Contents
-
Imperfect Congruence
- A mathematical note
- About the author
- Part 1: Kepler and the regular polygon tilings
- Part 2: Rhomb/rhombus/rhombic tilings
-
Part 3: A universal tiler
- Fans and other rhombic structures
- Regular polygon decomposition
- Regular polygon decomposition (odd)
- Worms and zonagons
- Decomposition algorithm
- Decomposition mathematics
- Variations on the universal tiler
- The regular polygon extension theorem
- The triangle extension theorem
- The rule of two
- The vertex figure extension conjecture
- Vertex figure extension examples
- More vertex figure extension examples
- Part 4: Five-fold symmetry
- Conclusion
- Tiling Gallery
- Appendix: Bridges conference
- Appendix: Proof of the Gap Theorem
- Appendix: Tiling source code
Part 2: Rhomb/rhombus/rhombic tilings
Tilings from Robert Fathauer
In June 2011, after looking at an earlier version of this website, the artist-mathematician Robert Fathauer sent me eleven new tilings extended from the “forbidden” vertex figures.
Not only do these tilings often have smaller translational units than the tilings extended from these vertex figures elsewhere on this site, but seven of them are nutshell tilings!
I’ve described these tilings on the pages below using the same format I’ve used for the rest of this site.
While Robert discovered these tilings, he has stressed to me that he hasn’t made an exhaustive search to determine if they had been found previously by others. In particular, the 3.9.18 Art tiling, or at least related tilings, appear in Islamic art.
It is wonderful to get more examples of tilings extended from the “forbidden” vertex figures - the more examples we find, the more likely it is that we may find beautiful patterns and perhaps new mathematics.
I’d welcome more such tilings extended from the 3.7.42, 3.8.24, 3.9.18, 3.10.15, 4.5.20 or 5.5.10 vertex figures. Please send descriptions to kevinjardine at yahoo dot com.