A362938 a(n) = minimum number of pieces needed to dissect a regular n-sided polygon into a monotile, i.e. a polygonal tile which tiles the plane (conjectured).

1, 1, 2, 1, 3, 2, 3, 2, 4, 3, 4, 3, 5, 4, 6

Offset: 3

Author

N. J. A. Sloane, Aug 29 2023

Keywords

• nonn
• more
• hard

Comments

I do not know which of these values have been proved to be minimal. Probably only a(n) for n = 3, 4, 5, 6, 8, and 10.

The three related sequences A110312, A362938, A362939 are exceptions to the usual OEIS policy of requiring that all terms in sequences must be known exactly. These sequences are included because of their importance and in the hope that someone will establish the truth of some of the conjectured values.

More formally, a(n) = minimum number of pieces needed to dissect a regular n-sided polygon into a monotile, that is, a prototile for a monohedral tiling of the plane.

Turning over is allowed. The pieces must be bounded by simple curves to avoid difficulties with non-measurable sets.

The definitions imply that a(n) <= A362939(n) <= A110312(n).

On Aug 31 2023 Gavin Theobald sent me two different solutions for a(9) = 3 and one solution for a(11) = 4 (see links). He reports that he found these dissections in the 1990's. In his email and in a later email (Sep 04 2023) he also gives the values a(13) = 5, a(14) = 3, a(15) = 5 (with one piece turned over), a(16) = a(18) = 4, a(20) = 5. He conjectures that a(2t) = floor(t/2) for all t >= 2.

References

• Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. [The sequence is defined in Section 2.6, pp. 91-95.]
• Harry Lindgren, Geometric Dissections, Van Nostrand, Princeton, NJ, 1964. Plates B6, B7, B8, B9, B10, and B12 illustrate n = 6, 7, 8, 9, 10, and 12, respectively. One would expect that plates B11 and B13 would refer to n = 11 and 13, but in fact they appear to give alternative solutions for n = 10 and 12, respectively.
• Harry Lindgren, Recreational Problems in Geometric Dissections and How to Solve Them, Revised and enlarged by Greg Frederickson, Dover Publications, NY, 1972.

Links

• Branko Grünbaum and G. C. Shephard, Tilings and Patterns, W. H. Freeman, New York, 1987. Annotated copy of Fig. 2.6.1, illustrating a(5), a(7), a(8), a(10), and a(12). (Their value for a(9) is out-of-date.)
• Harry Lindgren, Geometric Dissections, Annotated scan of Plate B12, showing tiling of plane arising from the conjectured a(12) = 3.
• N. J. A. Sloane, Illustration for a(5) = 2, after Grunbaum and Shephard, Fig. 2.6.1. Left: The 2-piece dissection of the pentagon. Right: Shows how the hexagonal tile made from those two pieces tiles the plane.
• N. J. A. Sloane, An illustration for a(12) = 3, based on Lindgren's plate B12.
• N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: Video, Slides, Updates. (Mentions this sequence.)
• N. J. A. Sloane and Gavin A. Theobald, On Dissecting Polygons into Rectangles, arXiv:2309.14866 [math.CO], 2023.
• Gavin Theobald, Illustration for a(5)
• Gavin Theobald, Illustration for a(7)
• Gavin Theobald, Another illustration for a(7) <= 3, after Lindgren.
• Gavin Theobald, Illustration for a(8)
• Gavin Theobald, A 9-piece dissection of a 9-gon into a monotile, with the monotile outlined in red, illustrating a(9) = 3.
• Gavin Theobald, A 9-piece dissection of a 9-gon into a monotile, showing how the monotile is obtained from the 9-gon.
• Gavin Theobald, An alternative illustration for a(9) = 3.
• Gavin Theobald, Yet another illustration for a(9)
• Gavin Theobald, Illustration for a(10)
• Gavin Theobald, Illustration for a(11)
• Gavin Theobald, Illustration for a(12)
• Gavin Theobald, Illustration for a(13) <= 4
• Gavin Theobald, Illustration for a(14)
• Gavin Theobald, Illustration for a(15) (5 pieces)
• Gavin Theobald, Another illustration for a(15) <= 5
• Gavin Theobald, Illustration for a(17) (The piece marked X must be turned over)
• Gavin Theobald, Illustration for a(19) <= 7
• Gavin Theobald, The Geometric Dissections Database

Crossrefs

Cf. A110312, A362939.

Extensions

a(9) = 3, a(11) = 4, a(13) = 5, a(14) = 3, a(16) = 4 from Gavin Theobald, Aug 31 2023 - Sep 11 2023.

Updated with many further illustrations from Gavin Theobald. - N. J. A. Sloane, Sep 19 2023