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
A thick 4.5.20 tiling?
Following the Dürer construction, we can rearrange four copies of the translational unit for the 4.5.20 thin tiling to create a gap that could be filled by a thick patch. As for the 5.5.10 thick tiling, it makes sense to first put a rose between the two vertical blocks.
This gap can be filled using four copies of the top half of the thin patch (above the watermelon), inflated squares and a few extra polygons on the edges.
However, the thin patches overlap. It may be possible to delete the overlapping tiles and perhaps some others and complete the thick patch but I have not yet been able to figure out how to do it.
As you can see from the image below, this shape acts just like a thick rhomb. Five copies can be arranged around a central point (with a 4.5.20 rose gap) and the shape fits together with the thin rhomb patch (leaving only an inflated square gap).
If the thick patch could be properly completed, then it would allow a substitution tiling which replaces a Penrose rhomb tiling with one that has an icosagon (20-sided polygon) at every vertex of the original tiling.
You can experiment yourself using this thick patch SVG file. The file also includes the 3π/10 fractional rhomb (the salmon-coloured tile) in case that is needed to complete the tiling.
If you find a solution (or a proof that one is impossible), then please email me at kevinjardine at yahoo dot com.