Some tilings of the plane whose singular points are uncountable and unbounded

✍ Scribed by Marilyn Breen

• Publisher: Springer
• Year: 1984
• Tongue: English
• Weight: 283 KB
• Volume: 17
• Category: Article
• ISSN: 0046-5755
• DOI: 10.1007/bf00151501

✦ Synopsis

Let 2: be a tiling of the plane such that for every tile T of 2: there correspond a tile T' of 2: (not necessarily unique) and an integer k(T, T') (depending on T and T'), k(T, T') > 2, such that T meets T' in k(T, T') connected components. Tiles T and T' satisfying this condition are called associated tiles in 2:. Various properties concerning 2: and its singular points are obtained. First, it is not possible that every tile in 2: have a unique associated tile. In fact, there exist infinite families of tiles {F'} u {F,: n >/1 } such that F' is the unique associated file for every F,. Next, if x is a singular point of 2:, then every neighborhood of x contains uncountably many singular points of 2:. Finally, the set of singular points of 2: is unbounded.