#### Document Type

Article

#### Publication Date

Fall 2005

#### Publication Title

Illinois Journal of Mathematics

#### ISSN

0019-2082

#### Abstract

The set *V*_{n} of *n*-vertices of a tile *T* in *\R ^{d}* is the common intersection of

*T*with at least

*n*of its neighbors in a tiling determined by

*T*. Motivated by the recent interest in the topological structure as well as the associated canonical number systems of self-similar tiles, we study the structure of

*V*for general and strictly self-similar tiles. We show that if

_{n}*T*is a general self-similar tile in

*\R*whose interior consists of finitely many components, then any tile in any self-similar tiling generated by

^{2}*T*has a finite number of vertices. This work is also motivated by the efforts to understand the structure of the well-known L\'evy dragon. In the case

*T*is a strictly self-similar tile or multitile in

*\R*, we describe a method to compute the Hausdorff and box dimensions of

^{d}*V*. By applying this method, we obtain the dimensions of the set of

_{n}*n*-vertices of the L\'evy dragon for all

*n*≥1.

#### Recommended Citation

Deng, Da-Wen, Sze-Man Ngai.
2005.
"Vertices of Self-Similar Tiles."
*Illinois Journal of Mathematics*, 49 (3): 857-872.
source: https://projecteuclid.org/euclid.ijm/1258138223

https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/608

## Comments

© 2005 University of Illinois. This is an open access article retrieved from the Illinois Journal of Mathematics. Articles older than 5 years are open access.