A New Multidimensional Slow Continued Fraction Algorithm and Stepped Surface

Abstract

We give a new algorithm of slow continued fraction expansion related to an arbitrary real cubic number field as a 2-dimensional version of the Farey map. Using our algorithm, we can find the generators of dual substitutions (so-called tiling substitutions) for any stepped surface for any cubic direction.

2010 AMS Subject Classification::

• 11A55
• 11Y16
• 11Y65
• 37B10

Keywords::

• continued fraction
• Farey map
• stepped surface
• multi-dimensional continued fraction algorithm
• Pisot number
• sturmian sequence
• real cubic number field
• tiling

Notes

¹GiNaC is available at http://www.ginac.de/. The routine that was written for this purpose can be downloaded from the website http://www.lab2.toho-u.ac.jp/sci/c/math/yasutomi/mfarey.html.

²The routine that was written for this purpose can be downloaded from the website http://www.lab2.toho-u.ac.jp/sci/c/math/yasutomi/mfarey.html.