I don’t understand the problem well enough to understand why this is a problem.
Am I alone on this?
Geometry has always been a mystery to me.
Most shapes that are able to completely cover an area with no gaps result in patterns that are periodic, i.e. that repeat themselves perfectly over and over. For example, an array of squares, triangle or hexagons, can cover a surface with no gaps, but if you make a transparency of the pattern, this pattern can be moved over by one tile or two or three, or rotated by different angles, and still overlap perfectly with what it was before the translation or rotation. Being able to cover a surface with only two different shapes (the Penrose tiling) or with only one shape (the shape in this article) with no gaps, but without ever repeating, is rare. According to the article, it wasn’t even known if it was possible to use one single shape to completely cover a plane with no gaps and without any periodic repetition.
“I don’t understand the problem well enough to understand why this is a problem.
Am I alone on this?”
No. I don’t get it. And I guess I lack imagination because for me, I wonder why anyone has been thinking about this since the 60s.