I have made.

This pentomino can be translated to form a diagonal pattern and then the diagonals can fit together and this repeated pattern will tessellate the plane.

Is there a more mathematical reason behind this? Could I draw on the fact that at the vertex of a tessellation the sum of the angle measures must be $360°$? In this tessellation, looking at any vertex where two different colored pentominoes meet the measures of angles will sum to be $270°+90°=360°$ or $180°+180°=360°$.

I'm looking for advice on what a mathematically coherent explanation would be for this situation.

Answer 1

Thanks for your question. It's sometimes really hard to prove something that seems to be obvious at first glance.

I have three ideas:

1. Observe a pythagorean tiling inside yours. Then derive from it that your pattern is indeed a tessellation.
2. You can explain it using the fact that your pattern is invariant to translation by a vector $[2,1]$. Therefore it covers the whole plane. (It's similar to your idea of diagonals formed from the tiles).
3. I stumbled upon something called "Wallpaper groups". Maybe something from that area would help you?